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Point-Contact Spectroscopy

Translation from the russian edition: Moskwa, Izdatelstwo ”Znanije”, Serija ”Fizika”, N12, (1989)

Yu. G. Naidyuk, I. K. YansonB. Verkin Institute for Low temperature Physics and Engineering

National Academy of Sciences of Ukraine,

61103 Kharkiv, Ukraine

February 2, 2008

Abstract

The history is described of how one of the mostcommonly used electric circuit components, an ordi-nary electric contact, has become a powerful tool forthe physicists to study various mechanisms of elec-tron scattering in metals. The physical principles ofspectroscopy of quasi-particle excitations in metalsby means of point contacts (PCs) whose dimensionsrange from only a few to tens of nanometers are pre-sented in a popular form.

PACS 73.40.Jn

Contents

1 Introduction 1

2 Contacts Around Us 2

3 Electrons and Phonons 3

4 Waste recovery 6

5 What happens inside the contact 7

6 Some theoretical fundamentals 9

7 Procedures for PCs creating 11

8 How to measure the second derivative 13

9 Point contact instead of reactor 14

10 Study of electron-phonon interaction 15

11 Magnons and excitons 16

12 From pure metals to alloys and compounds 18

13 Contact heating 20

14 Electrical fluctuations in PCs 24

15 PCs in a high-frequency electromagnetic field 25

16 Study of PCs on semimetals and semiconductors 26

17 Spectroscopy of superconductor 28

18 New trends of research 29

19 Conclusion 31

1 Introduction

The most interesting events and phenomena both ineveryday life and science are those which should becharacterized by the prefix ’super’. This is usually the

1

case when the objects in question are under extremeconditions. Thus, on the verge of XX century, a greatimpression was made, for example, by the discoveryof all-penetrating invisible X-rays or the phenomenonof resistance loss (superconductivity) in some metalsat low temperatures. Today the problem of controlledthermonuclear fusion is constantly kept in the focalpoint. Its solution, however, involves heating andconfinement of plasma at a superhigh temperature ofhundreds million degrees.

As to the solid state physics, one of the aspects ofwhich is dealt with in this booklet, the most strikingresults have been obtained when studying superpureand perfect or highly disordered (amorphous), high-quality laminated structures and other compounds ofcomplex composition. In the first place, the greatestachievements are gained if the substance under in-vestigation undergoes the extreme affects of a highmagnetic field, low and super low temperatures, ex-tra high pressure or electromagnetic radiation.

If all the events described above are associated withthe prefix ”super”, how to make them compatiblewith the word ”point” used in our case ? It becomesclearer if one deciphers the word ”point” as supersmall, i.e. a point contact (PC) stands for a contactbetween two substances over a very small area. Aquestion arises, how small is that area and what it canbe compared with. Naturally, a lower bound is set bythe interatomic spacing in a solid, an upper boundbeing less clearly defined to within one micrometer.Since in this case we are interested in the currentpassage through such contacts, the term ”super” canbe also employed to estimate the density of current.Thus the current densities of 109 to 1010 A/cm2 areachieved in PCs without substantial heating, thoughbulk conductors would momentarily evaporate underthese conditions.

The other word, spectroscopy, probably, does notrequire any detailed explanation. In a more generalsense, spectroscopy is a method of measuring the en-ergy spectrum of a given object. For example, whenanalyzing the emission or absorption spectra of anatom, one can determine the positions of its energylevels. A spectrum of different collective excitationsalso exists in solids since the atoms are united into asingle ensemble. Such expressions as the energy spec-

trum of electrons, phonons, magnons, etc., are well-known in the solid-state physics. Therefore, the term”point-contact spectroscopy” can be interpreted as amethod of measuring the elementary excitation spec-tra in conductive solids with the aid of PCs of supersmall size. In this method, the contact itself seems toplay the role of analytical instrument, a certain kindof spectrometer.

The point-contact spectroscopy has proved to be acomparatively simple, available and highly informa-tive method. Within the last decade, very impressiveresults have been obtained in this rapidly developingfield by physicists in many countries of the world, aswill be described below.

2 Contacts Around Us

In our everyday life, we permanently deal with differ-ent electric contacts by means of which current fromthe source is transmitted to the user and distributedover the circuit. Undoubtable, in number of cases theoperation of electrical appliances depends on reliablecontacts connecting different sections of electric cir-cuits. Consequently, the requirements for trouble freeand stable operation of the contacts are growing fromday to day. This problem has become especially top-ical due to the rapid growth of microelectronics whenthe number of separate elements per unit volume hasincreased by a factor of tens of thousands.

From the very inception of electrical engineering,researchers began to study the electric properties ofcontacts between conductors. The results of theseactivities were summed up by the German scientistR.Holm in his basic work ”Electric Contacts Hand-book” published in 1961. Scientists and engineershave noticed that the electrical resistance of twopieces of metal tightly pressed to each other is appre-ciable higher than could be expected assuming thatthe electric conductivity at contact is the same asthat inside the conductors. The following reasonshave been advanced. The first one is connected withthe fact that any real surface, even when polishedmirror-like, has some microscopic irregularities, as aresult of which in the contact region the electrodescome in contact only at separate sections. Hence, the

2

true contact area can be considerably smaller thanthe apparent size of contact. The other reason is thatthe surfaces of metallic electrodes are always coatedwith a thin layer of oxides or other compounds ad-sorbed by atoms from the air, organic molecules andother ’dirt’. All these substances, as a rule, possesseither dielectric or semiconductor properties. As a re-sult of it a direct electric contact is far less than thetotal area of mechanical contact between two conduc-tors. Current will not flow in those places where thedielectric film is sufficiently thick; if the film thick-ness does not exceed a few nanometers, part of thecurrent will pass due to the tunneling effect; metallicconductivity will be effected only at the part of con-tact area which is free from film and ’dirt’ becauseof their damage or absence. No matter how simplethe contact between two electrodes may look, the realpicture can be rather complicated.

The interest in the study of small-size contacts isconnected not only with their technical applications.They proved to be a most sensitive tool in the handsof scientists. Thus, in the middle of sixties Academi-cian Yu. V. Sharvin from the Institute of PhysicalProblems of the Russian Academy of Sciences pro-posed to make experiments in which PCs were usedto inject electrons into thin metallic plates. Then amagnetic field was applied, the injected beam wasfocused on the opposite surface of the plate wherethe second similar contact was placed to act as a de-tector. By changing the magnetic field intensity andrelative positions of contacts on the sample surface,one can observe the focusing of electrons both alongand across the field lines. As to the later case, suchexperiments were first made by V. S. Tsoi at the Insti-tute of Solid-State Physics of the Russian Academyof Sciences. The detector was capable of register-ing both the electrons emitted directly by the injec-tor and those once or repeatedly reflected from thesurface. The results of such studies yielded helpfulinformation on the Fermi surface geometry and elec-tron scattering by the sample boundary. However,to describe further investigations more substantially,we will present some fundamentals of the electronictheory of metals in the next chapter.

3 Electrons and Phonons

An atom is known to possess a definite set of en-ergy levels at which the electrons reside. When theatoms combine to form a solid, the wave functions(or orbits for simplicity) of the valence electrons ofneighboring atoms overlap. Thus, the electrons canmove along a crystal, i.e. they actually become col-lective and are no longer bound to a definite atom.According to the quantum picture, the product ofparticle energy uncertainty δε and the lifetime in aparticular state δτ is a constant proportional to thePlanck’s constant h, i.e. δεδτ ≃ h. Therefore, thelifetime δτ of an electron on a stationary atomic or-bit is large, and thus the energy uncertainty δε issmall, i.e. the energy level is narrow enough. Thevalence electrons in a solid are no longer bound withspecific atoms, δτ becomes the smaller, the closer arethe ions and the easier it is for the electrons to passfrom one site to another. As a result of it the value ofδε increases respectively and the atomic energy lev-els become the energy bands δε wide. If a substanceconsists of N atoms, this band will contain N energylevels at which all the collective electrons reside.

In the quantum mechanics, the microparticle be-havior depends on whether they have an integer (0,1, 2, ...) or half-integer (1/2, 3/2, 5/2, ...) spin. Aspin is a quantum characteristics of a particle whichis associated with its intrinsic angular momentum.The number of particles being in the same state withthe same energy is unlimited for those with an inte-ger spin named bosons. On the contrary, only oneparticle with a half-integer spin, a fermion, can ex-ist in a state with similar parameters. Electrons arefermions with a 1/2 spin, as a result of which onlytwo electrons with opposite spins can be located atthe same energy level. According to this principle, ata temperature close to absolute zero all the electronscannot reside at the minimum energy level and areforced to successively fill the higher-lying energy lev-els in the allowed band. Coming back to the band ina monovalent solid consisting of N atoms comprisingN states, it becomes clear that this band can hold Ncollective electrons starting from the minimum en-ergy level (band bottom) and up to the half-height(Fig. 1). The highest level occupied by electrons at

3

Figure 1: a) Electron distribution over the energy levelsin a solid consisting of N atoms on condition that eachatom gives away one electron into the conduction band.According to the Pauli principle, each level can accom-modate two electrons with opposite spins, then the con-duction band will be half-full and Fermi level separatingthe filled states from the free ones passes in the middleof the band. b) The Fermi surface for free electron gasin a solid in the momentum space. At T = 0 it is anisoenergetic surface separating the filled electron statesfrom the free ones. In real metals the Fermi surface isclose to a sphere only in some alkali metals, in the otherit has more complicated topology and can consist of sev-eral parts. The geometrical characteristics of the Fermisurface, its shape, curvature, extreme crossections valuesdetermine the main electronic properties of metals, thusenabling reconstruction of its form from the experimentaldata.

T = 0 was called the Fermi level, and the energy ofthis level - the Fermi energy εF which correspondsto the Fermi momentum 1 pF = (2melεF )1/2. Inthe momentum space, the Fermi level is an isoen-ergetic surface (Fermi-surface) which separates theoccupied states from the vacant ones. For the caseof the isotropic dispersion law, when the energy de-pends only on the momentum magnitude, but not onits direction, the Fermi surface is a sphere with theradius pF (Fig. 1(b)).

The electrons can move within the lattice, thenumber of particles moving, for example, from right

1It should be noted that the electron behavior in a latticeis quite different from that in a vacuum. In fact, one can-not consider a separate electron in a solid, but rather refer toelectron-like excitations, i.e. quasi-particles with a charge sim-ilar to that of a free electron and a close mass. This is why thedispersion law includes the effective mass value. Further on weshall use the term ’electrons’ implying thesequasi-particles.

to left, being exactly equal to the number of thosemoving in the opposite direction, while the state ofthe system remains unchanged due to the quantum-mechanical indistinguishability of the particles. Tocreate current in the crystal, it is necessary to trans-fer part of electrons into a state in which they canmove under the action of the field. Since the num-ber of atoms in a solid is about 1022 per 1 cm3 andthe Fermi energy is 1 to 10 eV or 104 to 105K, thedistance between the nearby levels proves to be verysmall, equal to εN ≃ 10−21 to 10−22 eV or 10−17 to10−18 K. Thus, the electrons residing at the Fermilevel under the action of the field move to the near-est free higher lying level separated by an extremelysmall energy gap of about 10−18 K, i.e. can easilyincrease their energy or accelerate. Such substanceswith unfilled electron band readily conduct the elec-tric current and are metals. As to the dielectrics,their bands are filled and, as a rule, the nearest va-cant band is separated by an energy gap of a fewelectron-volts. This is possible when two electrons ofeach atom enter the band. Now the electrons cannotchange their states under the action of the field as inthis case they have to ”jump over” a large energy gapand there will be no current. Note that the contri-bution into conduction is made only by the electronsresiding near the Fermi level while the lower-lyingones cannot change their states since the higher lyinglevels are occupied. Therefore, all the kinetic proper-ties of conductors are mainly determined by a smallgroup of electrons residing near the Fermi level in alayer with a width about the thermal smearing kBT .

Now we will try to obtain an expression for calcu-lating the conductivity of metals. The current den-sity j can be written as a product of average velocityof ordered electrons motion 〈v〉, their density n andcharge e: j = ne〈v〉. In a field with the intensity Ean electron acquires acceleration a = F/m = eE/m,whence 〈v〉 can be estimated as 〈v〉 ≃ aτ = eEτ/m,where τ is the mean free time of electrons betweentheir scattering by impurities, defects, phonons, etc.As a result, the current density can be written asj = ne2τE/m = E/ρ, where ρ = m/ne2τ is the re-sistivity of metal. It can be seen that the current inmetals is proportional to the applied field, i.e. thewell-known Ohm’s law holds. Let us rewrite the ex-

4

pression for ρ in the form

ρ = m/ne2τ = mvF /ne2τvF = pF /ne2l (1)

where vF is the Fermi velocity, l = vF τ is the meanfree path (mfp) of electrons. It evidently follows fromEq. (1) that the less is the mfp of electrons in thelattice, the higher is its resistance.

On which factors does the mfp depend? It hasbeen already mentioned that the electrons are scat-tered by the impurity atoms, defects, sample surface,etc. At first glance, it appears that electrons shouldrather ”stumble” on ions in the lattice. However, itturns out that according to the laws of quantum me-chanics the electrons are scattered only by thermalvibrations of atoms - the phonons. In a strictly pe-riodical perfect lattice at T = 0 the electrons movewithout collisions as if they do not notice the ionicframework. This is another consequence of the factthe properties of conduction electrons in a solid dif-fer but radically from those of free electrons in thevacuum.

The probability of electrons scattering is propor-tional to the frequency of their collisions, i.e. tothe inverse relaxation time τ−1. Considering all thescattering mechanisms as independent ones, the totalprobability of scattering under the actions of all fac-tors will be equal to the sum of separate probabilitiesτ−1 = τ−1

i + τ−1p + τ−1

e + . . ., or, which is the same,ρ = ρi + ρp + ρe + . . . as τ−1 ∼ ρ. Here ρi, ρp and ρe

are resistivity associated with the scattering by im-purities and defects (i), by phonons (p), by electrons(e). We have obtained the so-called Matthiessen ruleaccording to which the total electrical resistivity isthe sum of contributions due to separate scatteringmechanisms.

We will be more interested in the term ρp determin-ing the resistivity due to the electron-phonon interac-tion. First of all, we must make clear what a phononis. The vibrations of lattice atoms can be presented inthe form of excitation waves propagating in a crys-tal. It is appropriate here to draw on the analogywith waves on the surface of water. A float immersedin the water will move up and down during the wavepropagation, i.e. it will vibrate. Thus, the vibrations

εk

0

F(ε)ε

εD

(b)(a)

T1

T2

L

πh/a q0

Figure 2: a) Phonon dispersion law in metals with oneatom per unit cell. Two lower curves correspond to thetransverse vibration of atoms with two polarization, theupper curve corresponds to the longitudinal vibrations,compression-extension waves. b) Typical phonon spec-trum for a solid. The boundary Debye energy εD is de-termined by the maximum possible frequency of atomvibrations in the lattice. As a rule, the maxima associ-ated with low group velocity phonons v ∼ dε/dq → 0.In Fig. 2(a) it corresponds to the phonons with maximummomentum qmax = πh/a.

of atoms in a solid are described in the language ofwaves. According to the quantum mechanical dual-ism, each wave can be compared to a particle and thethermal vibrations of lattice atoms can be describedin terms of quasiparticles - the phonons. The wavevector of the wave k corresponds to the momentumof particle 2q = hk. In such a model the electronsscattering by thermal vibrations of the lattice can beconsidered as the interaction of quasi-particles, elec-trons and phonons, making use of the familiar lawsof conservation of energy and momentum.

The energy of phonon εph, as well as that of elec-tron, depends on its momentum. The dispersion lawfor phonons, i.e. the εph dependence on q, is morecomplicated than for electrons for which the formulaε = p2/2meff gives a fairly good approximation. Itis not easy to describe this law analytically over thewhole range from 0 to qmax and it is usually obtained

2Since the wavelength λ of atom vibrations in the latticecannot be smaller than the interatomic distance a, namelyλmin = 2a, the maximum momentum of phonons is qmax =hkmax = h2π/λmin = hπ/a. For metals qmax ∼ pF .

5

experimentally. Besides, the phonon dispersion lawdepends not only on the value of q but also on itsdirection. Fig. 2(a) shows the dispersion curves formetals with one atom in the unit cell and one di-rection in a crystal. A more convenient and moreoften used characteristics of phonons in a solid is theso-called phonon spectrum F (ε). This dependencereflects the phonon density of states or, in more de-tail, indicates which part of phonons has the par-ticular energy. Fig. 2(b) shows the typical form ofthe F (ε) curve in a crystal. The first two maximaare generally due to transverse vibrations of atomsin two perpendicular directions (polarizations) whilethe third maximum is due to more energized longi-tudinal vibrations since their excitation in a crystalcauses the compression and extension waves. As isseen from Fig. 2(a) there exists a maximum energy ofphonons, or in the wave language, the boundary (De-bye) frequency of vibrations ωmax(hωmax = εmax).The Debye frequency expressed in degrees is close toso-called Debye temperature. It has a very simplephysical meaning: This is a temperature at whichthe whole vibrational spectrum of metal, up to thehighest-energy phonons, is excited.

The phonon spectrum is an important character-istic of a solid. It enables determination of thermalcharacteristics of the lattice, the heat capacity andheat conductivity, plays a decisive role in kinetic phe-nomena stipulated by the electron-phonon interac-tion.

4 Waste recovery

Such a heading would be most suitable for popular-scientific literature on economy or ecology. It occurs,however, that sometimes the ”waste” can be utilizedin scientific research as well. Turning closer to thepoint, we will first describe one of wonderful phenom-ena - the tunneling effect. One of the paradoxical pre-dictions of quantum mechanics which appeared at thebeginning of this century was the possibility of mi-croparticle penetration through a potential barrier.In the common language it would mean that, for ex-ample, a ball strikes against a wall and passes throughit without causing any injuries. This phenomenon

was called tunneling, naturally bearing in mind thatno hole is left after the particle passage. The proba-bility of tunneling is the higher, the smaller are thethickness and height of potential barrier (wall thick-ness and hardness) to be overcome. The tunnelingeffect was immediately used to explain such micro-cosm phenomenon as α-decay of radioactive nuclei.However, the most striking manifestation of tunnel-ing could be observed in solid-state physics of the lastsixties.

Let us see what will occur when a potential differ-ence is applied to two metallic plates arranged like ina capacitor and then the interplate distance is grad-ually decreased. Let the plate potential be not veryhigh to avoid electric breakdown or suppose they arein vacuum. It turns out that at a certain plate separa-tion when they are not yet in contact, current appearsdue to electrons tunneling through the gap, which inthis case serves as a potential barrier. The interplatedistance at which these effects arise proves to be verysmall, not greater than a few nanometers (10−9m). Itis rather difficult to create and stationarily maintainsuch a vacuum gap between conductors 3 , thereforephysicists first took a different line. The gap betweentwo evaporated films of metal was formed from dielec-tric oxides, one being oxidized before evaporating theother. The oxide layer on a metal, as a rule, is fewnanometers thick and serves as a good and strong di-electric. This structure was called a sandwich or atunneling contact. The latter proved to be interest-ing not only due to the observation of the tunnelingeffect, but also as a tool of physical research.

When two metal come in contact, their Fermi lev-els coincide for otherwise the electrons would transferfrom the metal with a higher energy to the metal witha lower one. If a voltage V (Fig. 3) is applied to thetunneling contact, the Fermi level will shift with re-spect to one another by the value eV . The electrons

3It should be noted that in the beginning of eighties it be-came possible to make precision instruments where tunnelingproceeds through a vacuum or air gap between a flat surfaceand a sharp needle. As the tunneling current strongly dependson the amount of gap, by moving the needle over the surfaceand simultaneously approaching it to or withdrawing it fromthe sample at constant tunneling current, one can take surfacerelief with a superhigh resolution (not worse than 10−10m).Such instruments were called tunneling scanning microscopes.

6

Figure 3: Energy scheme of the tunnelling contactformed between two metals separated by a thin layer ofdielectric to which the potential difference V is applied.Two possible channels of electron tunnelling are show: theelastic one with energy conservation and inelastic withenergy loss and phonons or molecules excitation in thebarrier or on its surface.

residing on the left can tunnel into the right metalas there are free states on the right. The Ohm’s lawholds in such a contact even at not very high cur-rent voltages. The situation slightly differs if in theprocess of tunneling the electron changes its energyafter the interaction with, for example, molecules ab-sorbed on the dielectric layer surface, phonons in thebulk metal or in oxides separating the metals. Afterthe energy loss an electron tunnels to other states,i.e. it obtains a wide choice of places to get to. Suchprocesses will result in the tunneling current increaseand deviations from the Ohm’s law. Such correc-tion is small and for its exact observation one canusually plots not current-voltage characteristics buttheir derivatives which make it possible to determinethe energy of quasi-particles or excited molecules in-teracting with electrons. This method of study wascalled tunneling spectroscopy and has been widelyused in solid-state physics. The most complicatedthing in this technique is to create a thin dielectriclayer between the metals. If the oxidation is strong,the layer will be thick and the tunneling current negli-gibly small. If one tries to reduce the oxidation time,the appearance of ’holes’ becomes possible, i.e. thedielectric film will be discontinuous and conductivity

will be stipulated not by tunneling but by currentflowing throw the metallic shorts. Such contacts arenaturally to be rejected for they are no good for tun-neling investigations.

One of the authors (I. K. Yanson) has been engagedwith the problems of tunneling spectroscopy for along period of time. At a certain stage he paid atten-tion to this rejected tunneling contacts with metallicshorts ”wasted” from the tunneling measurements.It turned out that the current-voltage characteris-tics of shorted-out contacts also displayed deviationsfrom the Ohm’s law. In contrast to the tunnelingcontacts where current increased more rapidly thanlinearly, i.e. the contact resistance decreased withthe increase of the voltage, in shorted-out samplesthe resistance increased. One day, while investigateda shorted-out tunneling contact made of lead, thescientist found out that the second derivative of itscurrent-voltage characteristics looked amazingly sim-ilar to the phonon density of states in this metal(Fig. 4). It could not happen accidentally since whenshorting out this contact by current pulses, i.e. in-creasing the constriction dimensions, the picture didnot change and the maxima retained their positions.Later on this was checked on shorted-out contactsfrom other metals: copper, silver, gold, tin, etc. Sim-ilar pictures were observed in all cases : the sec-ond derivative of the current-voltage characteristicsreminded of the phonon density of states in metalsunder investigation.

5 What happens inside the

contact

The question arises why the above-mentioned shortsrevealed such spectroscopic properties, what kindthey are and what size they have. One may sup-pose that in shorted out tunneling contacts the sizeof ’holes’ is comparable to the dielectric thickness,that is a few nanometers. But the size of a metallicconstriction can be determined more precisely by theits resistance.

Consider PC model in the form of a thin, electron-opaque partition with a hole d (Fig. 5(a)). The ex-

7

0 5 10 15Voltage (mV)

LA0

0.5

1

d2 V/d

I2 (a

rb. u

nits

)

TA

Pb

314 Ω

0.3 Ω

5.8 Ω

83 Ω

Figure 4: Second derivatives of current-voltage char-acteristics of a shorted out tunnelling contact of leadrecorded with its resistance decrease from 314 to 0.3 Ohm.The spectral maxima correspond to those of the phonondensity of states in Pb marked by arrows.

pression for the resistance of such contact proved tobe dependent on the ratio of its size d and the elec-tron mfp l. The case of l ≪ d was considered bythe famous physicist J. Maxwell already at the endof the XIX century. The resistance of such contactswith large diameters or contacts formed by metalswith short electron mfp is expressed rather simply asRM = ρ/d. In the case of the reverse ratio, whenl ≫ d, the so-called ballistic regime of the electronsflight is realized, i.e. they pass through an orificealong rectilinear trajectories. In this case, as wasshown by Yu. V. Sharvin in the middle of sixties,R = 16ρl/3πd2. The expression for contact resis-tance at random l/d ratio was obtained in 1966 byG. Wexler

R = Γρ/d + 16ρl/3πd2 (2)

Here the value of Γ varies from 0.7 to 1 with thedecrease of l/d ratio from ∞ to zero. It can be seenthat the Wexler formula is actually the algebraic sumof two resistances, those of Maxwell and Sharvin.

Let us evaluate the size of lead contact mentionedin the previous chapter. The contact resistance was314 Ω and from Eq. (2) we will find that its diameteris about 2 nm or 20 A, i.e. it is really close to the

Figure 5: a) Model of PC in the form of a thinelectron-non-transparent screen separating two metallichalf-spaces with the orifice diameter d. b) Central sec-tion of the Fermi surface in the center of PC to whichthe voltage V is applied. c) The same in a homogeneousconductor with the electric field intensity E. here underthe field effect the whole fermi surface is shifted by eEl (lis the electron mfp) which is much smaller then eV.

thickness of oxide 4.

We have already considered the factors responsi-ble for the mfp l, but to compare it with the contactsize one should find its absolute value. In metalsat low temperatures, in the absence of phonons andother quasi-particle excitations, l is determined byscattering due to various lattice defects : impurities,dislocations, sample boundaries. In extremely pureand perfect single-crystalline samples at helium tem-peratures mfp may reach fantastic values of tens andhundreds thousands of interatomic distances, i.e. bemeasured in millimeters. With the increase of tem-perature and phonons excitation, the electron mfpdecreases due to scattering by phonons. At this timel can reduce to tens of nanometers.

It was noticed in the initial experiments on theinvestigation of PCs that the spectrum was mostdistinctly visible for contacts with higher resistancewhose size was of a few nanometers, respectively. Atthe same time, the spectral features smeared and

4When calculating, the value of ρl was set equal to10−11Ωcm2 and ρ ∼ 10−6Ωcm. These values can be usedfor most metals.

8

gradually disappeared in the case of large-size low-resistance contacts (Fig. 4). It become clear that dis-tinct features were observed in samples for which theelectron mfp was greater than the contact size, orwhen the electrons flew through contacts along theballistic trajectories.

If the electrons flow through the contact withoutscattering, the contact resistance is expressed by theSharvin formula, the current-voltage characteristicswill be linear, its second derivative will be equal tozero and, naturally, there will be no spectrum. Thesituation changes if we take into account that partof electrons are scattered by phonons in the con-tact neighborhood. According to the calculations,the probability of this scattering is proportional tod/le−ph, where le−ph is the length of electron-phononinteraction. At low temperatures, when the num-ber of phonons is small, the processes of excitation(generation) of phonons by electrons near the con-striction predominate. As a result of it part of elec-trons will not get into the hole and the total currentwill decrease. This means that the contact resistancewill rise due to the electron-phonon interaction, thuscausing deviations from the Ohm’s law and the signalappearance in the second derivative.

Qualitative considerations are also appropriatehere. At the increase of voltage by ∆V , the changeof resistance ∆R will be proportional to the prob-ability G(ε) of excitation by electrons in the con-tact zone of a phonon with the energy ε, i.e. ∆R ∝G(ε)∆V or, taking into account that R ∝ ∆V/∆I,we shall evaluate the second derivative as d2V/dI2 ∝∆R/∆V ∝ G(ε). Thus, the second derivative of thecurrent-voltage characteristics reflects the probabil-ity of phonons emission G(ε). The latter, in its turn,is proportional to the phonon density of states F (ε)multiplied by the so-called square of the electron-phonon interaction α2

PC(ε) taking into account the”force” of electrons interaction with one or anothergroup of phonons and the contact geometry. There-fore we can write

d2V/dI2 ∝ G(ε) = α2PC(ε)F (ε) (3)

This expression was later rigorously substantiated intheoretical works described below. It turned out that

the α2(ε) dependence is ”smoother” function of en-ergy compared with F (ε), so the shape and peculiar-ities (maxima, minima) of d2V/dI2 are to a greaterextent dictated by the phonon spectrum behavior.Now it is clear why the spectra for lead and othermetals remind us the phonon density of states.

6 Some theoretical fundamen-

tals

The theory of point-contact spectroscopy has beendeveloped by I. O. Kulik and his coworkers A. N.Omelyanchuok and R. I. Shekhter. Their principalresult was rigorous derivation of the expression ford2V/dI2 of PC in the ballistic regime and its relationto the function F (ε). A model chosen for PC was anorifice in a thin opaque to electrons partition separat-ing two metallic half-spaces (Fig. 5(a)). Later on ithas been demonstrated that the result is noncriticalwith respect to the contact geometry. The expressionobtained theoretically has the form

d2V

dI2≃ − d2I

dV 2∝ 〈K〉−1d3N(εF )G(ε) (4)

where 〈K〉 is the PC geometry factor equal to 0.25 forthe orifice model, d is the contact diameter, N(εF ) isthe electron density of states at the Fermi surface.

The nonequilibrium function of electrons distribu-tion in the center of contact is shown in Fig. 5(b).When voltage is applied to the contact, the electronsare separated into two groups arriving from the rightand the left regions, correspondingly. Maximum en-ergies of these groups at T = 0 differ precisely bythe amount of eV . It becomes important that theenergy difference between any two states of electronsat the Fermi surface is equal to eV or zero. It isthis circumstance that enables us to use PCs as anunusual energy ”probe” to determine the energies ofother quasi-particles.

Fig. 5 shows that the electrons distribution func-tion in PC differs but greatly from a similar one inthe case of current flowing through a homogeneoussample. In this case it is impossible to impart a con-siderable excess energy to the electrons that is com-parable to the energy of Debye phonons. It turns

9

I

dV/d

I

V

V

V

Vεph

0

5.4 kT

d2 V/d

I2

Figure 6: Current-voltage characteristics I(V ), its firstand second derivatives for PC of hypothetic metal inwhich all phonons have the same energy εph. The peakin the second derivative will have the width 5.4kBT dueto the Fermi level smearing (see Chapter 8).

out that for the electron energy acquired in the elec-tric field to reach the characteristic phonon energies,the current energy should exceed 109 A/cm2. Suchcurrent densities in the bulk metal are unattainablebecause of the Joule heating and melting. It is notthe case with PCs as due to their small size the heat isquickly ’sucked off’ into the cold bulk metal. In otherwords, the contact area remains cold as a result of alarge mfp of phonons compared with the contact side:the phonons have time enough to depart and transfertheir energy into the surrounding medium.

For better understanding of scattering processes inPC, consider the case of a certain hypothetic metalin which only the phonons with the same energy εph

are present. If the electrons passing through the ori-fice have the excess energy eV < εph they are unableto excite the phonons and the electron trajectoriesare straight lines without scattering. The contact re-sistance is precisely equal to the Sharvin’s one, and

the current-voltage characteristics will be rectilinear(Fig. 6). When the electrons obtain the energy εph,they begin to excite phonons, scatter, and will notget into the orifice, so the current through the con-tact will build up with voltage more slowly than inthe absence of scattering. It is clear that the current-voltage will have a break at this point. The differ-ential resistance Rd = dV/dI dependence will makea step, and the second derivative d2V/dI2 will dis-play a sharp peak at a specific energy (Fig. 6). Ifthe metal possesses several groups of phonons withdefinite energies, then the d2V/dI2 curve will have amaximum for each of them. In the general case atcontinued phonon spectrum the second derivative ofthe current-voltage characteristics will be similar tothis curve.

As has been already mentioned, d2V/dI2 does notmerely reflect the phonon spectrum but is propor-tional to the probability G(ε) of emission of phononwith the energy ε. The G(ε) function is gener-ally called the PC spectral function of the electron-phonon interaction. In the case of conductors,the transport and thermodynamic functions of theelectron-phonon interaction are also distinguished.The former is connected with the current flow in met-als, the latter determinates the effective mass, veloc-ity, and electron density of states changes due to theelectron-phonon interaction.

The PC spectral function G(ε) introduced by anal-ogy with the above ones is a variety of transport func-tion of the electron-phonon interaction and differs bya stronger dependence on the angle of electrons scat-tering in the contact. In the orifice model this depen-dence is determined by the expression (1− θ/ tan θ),where θ is the scattering angle of electron momen-tum. It is seen that at θ = π, when the electronreverses its motion direction this expression diverges,i.e. such processes give maximum contribution intothe PC spectrum and are most significant. This es-sentially distinguishes the situation in contacts fromscattering in a homogenous conductor resulting innormal resistance or from electron-phonon interac-tion responsible for superconductivity. In the formercase the scattering efficiency increases with the angleas (1−cos θ), in the latter case it is independent of theangle at all. Thus, all the functions in question can

10

be written in the form of Eq.(4), but with differentα2(ε) dependencies.

7 Procedures for PCs creating

We have already mentioned one of the techniquesused to obtain micro contacts in the tunneling struc-tures. The schematic representation of a contact witha short in the dielectric layer is shown in Fig.7a. Sucha conducting bridge often appears spontaneously as aresult of metallic dendrites intergrowth through weakspots or defects of the dielectric film. The bridge canbe created artificially by means of mechanical loadscausing the oxide film damage or by means of electricbreakdown. In the former case use is made of a sharpneedle creating high local stresses, in the latter casethe electric voltage is raised up to the level exceed-ing the breakdown value at the weakest spot of thedielectric. The latter case resembles electric microw-elding when metal is instantaneously melted in theconstriction and is purified from foreign inclusions.Certainly depending on the film material, dielectricinterlayer properties, magnitude and duration of thecurrent pulse, the degree of structure perfection andthe purity of metal in the constriction can vary in awide range. Using the film technology, one can ob-tain rather small PCs only of a few nanometers withsufficiently high mechanical stability and vibrationresistance.

However, the fabrication of film PCs involvesrather awkward technique of vacuum deposition ofmetals. Besides, the structure of metal in the films isusually less perfect than in the bulk samples. That iswhy the creation of pressure contacts between bulkelectrodes gave an impetus to the development of re-search in this field. A pressure contact of the needle-anvil type is shown in Fig.7b. It is obtained by grad-ually approaching two electrodes directly in liquidhelium by means of precision mechanism. Gener-ally, the radius of curvature of the needle tip is afew nanometers. Before measurement the electrodesare processed mechanically, then chemically or elec-trochemically, to obtain clean unperturbed surfacesof required shape. At this time the surface is coatedby a thin film of oxide or other non-conductive com-

Figure 7: Method of PC preparation a) A film of metalis deposited in vacuum onto a dielectric substrate, a thindielectric layer is formed on the film by oxidation or othermethods, on top of this layer the second film of metal isdeposited. The technology is selected in such a way thatcontacts should be formed spontaneously in the processof preparation or as a result of subsequent mechanical orelectrical effects. b) ”Needle-anvil” method: a speciallyprocessed metallic electrode in the form of thin needle isslightly pressed to the flat surface of the other electrodeby means of precision mechanism. Here one can con-trol the resistance or size of contact, decrease or increasethem. c) Shear method - the contact is formed by slightlybringing together the edges of two electrodes which canhave random shapes and then by shifting one electrodealong the other. As a result, contacts can be obtained inmany places and due to the shear stresses the metal de-formations and contaminations near the constriction areless than in the preceding case.

pound. Due to the dielectric layer, at the first lighttouch of electrodes the contact conductivity is oftenof the tunneling character, i.e. the contact resistancedecreases with the increase of voltage. A metallicbridge appears at the increase of pressing force eitheras a result of dielectric cracking or of breakdown bythe current pulse. The presence of dielectric film onthe electrode is necessary to stiffen the whole struc-ture. In this case the mechanical contact area is muchgreater than the size of metallic constriction.

In the needle-anvil configuration, the metal in thecontact area is obviously deformed and polluted withpressed-in oxide residues, as well as with various im-purities absorbed on the surface. Better results areobtained if the pressure contacts are created by thetechnique shown in Fig.7c. In this instance the elec-

11

trodes are brought together till touching, then theyare shifted relative to one another in the plane oftheir crossing. As a result of it, the deformation ofmetal here is much less while the oxide and surfaceimpurities are carried out of the contact area. Thus,constrictions are formed in those places where thecleaned areas of electrodes come in contact. Withthis technique, the samples can take the form of smallbulk cylinders or blocks of metal, as well as filmsdeposited on dielectric rods. By shifting the elec-trodes independently in the mutually perpendiculardirections or rotating them about their axes, one canchange the places of contact and obtain a numberof shorts with different resistances within the samemeasurement cycle.

The point-contact spectroscopy would be impossi-ble if the characteristics to be measured were criti-cally dependent on the constriction geometry whichcannot be controlled due to the small size of short.However, experience has shown that if the contactswere chosen properly by means of experimental crite-ria, the the reproducibility of PC spectra is not worsethan in tunneling spectroscopy where these choice isgenerally accepted. Let us formulate the quality cri-teria of PCs which meet the requirements of the the-oretical clear orifice model, the orifice being muchsmaller than the electron mfp. We shall take a PCspectrum of one of simple metals, sodium (Fig. 8) asan example.

1. For pure metals, the contact resistance is within1 to 100 Ω and the constriction has the metallicconductivity, i.e. its resistance increases with theincrease of voltage so that d2V/dI2 > 0 over thewhole energy range.

2. Sharp peaks or other distinct features are ob-served in the second derivative (PC spectrum)at the energies from zero to the Debye oneεD ≃ hωmax reproduced for different contacts ofthe particular metal. For the energies eV > εD

the spectrum assumes a certain constant valuecalled the background. The background param-eter γ = B/A (Fig. 8) should be small enough,different for each metal, but as a rule not greaterthan 0.5.

0 10 20

Energy (meV)

TA1 A

B0

3

6

d2 V /

dI2

(ar

b. u

nits

)

LA

TA2

hω max

Figure 8: Point-contact spectrum of one of simple metals,Na (solid curve). The curve clearly reveals three featuresdue to the electron excitations of phonons in two trans-verse (TA) and one longitudinal (LA) branches. Compar-ing the spectrum with the phonon density of the states(dashed curve) one can see that in this case electrons in-teraction with transversely polarized phonons is consider-ably suppressed which is due to the closed Fermi surfaceof this metal.

3. Relative variation of differential resistancewithin the spectrum should be maximum (fromones to tens per cent) at minimum γ. The ini-tial portion of the d2V/dI2 curve at the energieseV ≪ εD is smooth and has no features.

The above attributes make it possible, judging bythe form of the PC spectrum, to estimate the contactsuitability for further investigations of the electron-phonon interaction function. Applying these criteriato the spectra shown in Fig. 4, one can see that theyare satisfied by curves 1 and 2.

12

8 How to measure the second

derivative

One of the widely used methods of measuring thecurrent-voltage characteristics derivatives is the mod-ulation technique. It consists in the following. Be-sides the measuring direct current I, the sample issupplied with a definite frequency ω ac current. Inthis case the contact voltage can be presented as aTaylor series

V (I + i cos(ωt)) = V (I) +

+dV

dIi cos(ωt) +

1

2

d2V

dI2i2 cos2(ωt) + · · · = V (I) +

+dV

dIi cos(ωt) +

1

4

d2V

dI2i2(1 + cos(2ωt)) + · · · (5)

If the current i is sufficiently small, the higher-order in i terms can be neglected. It can be seen thatthe signal measured at the frequency ω will be pro-portional to the first derivative dV/dI of the current-voltage characteristics, while at the frequency 2ω itwill be proportional to the second derivative d2V/dI2.The next problem is to measure this weak voltageat the frequency ω and 2ω. This is achieved by aspectrometer whose schema is shown in Fig. 9. Thesample (1) is held in a cryostat with liquid heliumwhich is immersed into an outer cryostat with liquidnitrogen to reduce the heat input and coolant evapo-ration. Connected to the sample are four wires, a pairof which serves as an electric circuit, through whichthe contact is supplied with direct current from thesource (2) and the alternating signal from the har-monic voltage generator (3). The direct voltage Vcreated at the sample by current I is applied to the X-coordinate of the recorder (6) by means of the otherpair of wires. This pair is also used for measuring theac voltage of the second harmonic of modulating sig-nal which requires a special circuit for its amplifica-tion and separation from noise. The circuit comprisesthe filter (4) and lock-in amplifier (5). When register-ing this signal, proportional to d2V/dI2, the filter istuned to resonance to a required frequency 2ω. Thefilter also serves for matching the low-resistance sam-ple with the high-resistance input of the amplifier.

Figure 9: Schema of the spectrometer: 1 - sample, 2 -DC source, 3 - AC source, 4 - filter, 5 - look-in amplifier,6 - plotter.

Besides, the amplifier receives a 2ω generator signalused as a reference voltage. As a result of it, not onlythe signal is amplified but also the required frequencyis separated. The voltage is converted into the directone and applied to the Y-coordinate of the recorder.Thus, when a linearly increasing current from source(2) passes through the sample, the recorder will auto-matically record the second derivative of the current-voltage characteristics as a function of voltage.

As is seen from Eq.(5) the signal at the frequency2ω is proportional to the square of the modulat-ing current or first harmonic voltage amplitude Vω .Therefore, it seems there are no special problems inmeasuring this signal by merely increasing the alter-nating component Vω . However, it cannot be madearbitrarily large as the ac voltage amplitude deter-mines the instrumental resolution of this technique.Exact calculations indicate that an infinitely nar-row spectral feature when measuring d2V/dI2 by themodulation technique will be smeared to form a bell-shaped maximum 1.7Vω wide. In real experiments,the value of Vω is as a rule not greater than a fewmillivolts, and in separate cases it can be equal toseveral tens of millivolt. In this case the value ofmeasured signal which characterizes the nonlinearity

13

of current-voltage characteristics associated with theelectron-phonon interaction is of the order of micro-volts. At the same time the measuring circuit sensi-tivity is two orders of magnitude higher which makesit possible to reliably register the PC spectra.

It should be also noted that the point-contact spec-troscopy resolution is dependent on the temperatureT due to the Fermi level smearing by about kBT .Therefore, to obtain distinct spectral features, thecondition kBT ≪ εD should be satisfied, i.e. mea-surements should be taken at temperatures consider-ably lower that the Debye ones. Under the action ofthe instrumental and temperature factors, the result-ing resolution is expressed by

δ = [(5.4kBT/e)2 + (1.7Vω)2]1/2 (6)

Generally, measurements are taken in liquid heliumat a temperature of 1.5 to 4.2 K and modulating sig-nal of 0.3 to 1.5 mV. As a result of it, the resolutioncalculated by Eq.(6) is 0.9 to 2.9 mV or, expressedin degrees, 10 to 34 K. This is at least an order ofmagnitude below the Debye temperatures of typicalmetals which enables obtaining a sufficiently detailedspectrum of electron-phonon interaction.

9 Point contact instead of reac-

tor

Obviously, the headline of this section is rather adver-tising than true to life. A long time will pass till a nu-clear reactor can be replaced by a competitive sourceof energy or research tool. Our headline is used inthe sense that PC can compete with the reactor incertain fields of solid-state physics. We shall considerthe experimental techniques of phonons investigationin metals and the determination of phonon spectrum.

One of the principal methods used for determiningthe phonon dispersion law ε(q) is the study of neutronscattering in a crystal. The source of neutrons is usu-ally a reactor. For this purpose use is made of the so-called thermal neutrons with the energy of 30 meV or3000 K and, respectively, the de Broglie wavelengthcomparable to the lattice constant. Knowing the mo-mentum and energy of incident neutrons, then mea-suring their energy and momentum after scattering,

on the basis of conservation laws one can determinethe energy and momentum of phonon by which theneutron was scattered. Thus one obtains the disper-sion curves shown in Fig.2a. Since the law of phononsdispersion in a crystal is anisotropic, the dispersion ismore often measured only for few principal crystallo-graphic directions as the measuring procedure itselfis rather long and requires a powerful source of neu-trons. To obtain a phonon spectrum, one needs thedispersions in the lattice, then one should calculateall the phonons whose energy falls within the inter-val from εk to εk + ∆ε = εk+1 (where ε ≤ εD and∆ε ≪ εD), then plot this value (Fig.2b). Then re-peat this procedure for the next interval from εk+1

to εk+1 +∆ε, etc. The less the interval ∆ε, the moreaccurate is the F (ε) curve. But, as mentioned above,it takes a long time to measure the dispersion curves,and they are available only for several principal crys-tallographic directions in the lattice. To obtain thedispersion law for a maximum number of directions,use is made of theoretical calculations describing thelattice thermal vibrations. The formulas employedfor this purpose include the parameters which havedefinite values for one or another solid. These param-eters are found by fitting the theoretically calculateddispersion curves to the experimental ones. The ob-tained parameters make it possible to calculate thedispersion curves for any direction and, therefore, tocalculate the energy of phonons with any momentumvalue both in magnitude and direction, and to plotthe F (ε) curve. The required calculations are ratherbulky, voluminous and need the use of a powerfulcomputer. Besides, the result depends on the modelselected, i.e. on how accurately the theory takes intoaccount the real processes of atoms interaction in thelattice. In principle, it is possible to obtain the in-formation on the phonon density of states in a moredirect ”modelless” way. For some metals one canuse the one-phonon incoherent scattering of neutronswhen the directly measured scattering cross-sectionof neutrons with a specific energy is proportional tothe phonon density of states. However, the lattertechnique is not suitable for all metals and does nothave high resolution.

How can the point-contact spectroscopy be of anyhelp here and how can it substitute the reactor? We

14

Figure 10: Mendeleev’s periodic table containing onlythose elements which have been studied by point-contactspectroscopy (until 1988).

have demonstrated that the form of electron-phononinteraction function G(ε) is dictated by the phonondensity of states. It turn out that for practically allmetals under investigation the spectrum extrema arestipulated by their presence in F (ε). Moreover, forsome metals, mainly polyvalent ones, such as Pb,In, Sn, Mg, Be, etc., these two dependencies G(ε)and F (ε) practically coincide in form. Therefore, thepoint-contact spectroscopy can be referred to as anew method of phonon spectroscopy in metals. Itshould be mentioned that the PC experiments aresimple and accessible for any physical laboratory.One should only have liquid helium for low tempera-ture measurements and ordinary electronic engineer-ing equipment. It is noteworthy that PCs can be cre-ated by means of very small amounts of a substance,a few cubic millimeters of which being sufficient. Itbecomes especially important for studying rare sub-stances available only in small quantities.

To date, the PC spectra have been obtained for al-most four tens of pure metals indicated in Fig. 10. Ifwe drop the rare-earth elements, we have to measureyet of about ten metals, basically highly active alkali-earth and alkali ones. By means of point-contactspectroscopy, it became possible to refine the posi-tions of extrema and other features of the F (ε) func-tion. For a number of metals it was possible to de-termine the shape of the whole curve, i.e. to find out

the number of maxima, their positions with respectto energy and relative intensities.

10 Study of electron-phonon

interaction

It has already been mentioned that the spectral func-tion G(ε) is a kind of transport function of theelectron-phonon interaction. By comparing G(ε)with phonon spectra for various metals, one can studythe Fermi surface effect on the dynamics of electron-phonon interaction. Thus, in alkali metals Na andK the Fermi surface is practically a sphere. At suchan isoenergetic surface, it is difficult for electrons tointeract with transversely polarized phonons. In fact,the features associated with scattering by transverse(T) phonons are strongly suppressed in the PC spec-tra of sodium (Fig. 8) and potassium. On the con-trary, the Fermi surface in noble metals is open whichis favorable for interaction with transverse phonons,so in the PC spectra of copper (Fig. 12), gold, silverthe peaks corresponding to these phonons prevail. Fi-nally, in the polyvalent non-transition metals: lead,tin, magnesium, berillium (Fig. 12), etc., the Fermisurface is of intricate configuration and few phononsof one or another polarization are released. In thiscase the G(ε) function shape is closed to the depen-dence of phonon densities of state, i.e. one can as-sume that the parameter α2(ε) in the formula (3) isweakly dependent on energy.

On the other hand, the electron-phonon interac-tion function itself is of fundamental importance. Itdetermines the properties of metals stipulated by theelectron-phonon interaction, in particular the possi-bility of superconducting transition, as well as variouskinetic parameters characterizing the charge and en-ergy transfer. The point-contact spectroscopy yieldsdetailed information on the relative force of elec-trons binding with some groups of phonons, e.g.with those transversely or longitudinally polarized.Thus, it becomes clear which phonons play more im-portant role in various processes. For normal non-superconducting metals, the study of PC spectra isthe only experimental technique which enables direct

15

measurement of energy dependence of the spectralfunction of electron-phonon interaction.

11 Magnons and excitons

From the very childhood we have used to metalsbreaking down into magnetic and nonmagnetic oneswithout thinking on the nature of such difference.This classification of metals has been explained interms of quantum theory. Generally speaking, mag-netism is a purely quantum-mechanical phenomenonthat cannot be explained in the framework of classi-cal theory. As one of the well-known physicist said,if the Planck constant had gone to zero, there wouldhave been no science of magnetism.

As is known from the atomic physics, an electronhas its moment of momentum, the spin. Besides, anelectron in an atom has an orbital magnetic momentdue to the motion about the nucleus. To determinethe total magnetic moment of an atom, one shouldconsider its electronic structure. If the atoms witha magnetic moment are united in a lattice, it is notenough to make the whole material magnetic. Ev-erything will depend on the actual interaction of mo-ments in which the so-called exchange forces play animportant role. These forces arise due to quantumindistinguishability of particles and the Pauli prin-ciple. The exchange forces are not connected withdirect interaction of spin magnetic moments which isweak but are dependent on their relative positions.The exchange interaction just makes the magneticmoments of separate atoms feel one another and lineup in a certain way: in parallel or in antiparallel di-rection. In the former case we obtain a ferromagnet,in the latter case antiferromagnet.

Consider, for example, a ferromagnetic metal inwhich the magnetic moments of atoms have the samedirection. Let for some reason, for example due tothermal vibrations, one atom reverse the directionof its moment. Then the neighboring atom will beaffected by a force tending to change over the direc-tion of its magnetic moment too, etc. Hence, a waveof magnetic moment reversals will pass through thecrystal. This wave was called the spin wave. The spinreversal events in magnetic materials are usually con-

sidered in the language of spin waves convenient forthe theory. According to the quantum mechanicalprinciple of dualism, the waves can be substituted byquasi-particles and, by analogy with a phonon, thespin oscillation wave was called a magnon.

Thermal vibrations ”rock” the magnetic momentsand with the increase of temperature the number ofatoms with reversed spins, or, in other words, thenumber of magnons increases. When the energy ofthermal vibrations becomes equal to that of exchangeinteractions, there will be no ordering of magneticmoments and the magnetic metal will become non-magnetic. The temperature at wich it occurs wascalled the Curie temperature Θc. The latter is simi-lar to the Debye temperature for phonons: there canbe no magnons with the energy above kBΘc in metal.

Magnons are quasi-particles similar to phonons.They carry heat, scatter the conduction electrons, i.e.affect the kinetic properties of metals. Therefore, it isquite reasonable to suppose that the electrons scat-tering by magnons or, more precisely, the magnonsexcitation by electrons in a PC should cause non-linearity of current-voltage characteristics, as is thecase with phonons, and the PC spectrum will displaythe features associated with the electron-magnon in-teraction. Indeed, such features were registered wheninvestigating the PC spectra of rare-earth metals Gd,Tb, Ho which are ferromagnetics at low tempera-tures. A question arises why rare-earth metals wereinvestigated but not less exotic well-known ferromag-netics, such as Fe, Ni, Co. The matter is that in thesemetals the Curie temperature, and consequently theenergy of magnons, is considerably greater than thoseof phonons. At this time it turns out that theelectron-phonon interaction essentially decreases theelectrons mfp and the ballistic regime in a contact isdisturbed before reaching the energy of characteristicmagnon frequencies. On the contrary, in the abovementioned rare-earth metals the Curie temperatureis less than the Debye temperature and the electron-magnon interaction features show up at lower ener-gies than the phonon ones.

In a solid, the conduction electrons can also in-teract with other elementary excitation or ”-ons”.Let us dwell on one of such quasi-particles, an exci-ton, representing the electronic excitation in a crystal

16

of semiconductor or dielectric. As a result of inter-atomic interaction, an exciton propagates over a crys-tal in the form of a wave without carrying charge ormass. Excitons are most intensively emitted fromsemiconductors and dielectrics where they presenta bound state of the conduction electron and hole.However, there exist some exciton-like excitations inmetals as well.

Thus, in the compound PrNi5 the main nine-folddegenerated state of Pr3+ ion is split by the intracrys-talline electric field of lattice atoms into six energylevels, three of which remain doubly degenerate, theother three presenting a separate level of energy (seediagram in Fig. 11. When the Pr3+ ion is excited, itcan reside in one of the higher-lying states, for exam-ple Γ5A, in which it has a magnetic moment. Suchan excited state is transmitted from ion to ion, i.e. itcan be treated as a magnetic exciton. The processesof Pr3+ ions excitation by electrons should obviouslycause the PC spectrum features at corresponding en-ergies.

At low temperatures, the Pr3+ ion is in the groundstate Γ4 with minimum free energy. When the ionis excited, it can move to the above-mentioned levelΓ5A, as well as to the higher-energy levels Γ3 and Γ5B.Other transition to the levels Γ1 and Γ6 are forbid-den by the selection rules. However, in an externalmagnetic field this exclusion is lifted (it remains onlyfor one high-symmetrical direction along the hexago-nal axis), and the Zeeman splitting of spin-degeneratelevels occurs (see Fig. 11).

The processes of Pr3+ ion excitation by electrons,as a result of which the ion moves to a higher-lyinglevel, bring about their inelastic scattering and man-ifest themselves in the PC spectrum as peaks at thecorresponding energies. In the zero field the PrNi5spectrum distinctly reveals three features associatedwith the transitions shown in Fig. 11(lower panel).The most intensive peak corresponds to the Pr3+ ionexcitation to the state Γ5A. Evidently, these pro-cesses will make maximum contribution into the re-sistance of this substance. The probability of othertransitions and, respectively, their influence on elec-tric conductivity and other kinetic and thermody-namic properties of PrNi5 is appreciable less. Theseare the simplest qualitative conclusions that can be

Figure 11: Above: Point-contact spectrum of PrNi5 ina zero magnetic field and in a field of 11 Tesla. Ar-rows indicate the peculiarities due to the Pr3+ ion tran-sitions between the energy levels split by the intracrys-talline field. Below: scheme of the Pr3+ energy levelsin the intracrystalline electric field and their additionalsplitting in the external magnetic field. Arrows indicatethe allowed transitions marked on the PC spectra. Thetransitions marked by dashed lines are weakly probablein the magnetic field and are not visible on the spectra.

17

made directly from the form of PrNi5 PC spectrum.

The probability of transition to previously forbid-den levels manifests itself in the magnetic field andZeeman splitting of the remaining degenerate levelsoccurs. Consequently, the PC spectra should changewhen the magnetic field is switched on,which can beseen in Fig. 11(upper panel). An especially notice-able evolution occurs with most intensive maximum.Here, together with the peak splitting at 4 meV, onecan see additional features connected with the tran-sitions to the levels Γ6 and Γ1. Hence, instead ofone maximum several maxima appear in the mag-netic field. Note that these features can be distinctlyobserved only in a sufficiently high magnetic field.Thus, for example, the amount of splitting evaluatedby the formula E = 2gµH (where g is Lande factor,µ is the Bohr magneton) should be greater than theresolution of this technique which is about 1 meV.As a result of it, the magnetic field amplitude shouldexceed 4-5 Tesla and the higher the amplitude, themore distinct are the features. That is why PrNi5was studied in a field of up to 21 Tesla, while thefigure, by way of example, shows the spectrum for anintermediate field.

It should be noted that measurements were carriedout on single-crystalline samples of PrNi5 for threeprincipal crystallographic directions, which made itpossible to detect also the anisotropy of Zeeman split-ting. When doing so, we for the first time obtainedthe experimental data on the exact positions of levelsthe transitions to which are forbidden in zero field.These data were used for comparison with theoreticalcalculations which allowed us to determine the pa-rameters characterizing the intracrystalline field in aparticular compound. The above results can serve asan example of how new data were obtained by point-contact spectroscopy which enabled the constructionof a quantitative model of the intracrystalline effectsinfluence on the PrNi5 properties.

12 From pure metals to alloys

and compounds

In principle, any energy-dependent mechanism ofelectrons scattering in metals can be studied bymeans of point-contacts. To put it in another way,the processes influencing the inelastic mfp of elec-trons in the lattice will be reflected in the point-contact spectra. This can be easily proved using thealready mentioned Wexler formula (2) for the con-tact resistance. The second term in this formula, theSharvin resistance, is a constant for the product of ρlis constant. The Maxwell resistance can be rewrittenas RM = ρl/d (1/l) where the first cofactor is alsoconstant and the second 1/l, is energy- dependent ifthere exist the electron scattering processes affectingthe electron mfp, i.e. l = l(ε) = l(eV ). For the sec-ond derivative of the current-voltage characteristicswe shall write d2V/dI2 ∼ R dR/dV . Substituting in-stead of R its expression from Eq.(2) we obtain that

d2V/dI2 ∼ ρl/d d/dV [1/l(eV )] (7)

Thus, the second derivative will have extrema atthose energies at which the mfp of electrons changesrapidly. In the case of electron-phonon scattering theelectron mfp considerably decreases at those energiesat which there exist a large number of phonons, i.e.at the phonon spectrum maxima. That is why a peakcan be expected in this place of d2V/dI2.

It has been previously noted that the spectralregime is realized only in very small contacts, suchthat the electron mfp should exceed the constrictionsize. There are two mfp of electrons in conductors:the elastic mfp le after passing which the electronchanges its momentum or direction of motion, andthe inelastic mfp li at which the electron changesits energy, le is independent of the excess energy ofelectrons and, according to Eq.(7), should not affectthe point-contact spectrum form. A question ariseswhether the point-contact spectroscopy is possible ifthe elastic mfp becomes less than the short size. Thisquestion is not idle since many alloys and compoundshave rather small le. The positive answer makes itpossible to apply the point-contact spectroscopy tech-nique not only to pure metals whose number, though

18

large, is limited, but also to practically unlimited col-lection of more or less complex alloys or compoundsof different metals. It turned out that the techniqueworks under less strict conditions, namely if the in-elastic relaxation length during the diffusion motionof electrons in the contact lD which in the case ofalloys is determined by lD = (leli/3)1/2 is greaterthan the size of contact. The condition d ≪ lD canin many cases be satisfied for PCs made of complexsubstances and the spectral information on electronscattering mechanisms can be obtained. The regimeat which the condition le ≪ d ≪ lD is met was calledthe diffusion regime. In this case, as was shown bytheorists, the proportionality between d2V/dI2 andthe spectral function G(ε) is preserved, only in theright-hand part of Eq.(4) an additional factor ap-pears that is proportional to the le/d ratio and leadsto the decrease of point- contact spectrum intensity,for le/d ≪ 1.

If impurities are introduced into a perfect lattice,it will cause the decrease of elastic electron mfpli < le, d, which is the smaller, the higher concentra-tion of the impurity. At this time the point-contactspectra will not change at low impurity concentra-tions (up to several per cent), only their absolute in-tensities drop. A more interesting case is when themass mi of the impurity atom substantially differsfrom the mass M of lattice atoms. Thus, for exam-ple, if mi ≪ M , there arise the so-called local oscil-lations of the impurity atom with a frequency higherthan the maximum one of the host lattice vibrations.At the inverse relation mi ≫ M , the quasi-local os-cillations with a frequency essentially lower than theDebye one appear in the spectrum of lattice frequen-cies. It can be readily understood qualitatively if werecall that the frequency of oscillator (pendulum) os-cillations is inversely proportional to the square ofits mass. It appears that the electrons scattering bylocal and quasi-local oscillations occurs inelastically,i.e. with the change of energy. Hence, the peculiari-ties connected with such oscillations can be expectedin the point-contact spectra.

Fig. 12 shows the spectrum of beryllium, as wellas the CuBe alloy containing 4.2 atomic percent ofBe. One can see that an additional feature has ap-peared in the d2V/dI2 curve. This feature cannot be

0 25 50 75 100

Voltage (mV)

0

1

2

d2 V/d

I2 (a

rb. u

nits

)

CuBe

0 2 4

c (at%)0

1

LV/L

A

LVLA

Figure 12: Point-contact spectrum of copper alloy with4.2 at.% of beryllium. It is seen that in the Cu spectrum(between 0 and 30 mV) has developed a new maximum(LV) at 40 mV which is absent both in the spectrum ofpure copper and beryllium (dashed curve). It is connectedwith local vibrations of the Be atoms in the Cu lattice.Inset shows increasing of LV maximum intensity relativeto LA phonon peak in Cu with beryllium content rise.

found either in the spectrum of copper or in that ofberyllium. This is exactly the band of local oscilla-tions. In this case the impurity atoms are seven timeslighter than the copper ones. Another example of analloy with quasi-local oscillations is magnesium withthe admixture of lead atoms. Here the difference inmasses is 8.5 times. The point-contact spectra of thisalloy also reveal a feature that could be expected inthe region of energies essentially lower than the De-bye one.

The other impurities which strongly scatter theelectrons are the atoms of rare-earth metals of irongroup which retain their magnetic properties whenadded to a normal metal. The electrons scatteringby such elementary magnetics in a bulk conductorcauses an abnormal for metals rise of resistance at

19

low temperatures known as Kondo effect. PC hasproved to be a very sensitive indicator for detectingmagnetic impurities in metal. Thus, when addingonly 0.01% of iron or manganese atoms to copper,the point-contact spectrum will display distinct addi-tional features in the energy range 1-2 meV which areconnected with the electrons scattering by localizedmagnetic moments. One can evaluate the number ofimpurity atoms getting into the contact area at theabove concentration c = 10−4. The total number ofatoms in a constriction with size d will be V/θ, whereV ≃ d3 and θ is the volume per atom. The number ofimpurity atoms Ni is estimated as a value of the orderof cV/θ. For copper θ = 1.17×10−23cm3, and the sizeof PCs in question is about 1 to 2 ×10−6 cm. There-fore, we obtain the value of Ni equal to 8-70 atoms.It is of interest that the PC area contains a mea-gre number of atoms whose effect is, however, notice-able in the measured characteristics. Thus, we cansay that the point-contact spectroscopy is a uniquemethod of studying the electrons scattering by singlemicroscopic centers.

13 Contact heating

So far we consider PCs whose dimensions are smallin comparison with the electron mfp. As has beenshown, in this case a highly nonequilibrium functionof electrons distribution is realized and there existtwo groups of electrons whose energies differ by eV .Relaxation of such a nonequilibrium state leads to theexcitation of quasi-particles with the energy ε ≤ eVand to the nonlinearity point-contact current-voltagecharacteristics which contains spectral informationabout the energy distribution of quasi-particles un-der investigation. It turned out that even in the caseof small elastic mfp the spectroscopic capabilities ofcontacts remain. For this purpose it is only necessarythat the diffusion length of energy relaxations shouldbe greater than the size of constriction. It is of in-terest what will happen if the elastic and inelasticlengths will be considerably smaller than the size ofcontact. In other words, what will happen if the con-tact dimensions are large or when studying the sub-stances with sufficiently short relaxation lengths. It

appears that in such constrictions the flow of currentof the required density leads to conventional Jouleheating. The heat removal from such large contactswith small mfp is less efficient than in the ballisticregime. Since in metals the heat removal from thehot region to the bulk of the sample is principally ef-fected by the conduction electrons, it can be shownthat when the Wiedemann-Franz law holds, the volt-age drop at the contact and the metal temperatureat the center Tpc are related as

(kBTpc)2 = (kBT0)

2 + (eV )2/4L (8)

where T0 is the measurement temperature, L is theLorentz number. If measurements are taken at lowtemperatures, i.e. T=0, the contact temperaturewill be linearly dependent on its voltage kBTpc =

eV/2√

L. Substitute the known Lorentz number andobtain that Tpc ≃ 3.2 K/mV, i.e. a voltage of 1 mVraises the temperature at the contact center by 3.2degrees. The situation seems paradoxical: the sam-ples are at a temperature of liquid helium, but if weapply to the contact a voltage of only 100 mV or0.1 V, the temperature of the contact becomes higherthan the room temperature. Is it possible and howto verify this conclusion? Yes, it is, and rather sim-ply. If we take the above-mentioned ferromagneticmetals, at the Curie temperature the transition fromferromagnetic into paramagnetic state occurs. As aresult of it, an inflection appears in the electrical re-sistivity curve, i.e., magnons and associated scatter-ing disappear. As noted above, the contact resistanceis expressed by the Maxwell formula RM = ρ/d, sowhen reaching the Curie temperature in the contact,its resistance curve will have features similar to thatof ρ(T ). These peculiarities, which must show up stilldefinitely in the second derivative of current-voltagecharacteristics d2V/dI2 ≃ dR/dV . Such features ofthe point-contact spectrum for ferromagnetic met-als were observed at voltages equal, for example forNi, to about 200 mV. This corresponds, accordingto Eq.(8), to the known Curie temperature of nickelequal to 630 K (Fig. 13). When adding berylliumatoms to Ni, Tc of this alloy decreases and the featureof d2V/dI2 also shifts to the region of lower voltagesin full agreement with the above formula. This testi-fies of the fact that, actually, the temperature reached

20

0 100 200Voltage (mV)

d2 V/d

I2 (

arb.

uni

ts)

0 3 6

c (at%)500

600

Tc

(K)

Figure 13: Second derivative for PCs of nickel (up-per curve) and its alloys with beryllium Ni1−xBex x ≃

0.01, 0.03, 0.06 (bottom curves) far above the phonon en-ergies. The observed sharp features are due to the near-contact metal transition from the ferromagnetic state tothe paramagnetic one when heating by current up to theCurie temperature. The shift of features towards smallervoltages (see inset) when adding beryllium to nickel isdue to the Curie temperature decrease in the alloy andproceeds according to Eq.(8).

at the contact center causes the metal transition fromthe magnetic to the nonmagnetic state.

The criterion of thermal regime is also the depen-dence of the above anomaly position on the bath tem-perature T0. At this time it can be observed with-out making measurements in helium. The anomalieswere found out for nickel contacts at room tempera-ture, only they were shifted towards smaller voltages.It follows from Eq. (8) that the required temperaturein the contact is reached at the lower voltages, thehigher the measurement temperature T0. Thus, allkinds of d2V/dI2 anomalies, if they are of thermalnature, will shift over the voltage scale depending onthe measurement temperature. At the same time the

spectral features of d2V/dI2 do not change their posi-tions with temperature, they only broaden accordingto Eq. (6).

One of the straightforward and easy-to-grasp ex-amples of the experimental evidence of temperatureincrease inside the contact was its critical voltage ob-servation. This critical voltage cannot be exceeded nomatter how the current rises. The nature of this phe-nomenon consists in that each metal has the so-called”softening” temperature, about one-third of the melt-ing one, at which the metal readily lends itself to plas-tic deformation. When the contact voltage reachesthe corresponding ”softening” value, under the ac-tion of pressing force the contact spot will increaseor the contact resistance will decrease, so that at thecurrent build-up a certain critical voltage cannot beexceeded. This was observed for many metals withdifferent melting temperatures in which they corre-lated with the contact critical voltages.

In the above thermal regime, the current-voltagecharacteristics nonlinearity is governed by the tem-perature dependence of resistivity, as the contact re-sistance is ρ(T )/d and its temperature is voltage-dependent. This enables obtaining the ρ(T ) behav-ior by low-temperature measurements of the con-tact resistance as a function of voltage, i.e. with-out direct measurements of ρ(T ). The advantages ofthis technique are the possibility of direct measure-ments dρ(T )/dT derivatives proportional to dR/dVor d2V/dI2, the possibility of using small quantitiesof substances, the absence of need in the systems forcontrolling and stabilizing the sample temperatures.The above technique was called modulation tempera-ture spectroscopy since when measuring the current-voltage characteristics the sample is supplied with alow modulation voltage and the contact temperaturein the thermal limit is changed (modulated) in stepwith the voltage. Note that in this case the contactsize has no upper bound and the result is independentof the constriction geometry. It is only necessary thatthe heat should be removed from the contact mainlyby electrons.

Another convincing proof of metal heating insidethe contact in the thermal regime was the detec-tion and explanation of current-voltage characteris-tics asymmetry in a number of heterocontacts de-

21

pending on the applied voltage polarity. Consider anelectric circuit comprising a heterocontact formed be-tween different metals and a potentiometer connectedto it by two wires. Let the sample temperature be T0,the instrument is kept at room temperature Troom

and the wires have the temperature gradient fromTroom to T0. This will cause the electron diffusionfrom a hotter end to a colder one and the appear-ance of thermoelectromotive force 5. However, thegradient exists only in wires which are identical andequal quantities of electrons per unit time will ar-rive at the cold ends of wires. The potentials will bethe same and the outside device will read the voltage(potential difference) equal to zero. The situationwill change if the contact region will heat up andthe temperature will become other than T0. A sim-ilar diffusion of electrons from the hot region to thecold one will start in metals making up the hetero-contact. As the metals are different, in one of themthe diffusion will proceed faster than in the other.As a result of it, the potentials at cold ends will dif-fer and the device will read some potential differencedue to thermoelectromotive force. To describe thethermoelectric phenomena quantitatively, we will in-troduce the Seebeck coefficient S. It indicates, forexample, what potential difference will exist at theends of a conductor if a temperature difference of onedegree is maintained between them. If one knowsS of each metal in a heterocontact, one can calcu-late the potential difference between the cold endsof metals making up the heterocontact. The poten-tial at one end will be S1(TPC − T0), at the otherend it will be S2(TPC − T0) and the potential dif-ference registered by the outside instrument will be(S1−S2)(TPC−T0). To be more precise, we shall notethat the above considerations are valid if the coeffi-cient S is temperature- independent. Otherwise, thesample under study should have been divided intosmall sections at which the temperature difference issufficiently small to treat S as a constant, and thento sum up the thermoemf of all these sections. Thisactually implies integration and finally one can write

5In real metals the thermoelectric phenomena are connectednot only with the electron diffusion but also with more compli-cated processes, such as phonon drag, etc., which should notbe necessarily taken into account in this consideration.

the well-known formula Vtemf =∫ TP C

T0

(S1−S2)dT todetermine the thermoemf in a contact of two metals.

What will be the effect of thermoemf when mea-suring point-contact characteristics of heterocontactsin the thermal regime? The answer is rather ob-vious: it causes the asymmetry of current-voltagecharacteristics depending on the polarity of voltageapplied to the contact. The reason of it is as fol-lows. In one polarity denoted by, say (+), the voltageapplied to the contact will coincide with Vtemf andadd to it while in the other polarity denoted by (-)the applied voltage will be subtracted. As a conse-quence, at current I through the contact, its voltagein the former polarity will be V (+) = IRPC + Vtemf ,and in the latter polarity V (−) = IRPC − Vtemf ,where RPC is the contact resistance at zero volt-age. Thus, the resistance dependence in one polarityR(+) = V (+)/I = RPC + Vtemf/I will differ from theother one R(−) = V (−)/I = RPC − Vtemf/I and willcause the asymmetry of current-voltage characteris-tics. This asymmetry can be more clearly observed onthe current-voltage characteristics derivative, for ex-ample differential resistance RD = dV/dI. Here, by

analogy, we shall write R(+)D −R

(−)D = 2dVtemf/dI =

2d/dI∫ TP C

T0

(S1 − S2)dT ≃ 2RPCd/dV∫ TP C

T0

(S1 −S2)dT . As mentioned above, in the thermal regimethere exists a linear relation kBT = eV/2L1/2 be-tween the contact temperature (at T0 ≪ TPC) andvoltage. That is why the above expression can besimplified after substituting the differentiation vari-able V by T as

R(+)D − R

(−)D

RPC=

e

kB

√L

[S1(TPC) − S2(TPC)] (9)

Thus, the asymmetry of differential resistance ofheterocontacts in the thermal regime is proportionalto the Seebeck coefficients difference of the contactingmetals. A more rigorous theory which appeared lateron confirmed this qualitative analysis.

The current-voltage characteristics asymmetry ofheterocontacts is especially clearly defined in sub-stances with short mfp and large Seebeck coeffi-cients. These systems include heavy fermion com-pounds, Kondo lattices, intermediate-valence sys-tems. The measurements of current-voltage charac-

22

teristics asymmetry in such conductors may give in-formation of thermoemf behavior therein. For thispurpose one should use the second electrode of nor-mal metal whose Seebeck coefficients are considerablysmaller than in the above compounds, i.e. S1 ≫ S2.Hence, the RD asymmetry will be directly propor-tional to the Seebeck coefficient (see Eq. (9)) in thesubstance under study. This is of not only the aca-demic interest, for it can be used for analyzing theS behavior in the sample, especially when studyingsubstances available in small quantities for which thetraditional methods of determining the Seebeck coef-ficient are hard to apply.

We have considered the cases of contact heatingunder the action of transport current. Naturally, thisdoes not exhaust all thermal effects which are inter-esting when investigating PCs. Another importanttrend is the study of heat conductivity in shorts wheninstead of (or together with) the potential difference atemperature gradient is applied to the contact. The-oretical treatment of heat transfer through PC in theballistic regime brought about a rather unexpectedresult. The second derivative of the heat flux versusvoltage proved to be directly related to the electron-phonon interaction function. It is not that simple tomeasure this derivative and it has not yet been doneexperimentally.

However, some studies confirm the specific featuresof heat conductivity in PCs. One of the contactsmaking electrodes was attached to a heat exchangerwhile the other was held in vacuum. Both electrodescarried miniature pickups to check the electrode tem-peratures depending on the contact voltage. Thetemperature was found to rise slightly with voltagein each electrodes, this temperature being differentfor various polarities. When the temperature differ-ence versus voltage curve was plotted, it was closein shape to the electron-phonon interaction functionof the contacting metals. Therefore, the above ex-periments evidence the electron-phonon interactioneffect on the heat conductivity of PCs and confirms,though not directly, some theoretical conclusions. Itshould be noted that the measured temperature dif-ference has a finer structure that can be related to thepeculiarities of phonon scattering and enables theirinvestigation.

Now let us see what happens if a contact is af-fected only by the temperature gradient. Let oneelectrode be held at helium temperature T0 and theother at T , the inequality T0 < T being held. Inthis case the heat will be transferred from the heatedpart through a constriction to the colder part. Inother words, phonons will propagate from the hotplace where their number is large to the cold placewhere their number is smaller. The electron diffusionwill proceed in the same direction, thus causing theemf. However, when the contact is formed by elec-trodes of the same metal, as stated previously, thepotentiometer connected to the ends of electrodesshould read zero voltage. Nevertheless, an experi-ment made according to this scheme demonstratedthat this not quite the case. Even if the contact ishomogeneous and the electrodes are maintained atdifferent temperatures, a potential difference will de-velop between their ends. This occurs because in thiscase there is a circuit of homogeneous in compositionbut strongly geometrically inhomogeneous materialsdue to PC where kinetic processes are specific. Thus,for example, the flow of phonons directed from thehot part to the cold one drags” the electrons. Thiscauses thermal emf of the so-called phonon drag. It is”hard” for the phonons to drag the electrons throughthe bottleneck of PC and the drag thermoelectro-motive force is suppressed 6. In the case concernedtotal thermoemf will be written as a difference be-tween the bulk and contact contribution. The lat-ter is strongly suppressed in the contact, so only thebulk contribution remains. Hence, the potentiometerconnected to the sample leads will read the voltageto a higher extent governed by the drag thermal emf.This offers probably a unique possibility of measuringthe absolute coefficient of thermal emf in a material-homogeneous electric circuit.

6As shown by theoretical treatment, the diffusion part ofthermal emf will be also suppressed in PC, but less than asso-ciated with drag.

23

14 Electrical fluctuations in

PCs

Noises present an important problem in science andtechnology since they dictate the sensitivity of exist-ing radio-electronic devices which, in its turn, limitsthe magnitude and accuracy of measurements, thequality of processing various signals. The investiga-tion of sources and nature of noise makes it possiblenot only to measure it, but also to suppress it, thusincreasing the signal-to-noise ratio for particular mea-surements or devices.

What is noise? It is an unwanted voltage or cur-rent of fluctuation nature generated by the electronicdevice or some circuit component. If this voltage isapplied to an audio-frequency amplifier, this voltagegives rise to a hiss in the loudspeaker. Hence thename ”noise” which is applied to such fluctuationprocesses, though in most cases we do not actuallyhear it.

What are the current or voltage fluctuations dueto? First of all, the current carriers in a conductorare in constant motion. As a result of it, fluctuationemf appears at the conductor ends. At each par-ticular moment of time this emf will have a definitevalue, but averaged over a longer time it will be equalto zero. This noise was called thermal or Nyquistnoise, while the average square emf 〈V 2

n 〉 arising atthe resistance R and temperature T in the frequencyrange ∆f can be written as 4kBTR∆f . The othertype of noise is connected with the discreteness ofcurrent carrier charge. Thus, for example, the eventsof electron flying out of the cathode (emitter) of elec-tronic tube (transistor) form a sequence of indepen-dent events, occur at random times and lead to theso-called shot noise. At this time 〈V 2

n 〉 ∝ eIR/C, ebeing the electron charge, C the system capacity, i.e.the shot noise is proportional to the current.

In an impurity-free semiconductor electrons andholes can appear and disappear at random due totheir generation and recombination. This also causesfluctuation emf in the sample called generation-recombination noise.

One more source of noise attracted attention firstin the electron vacuum tubes. It was attributed to

the average emission rate alteration causing micro-scopic current fluctuations. The emission rate al-teration can be treated as some effect of collectiveinteraction of electrons, as a result of which such os-cillations or flicker of current occur. This noise isknown as flicker noise. It is also characterized by thefact that the spectral power of flicker noise densitySV = 〈V 2

n 〉/∆f is frequency dependent close to the1/f law, that is why it is also called 1/f noise. Thenature of this noise is not quite clear to date. Dif-ferent models are suggested, particularly recently theflicker noise in solids has been explained in terms oftwo-level system which serve as elementary disordercarriers in the crystal lattice.

The noise in electric contacts was studied ratherin detail. In 1969 Hooge obtained an empirical for-mula for the spectral density of contact noise at thefrequency f : SV = αV 2/Nf where α is a constant,N is the total number of charge carriers in the con-tact. The latter value can be found from the formulaN = nv where v is the constriction volume, n is thecarriers concentration. v can be easily estimate fromthe resistance v ≃ d3 = (ρ/R)3 using the Maxwellformula mentioned above. As a result we will obtaina more convenient expression for SV , viz.

SV ≃ R3V 2

nρ3f

.The spectral densities of noise have been also mea-

sured for PCs. Thus, for the thermal regime of cur-rent flow it was show that SV ∼ V t where t ≃ 2 atcontact voltages up to a few millivolts, whereas athigher voltages SV exponentially increases. The lat-ter fact is attributed to highly nonequilibrium stateof electrons and phonons in the vicinity of the contactwhen the characteristic phonon energies are reached.The SV dependence on the sample resistance also ap-peared close to R3. Consequently, the noise behav-ior in PCs under thermal regime has no peculiaritiescompared to those already known.

However, the study of SV dependence for con-tacts in the ballistic regime yielded some new un-usual information. It turned out that in PC withle, li ≫ d the spectral density of noise does not in-crease monotonously but consists of a number of ex-

24

Figure 14: Above: point-contact spectrum of a tin con-tact recorded at the temperature 4.2 K. Below: ”noise”spectrum for the given contact. In addition to the mono-tonic growth of noise, the curve display a number ofmaxima and minima attributed to the peculiarities ofelectron-phonon interaction in the contact.

trema, both maxima and minima (Fig. 14). In thiscase the minima positions in 〈V 2

n 〉 at eV > εD corre-sponded to the energy of multiphonon scattering ofelectrons. It was suggested that such multiphonongeneration in the constriction corresponds to the es-tablishment of some coherent (ordered) regime ofphonon emission at the realization of which the noisedecreases. For the energies eV < εD the 〈V 2

n 〉 also hassome minima attributed to the one-phonon normalelectron scattering in the coherent regime. It turnedout that the positions of minima correspond to theenergy at which the phonon dispersion curves crossthe Brillouin zone boundary, i.e. the wave vector ofphonons is maximum and their velocity is minimum.The latter circumstance promotes the accumulationof phonons in the vicinity of contact and the estab-lishment of coherent regime of emission. On the other

hand, the positions of maxima in 〈V 2n 〉 at eV < εD

coincide with the minimum energy of phonons forwhich the scattering with Umklapp is possible. Suchscattering correspond to the Bragg reflection of theelectron wave from the atomic plane in the lattice.At this time the electron momentum changes by theinverse lattice vector 2πh/a, i.e. equal to the doublemaximum momentum of phonon. The above pro-cesses are non-correlated because not only phononsbut the lattice as a whole participate in the elec-tron scattering events and the noise increases. Theobserved reproducibility of extrema position on thecurve 〈V 2

n 〉 for contacts of different resistance andtheir relation to the characteristic phonon energiesmore precisely to the singular points of the phonondispersion curves, makes it possible to speak aboutthe noise spectroscopy of separate groups of phononsin metals.

15 PCs in a high-frequency

electromagnetic field

For a considerably long time PCs have been used inmicrowave engineering as detectors, mixers and har-monic signal sources. Such contacts are formed bymeans of thin wire pressed against a smooth metalsurface. The wire serves as a kind of antenna throughwhich the high-frequency current penetrates in thebulk of the contact. As is already known, the current-voltage characteristics of PCs deviate from the Ohmlaw both in the ballistic and thermal regimes. This iswhy the contact is a nonlinear element and possessesthe above detecting and other properties. However,the physics of processes proceeding in the contact un-der the action of microwave irradiation is of consid-erable interest.

The first thing done by researchers was to analyzethe form of current-voltage characteristics derivativesat high frequencies. The experimental technique inthis case differs a little from that described in Chap-ter 8 and used for audio frequencies. As is seenfrom Eq. (7), in addition to the signal proportionalto d2V/dI2 at the frequency 2ω, there is a dc sig-nal of the second derivative. To detect this constant

25

component of d2V/dI2, the microwave signal is mod-ulated by means of some chopper. For example, use isoften made of a rotating disc with holes to pass radi-ation. Then, using a circuit similar to that describedin Chapter 8, the signal proportional to d2V/dI2 isseparated at the chopping frequency.

Naturally, the PC resistance is much lower thenthat of the surrounding medium, so when microwavefield is applied, a high-frequency current will ap-pear in the constriction and will play the role ofmodulation signal. Therefore, for a contact in theballistic regime one can measure the second deriva-tive proportional to the electron-phonon interactionfunction, but only at high frequencies. Such ex-periments were carried out and showed that thed2V/dI2 obtained in this way was identical to thesecond derivative recorded at low frequency. Dis-tinctions appeared when the frequency became ashigh as giga- and tera hertz or the period of oscil-lation reached 10−9 − 10−12s. Here d2V/dI2 beganto change: first the background level decreased, then,at still higher frequencies, the spectral features beganto broaden. These change in d2V/dI2 curve can beunderstood if we compare the microwave signal pe-riod with times of quasi-particles relaxation in thecontact. Thus, the electron-phonon scattering oc-curs within 10−13 to 10−14 s which can be readilyobtained dividing the mfp the Fermi velocity of elec-tron. The phonon-electron scattering processes areas many times slower as sound (phonon) velocity issmaller than the Fermi one and are characterized bytimes 10−9 to 10−10 s. They also affect the point-contact spectrum causing the presence of phononcomponent background therein. Consequently, whenthe irradiation frequency approaches 109 Hz that iscomparable with the frequency of phonon-electron re-laxation, the phonon reabsorption processes will slowdown because they will not be able to trace rapidoscillations of the electron gas under the action ofemf induced by external radiation. As a result thebackground level in the d2V/dI2 curve recorded atmicrowave frequency should decrease. With furtherfrequency increase to the level comparable with thefrequency of electron-phonon interaction (in this casethe energy quantum of radiation is comparable to thephonon energies) the processes in the contact should

be considered in terms of corpuscular theory as theabsorbtion and emission of phonons under the actionof radiation quanta. As a result of it further increaseof frequency will cause smearing of d2V/dI2 phononspectral features and the appearance of satellites athigher frequencies.

When passing to the thermal regime, the nonlin-earity of point-contact conduction is connected withthe effect of heating. In this case the characteristictime is determined by thermal relaxation processes inthe contact. It depends on the contact size d, specificheat capacity c, heat conduction of material λ andcan be evaluated by the formula τ = cd2/λ. At char-acteristic size of contact 10−8 m, heat capacity andheat conductivity of metals the value of τ is about10−9 s. Therefore, in the frequency range 109 Hzand higher the d2V/dI2 curve should change its formas compared with that obtained at low frequencies.This is caused by the fact that at the frequenciesabove those of thermal relaxation the contact temper-ature no longer traces the modulating high-frequencycurrent. As a result the possibilities of temperaturesmodulation spectroscopy described in Chapter 13 arelimited by the above frequency.

Thus, the study of contact conductivity at highfrequencies makes it possible to clear out the natureof various non-stationary processes in metals, to de-termine the time of phonon-electron collision, to elu-cidate the quantum limit effect in detecting for theballistic regime. In the thermal regime it enable theinvestigation of thermal relaxation processes in small-size conductor.

16 Study of PCs on semimetals

and semiconductors

The distinctive features of substances relating tosemimetal and semiconductor is, first of all, lowconcentration of charge carriers as compared withcommon metal. Thus, in typical semimetals, bis-muth, antimony and arsenic, it changes from 3×1017

(Bi) to 2 × 1020 (As) per 1 cm3, whereas in met-als it is 1022 − 1023. In semiconductors these valuecan be still lower by several orders of magnitudes.

26

Accordingly, the Fermi energy decreases from sev-eral electron-volts in metals to millielectron-volts insemimetals. Secondly, the effective mass of chargecarriers in semimetals, due to the peculiarities of theirband structure, can be more then an order of magni-tude lower then its value in common conductors. Thelatter, together with low Fermi energy, causes the in-crease of the de Broglie wavelength in semimetals upto tens of nanometers, i.e. it becomes comparablewith the contact dimensions. Thirdly, small effectivemass leads to the fact that the Larmor radius of elec-tron trajectory decreases to hundreds of nanometersalready in an easily attainable magnetic field about1 Tesla. Thus, an interesting situation is created insemimetals when all the above lengths become com-parable with the constriction dimension. This leadsto the influence of quantum interference effects onthe conductivity of contact since due to a large wave-length the quasi-particles can no longer be consideredas point ones and their wave properties should be alsotaken into account. Besides, the magnetic field effecton the charge carrier trajectories in the vicinity ofthe contact should be appreciable, which will also af-fect the characteristics being measured. Thus, forexample, under the field effect we observed transfor-mation of the electron-phonon interaction spectrumin antimony: the change of intensity, half-width andenergy position of maxima depending on the relationbetween the Larmor radius of carriers and their mfpand contact size.

On the other hand, a number of antimony contactdisplayed the change of spectrum sign, i.e., d2V/dI2

became negative but the extrema due to the electron-phonon interaction were retained. Such contacts weredistinguished by small elastic mfp which was of thesame order as the de Broglie wavelength. As a result,due to the quantum interference of electron waves,the current carriers are localized and the system con-ductivity decreases. The electron-phonon interactionprocesses destroy localization and increase conduc-tivity which corresponds to the decrease of differen-tial resistance and, consequently, to the negative signof d2V/dI2. This mechanism is most effective at theenergies with many phonons, i.e. at the phonon spec-trum maxima. Finally, under the localization regimefor the above contacts the recorded spectrum shape

will be close to the common one as if turned upsidedown. As stated above, this interesting effect was ob-served for contacts made of antimony which provideto be sufficiently convenient and workable materialfor making shorts.

Let us dwell on the point-contact study of semi-conductors where so far only the first steps are made.Note that this problem is much greater tackled the-oretically than experimentally where the investiga-tion can be counted on the fingers of one hand. Thematter is that the above technology of making PCsbetween metals is not well suitable for semiconduc-tors. This is due to the much stronger sensitivity ofthe surface properties of semiconductors to deforma-tion, pressure, imperfect structure, impurities whichare always present to some extent when the contactis formed mechanically. Even a perfect surface is a”defect” relative to the bulk material because theboundary atoms on the one hand interact with thelike ones, and on the other hand interact with theatoms of the surrounding medium. In this way thesurface state appear on the atomically pure surfaceof the semiconductor whose energy levels lie in theforbidden zone (Tamm or Schottky levels). In realcrystals, along with the above levels, the additionalones appear due to the impurities, defects, adsorbedatoms, etc. The density of such levels depends onsurface boundary and can vary in wide range. Thepresence of surface energy levels result in the forma-tion of near-surface layer of the space charge whosethickness is determined by the screening distance rS7 depending on the carriers concentration. At thistime layer with the main carriers concentration otherthan in the bulk appears on the surface. As a resultof it, when mechanically contacting two semiconduc-tors, the point-contact properties will be to large ex-tend determined by the surface layer properties anddiffer from the bulk ones. To avoid it, the latticesshould perfectly coincide during the mechanical con-tact which hardly probable. For this reason the char-

7In conductor, the electron gas screens separate chargeswhose radius of action is in this case limited by the distancerS depending on electron concentration as rS ∼ n−1/2. Inmetals the screening distance is only a few decimal fractionsof A, while in semiconductors it can reach tens and hundredsof A.

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acteristics of semiconductor PCs strongly vary fromcontact to contact and are poorly reproducible, thusimpeding the studies. Nevertheless, attempts to ad-vance in this direction are continued. Thus, techniqueof creating micro-constrictions has been implementedin semiconductor heterocontacts. Two-dimensionalmicro-constrictions less then one micron in size havebeen obtained by electron-beam lithography.

17 Spectroscopy of supercon-

ductor

So far we considered the current-voltage character-istics of contacts formed by normal metals. How-ever, at low temperatures many metals and alloysbecome superconducting. It should be noted thatresearchers began to study contacts between two su-perconductors long before the advent of point-contactspectroscopy. To tell the truth, the scientists weremore interested in the properties of superconductingcondensate formed by Cooper pairs. To break pair,it should be given some energy equal to or greaterthen the energy gap 8 in the electron density ofstates equal not more than several millielectron-volts.The current-voltage characteristics of superconduct-ing contacts first of all reveal the features connectedwith the presence of the energy gap. Such featureswere studied by many experiments in all simple su-perconducting metals.

A question arises whether the phonon featureswill show up in the second derivatives of current-voltage characteristics of superconducting contacts,i.e. whether the point-contact spectroscopy of super-conducting state is possible. The answer proved tobe positive. For simple superconducting metals (Pb,Sn) it was demonstrated that the point-contact spec-tra taken bellow the temperature of superconducting

8In common superconductors the electrons interact viaphonon and form the so-called Cooper pairs which ensure anon-dissipative charge transfer. As the pairs are Bose-particleswith the spin equal to zero, they are all ”condensed” at Fermilevel. The energy required for breaking a pair is equal to theenergy gap arising in the electron density of states in super-conductor at the Fermi level. The energy gap value, as well asthose of critical temperature and other parameters, is one ofthe most important superconductor characteristics.

transition Tc displayed the features associated withthe electron-phonon interaction. At this time theform of the spectrum practically coincide with simi-lar dependence for the same contacts in the normalstate. Only at the energies low compered with theDebye one there appear sharp features due to the en-ergy gap manifestation. But they do not greatly dis-tort the electron-phonon interaction spectrum sincethe phonon are located at higher energies.

At the first sight it seems that point-contact spec-troscopy in the superconducting state yield no newinformation. However, not all the metals, especiallyalloys and compounds with high critical parame-ters: superconducting transition temperature, criti-cal magnetic field, can be studied at low tempera-tures in the normal state. Therefore, the principalpossibility of the point-contact spectroscopy appli-cation for investigation the superconductors consid-erably extends its scope of use to the study of thehigh-temperature superconductors.

It is noteworthy that the point-contact spec-troscopy of superconductors has offered new oppor-tunities. It tuned out that in some metals and alloyswhose point-contact spectra in the normal state donot display any spectral features due to the shortelectron mfp, in the superconducting state there ap-pear peaks arranged in the region of characteristicsphonon frequencies. This was found out experimen-tally when measuring contacts of Nb, Ta, Nb3Sn,NbSe2. The nature of these features is somewhatdifferent from that in the ballistic contact. Theyare connected with the influence of nonequilibriumphonons generated by electrons on the supercon-ducting properties of metal near the constriction.The phonons with small group velocities more slowlyemerge from the contact vicinity, cause local over-heating and decrease of the energy gap. Therefore,at these energies features will appear in the current-voltage characteristics derivative. This makes itpossible to carry out the spectroscopy of ”slow”phonons which correspond to the flattening of dis-persion curves where dε/dq tends to zero (see Fig.2a).

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18 New trends of research

In the preceding chapters we discussed the possi-bilities of point-contact spectroscopy when studyingmainly simple metals and some alloys. It was shownthat this technique makes it possible to obtain de-tailed (spectroscopic) information on the mechanismsof electron scattering in conductor. At the same timethe recent advances of solid-state physics are asso-ciated with the synthesis and study of new classesof compounds. These include the heave-fermion andmixed-valence systems, so-called Kondo lattices, newhigh-temperature superconductors and many others.Naturally, the most promising trends in point-contactspectroscopy are also connected with their study.

The substances in question are noted for the factthat the conventional band scheme proceeding fromthe concept of free electrons in the conduction bandand the localized nearly atomic states of lower-lyingshells is not suitable for their description. In thesesubstances the inner shells near the Fermi level losetheir stability and their electrons acquire a partiallyband character. As a result the electron density ofstates strongly increases and can have a sharp max-imum near the Fermi level. In a number of case thepresence of such narrow maximum in Kondo latticesis attributed of many-particle Abrikosov-Suhl reso-nance caused by the interaction between localized f-or d-electrons and conducting electrons. A high in-crease of the electron density of states correspondsto the appearance of electrons with a larger effec-tive mass. In simple terms, these conduction elec-trons are not fully ”untied” from the atoms and theirfield-effected motion is more hindered. The presenceof such electrons stipulates the unusual properties ofthese materials.

Theoretical treatment of the kinetic properties ofsuch substances is based on taking into account theirelectronic spectrum peculiarities near the Fermi level.It is impeded by the absence of direct experimentaldata on the electron density of states in a narrow layeron the Fermi surface. The scientists were tempted bythe idea to use PC for recording the energy spectranot only of phonons but also of electrons.

A number of research groups abroad are investigat-ing the compounds with heavy fermions and interme-

diate valence in heterocontact with normal metal. Itwas discovered that for the substances with a maxi-mum electron density of states N(ε) near the Fermienergy εF the point-contact differential resistanceRD(V ) curve has deep minimum at V = 0. At thesame time, a sharp maximum of RD at V = 0 appearin the substances with a low density of states or withgap, i.e. N(ε) = 0 at εF −∆ < ε < εF +∆ where 2∆is the gap width. It also appeared that the widthsof RD(V ) maxima or minima is close to the assumedwidths of N(ε) features. This suggested the possi-bility of registering the N(ε) features near the Fermilevel in such substances using the point-contact spec-troscopy. However, not everything is so simple inthis problem. Firstly, the electron mfp in the sys-tems under consideration are small enough and oneshould take into account the thermal effects occur-ring in PCs. Secondly, there is not microscopic the-ory that allows for the effect of electronic spectrumpeculiarities on the point-contact kinetic phenomena.The experiments carried out by other groups showed,however, that the thermal regime is realized in thecontacts of such materials and that the main non-linearity of the current-voltage characteristics is de-termined by the temperature-depended part of sub-stance resistance. Thus, for example, the RD(V ) be-havior for contacts made of number of heavy-fermioncompounds, such us CeCu2Si2, UPt3, UBe13 and oth-ers, precisely duplicates the ρ(T ) dependence in bulkmaterial. Thus, the relation of V and T in a con-tact, typical on the thermal regime, is evident. Be-sides, it was shown that one should be very carefulwhen treating the result obtained on heterocontacts 9

between a normal metal and specific substances, forhere one should also successively take into accountthe thermoelectrical effects considered in Chapter 13.

There is probably only one convincing exampleof how the authors managed to obtain well-definedspectral features in the point-contact spectra ofintermediate-valence compound CeNi5. Such sub-stances contain rare-earth atoms with partially filled4f shell and +3 valence. Since the ion states with

9This experimental geometry is often used since the abovecompounds are frequently rather small samples and cannot beadequately shaped, so it is simple to make a needle of anothermaterial.

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filled or empty shell are most stable, the state Ce3+

will compete with the state Ce4+ when the f-shell incerium is empty. Thus, the Ce3+ ions in the com-pound CeNi5 can also exist in the Ce4+ state, givingout the excess electron. As a result, there is con-stant competition between these two state, Ce3+ andCe4+, so the valence of Ce ions is between 3 and 4is called intermediate. There exists a model accord-ing to which the energy levels of the Ce3+ and Ce4+

states are arranged near the Fermi level but on theopposite sides, thus increasing the density of stateat the corresponding energies. The electrons takepart in the transitions between these states or lev-els with difference valence. These transitions affectthe conductivity of substance and should be observedon the point-contact spectra. Actually, among withthe phonon features, the d2V/dI2 curves of CeNi5contacts reveal small maxima in the low energy re-gion. These maxima can be related with scattering bydifferent-valence configuration of the Ce ion. In thiscase the d2V/dI2 curve of CeNi5-normal metal het-erocontact displays a large asymmetry of features inquestion depending on the voltage polarity. In termsof the model under consideration, this is attributedto the resonance scattering by different-valence stateof the Ce ion with allowance for the specific form ofthe electron distribution function in PC (see Fig. 5b).

On the other hand, there is an alternative approachto the explanation of mentioned features observed inthe PC spectra of CeNi5. This approach is connectedwith taking into account strong spin correlations inthis compound. Their important role was indicatedby the low temperature behavior of CeNi5 resistanceproportional to T 2 attributed to the scattering byspin fluctuation or paramagnons. The latter shouldlead to the linear dependence of the initial d2V/dI2

section that was observed on all curves in the absenceof negative ”zero-bias” anomaly, which, as noted inChapter 7, indicates low quality of contacts.

As to the d2V/dI2 asymmetry in heterocontactsbased on CeNi5, the detailed study on single crys-tals has demonstrated its greater relation to the ther-moelectric effects. This evidence by the fact thatthe asymmetry corresponds to the thermoemf sign inCeNi5 and its value for the two principal directionscorrelates with that of the Seebeck coefficient. Thus,

0 20 40 60 80 100Voltage (mV)

0

d2 V/d

I2 (

arb.

uni

ts)

F (

ω)

(arb

. uni

ts)

La1.8Sr0.2CaCuO

Figure 15: Point-contact spectrum of high-temperaturesuperconductor La1.8Sr0.2CuO4 (solid curve). A numberof narrow spectral lines and the spectrum boundary near80 mV are well defined. The initial spectral part up to 10mV contain powerful bursts due to the manifestation ofthe superconductor energy gap not shown in the figure.Neutron phonon density of states (symbos) is shown forcomparison.

further investigation of CeNi5 by means of PC willenable better understanding of these interesting sys-tems.

Nevertheless, it should be noted that the dis-cussed possibility of refining the band structure of thecompounds under consideration by means of point-contact spectroscopy is of interest due to a high reso-lution about 1 meV. The traditional X-ray or photo-electron techniques used for the study of band struc-ture operate in significant larger energy scales withresolution not higher than hundreds of millielectron-volts.

The point-contact technique did not stand ofthe events associated with the discovery of high-temperature superconductors. One of the first com-pounds of this kind, La1.8Sr0.2CuO4 was studiedby means of point-contact spectroscopy. It wasrather simple to obtain the point-contact spectrumof phonons (Fig. 15). It was found that it extends upto the energies of about 80 meV and comprises six

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sharp peaks corresponding to the phonons with thesmall group velocities. The electron-phonon interac-tion estimated by the relative differential resistanceincrease of PC when the energy rises to εD has shownthis interaction to be almost the order of magnitudestronger than in common superconductors. Conse-quently, the superconductivity in the compound inquestion is more than likely due to the electron-phonon interaction. On the other hand, the analy-sis of current-voltage characteristics of contact fromsuch superconductors below the critical temperatureenables the determination of their energy gap. Theinformation obtained thereat relates to the substanceregion with the size about the contact diameter di-rectly adjacent to the constriction. This makes itpossible to investigate the properties of separate su-perconducting clusters whose sizes measure only hun-dreds A. The knowledge of the contact dimensions en-able us to estimate the critical current densities. Itturned out that this density reaches 108A/cm2 in thesuperconducting clusters investigated by the point-contact spectroscopy, whereas in the bulk sample thecritical current density does not exceed 103A/cm2.Apparently, the latter fact points to the presence ofweak contacts between superconducting granules insuch systems. Therefore, PC can be also used as anefficient probe which enables the study of cluster andother polyphase compounds.

It is necessary to note the even growing interestto the study of heterocontacts including those be-tween pure metals. It is clear that the point-contactspectrum of heterocontact between different metalsshould contain the peculiarities due to one and an-other electrodes, i.e. as a result the d2V/dI2 de-pendence can be presented as a sum of spectrumfrom each materials. Such experiments were madesoon after obtaining first point-contact spectra. Theresearchers observed the change the partial contri-bution into the spectrum from one or another sub-stance for various contacts with different resistances.This was attributed to the fact that the point-contactvicinity can be at random mostly filled with one oranother metal. However, theoretical works in thisfield have demonstrated that the heterocontacts havetheir own physics. Firstly the intensities of one oranother metal spectra in the heterocontacts are in-

versely proportional to the Fermi velocity, i.e. inother words, to the time of electrons transit or inter-action in the contact. Secondly, the effects of electrontrajectories reflection or refraction appear in the het-erocontact (due to the different electron momenta) asis the case when the light passes through the bound-ary between two media. As a result, as shown bythe estimates, it can lead to a considerable change inthe metal spectrum both in intensity and shape. Thedetailed comparison of theoretical calculations withexperiments on heterocontacts yields information onthe electron scattering at the metals interface.

Besides, the heterocontacts develop a whole num-ber of other interesting effects connected with theobservation of spectrum asymmetry 10 depending onthe applied voltage polarity. One of such effects de-tected recently is related to the electron drag in thecontact by the nonequilibrium phonons which enablethe study of their relaxation processes with the aidof PCs.

19 Conclusion

(Note, it was written 15 years ago.)The above examples explain the achievements and

prospects of the development of point-contact spec-troscopy for the study of the electron and phononsystem kinetics which determines the main electricalproperties of conductors. This technique is becominga powerful and efficient tool for the study of highlynonequilibrium phenomena in conductors, especiallythose with limited space dimensions. The latter isof interest for modern microelectronics where the di-mensions of separate components are already belowthe micron range.

Note that the point-contact measurements are alsohelpful in all the case when one deals with constric-tions of different types. Thus, they enable obtaininginformation on the state of metal in the regions ofcurrent concentrations, mean free paths, contact di-mensions, etc., i.e. to have some ”rated” characteris-tics of contact, the knowledge of what researcher orengineer is dealing with.

10In this case the asymmetry is not connected with the ther-mal emf since the contacts are in the ballistic regime.

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The best evidence of the efficiency and importanceof work carried out in this field is the wide geogra-phy of point-contact spectroscopy research. Thus,in the USSR both the experimental and theoreticalstudies are extensively carried out at the Institutefor Low Temperature Physics and Engineering of theUkrainian Academy of Sciences, which takes the lead-ing place in the world in this respect. This is thebirthplace of the point-contact technique where fur-ther fruitful research is under way.

Experimental groups of ”point-contact” physi-cists successfully work in the Kurchatov Instituteof Atomic Energy and the Institute of Solid-StatePhysics of USSR Academy of Sciences. Their princi-pal interests is connected with transition metals andalloys on their base. There are also groups of thetheorists dealing with the calculation of the point-contact spectra.

A number of important results have been obtainedby Dutch scientists from the University of Nijmegenwho, in particular, proposed the ”needle-anvil” tech-nique. Now this group has moved to Grenoble, tothe Laboratory of High Magnetic fields in the Insti-tute of Solid-State Physics of the Max-Planck Soci-ety. The principal attention here is paid to the studyof PCs on the base of compounds with rare-earthions and their behavior in high magnetic fields. Theresearchers of the University of Cologne (Germany)use PCs for the investigation of heavy-fermion com-pounds, Kondo lattices and mixed-valence systems.The group from the Higher Technical School (ETH)in Zurich (Swiss) works in the same direction.

The physicists in the USA and Canada are activelyworking on the theory of point-contact spectroscopyand calculation of point-contact spectra of simplemetals. Experimental study of PCs from some semi-conductors and pure rare-earth metals are carried outin the Cavendish laboratory in Cambridge (England).At the Institute of Experimental Physics in Kosice(Czecho-Slovakia) scientists are successfully develop-ing various trends in point-contact spectroscopy inclose collaboration with the researchers from the In-stitute for Low Temperature Physics and Engineer-ing of the UkrSSR Academy of Sciences. Recently,the scientists of Japan joined in the point-contactstudies. The group in Tokyo and Sendai are engaged

in the point-contact spectroscopy of Kondo lattices.There are number of other places where scientists arenot only interested in these technique but are tryingto use it. Within the last 15 years of its develop-ment, the point-contact spectroscopy, together withthe other spectroscopic techniques, has become in-separable part of physical research. More than twohundreds papers on this subject have been alreadypublished in the leading physical journals all over theworld. The papers devoted to point-contact spec-troscopy are presented at practically all conferencesdealing with the electronic properties of solid. It ishardly too much to say that the point-contact spec-troscopy method has been adopted by physicists allover the world.

Literature

Holm R., Electric contacts handbook. M., ForeignLiterature Publishers, 1961 (in Russian).

Kaganov M.I., Electrons, phonons, magnons. M.,Nauka, 1979 (in Russian).

Khotkevich A. V. and Yanson I. K., Atlas of Point

Contact Spectra of Electron-Phonon Interaction in

Metals, Kluwer Academic Publisher, Boston, 1995.Kittel C., Introduction to Solid State Physics by

John Wiley & Sons, Inc., 1971.Verkin B.I., Point-contact spectroscopy of metals

and alloys, Priroda, No. 10, 1983 (in Russian).

Topical reviews (added)

Duif A., Jansen A. G. M. and Wyder P., J. Phys.:Condens. Matter 1 3157 (1989).

Jansen A. G. M., van Gelder A. P. and Wyder P.,J. Phys. C 13 6073 (1980).

Naidyuk Yu. G. and Yanson I. K., J. Phys.: Con-dens. Matter 10 8905 (1998).

Yanson I. K., Sov. J. Low Temp. Phys. 9 343(1983).

Yanson I. K., Sov. J. Low Temp. Phys. 17 143(1991).

Yanson I. K. and Shklyarevskii O. I., Sov. J. LowTemp. Phys. 12 509 (1986).

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Figure 16: Original brochure cover.

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