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Transport Through Quantum Dots

Lecture Notes DPGSchool Electronic NanostructuresOct 812, 2001, Bad Honnef (Germany)

Dr. Tobias Brandes

February 16, 2007

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CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Quantum Dots: Basic Classifications . . . . . . . . . . . . . . . 1

1.1.1 Semiconductor Based Systems . . . . . . . . . . . . . . . 11.1.2 Metallic Systems . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Transport vs. Optics . . . . . . . . . . . . . . . . . . . . 21.1.4 Nonlinear Transport vs. Linear Transport . . . . . . . . 21.1.5 Single Dots vs. Coupled Dots/Dots in more complicated

structures . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Basic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Orthodox and Classics: Coulomb Blockade (CB), CBOscillations and CB staircase . . . . . . . . . . . . . . . 2

1.2.2 The Dot as a Many-Body System (Artificial Atom): LevelStructure, Selection Rules, Spin Effects . . . . . . . . . . 2

1.2.3 Kondo Effect: Many Body Theory at its best. . . . . . . 21.2.4 The Quantum World enters: Coherent Effects in Cou-

pled Dots, Artificial Molecules . . . . . . . . . . . . . . . 21.2.5 Best Regards from Quantum Optics: Spontaneous Emis-

sion of Phonons . . . . . . . . . . . . . . . . . . . . . . . 21.2.6 Not done here . . . . . . . . . . . . . . . . . . . . . . . . 2

2. The Single Electron Transistor . . . . . . . . . . . . . . . . . . . . . . . 32.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Charging Energy . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Linear Transport VL VR 0: Coulomb Blockade Os-cillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.3 Linear Transport: Conductance Formula . . . . . . . . . 52.1.4 Experiments: Coulomb Blockade Oscillations . . . . . . . 52.1.5 NonLinear Transport: Stability Diagram. Coulomb

Blockade Staircase . . . . . . . . . . . . . . . . . . . . . 6

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Contents ii

2.2 Nonlinear transport theory: Anderson Impurity Model . . . . . 6

2.2.1 Rate Equations . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Linear Transport Regime: Coulomb blockade oscilla-

tions and addition energies . . . . . . . . . . . . . . . . . 72.2.3 Linear Transport Regime: Peak Shape, Limit b, U 82.2.4 Nonlinear transport regime . . . . . . . . . . . . . . . . 92.2.5 Comparison to Experiments: Nonlinear Transport . . . 10

2.3 Symmetries, Many Body Effects, Spin . . . . . . . . . . . . . . . 112.3.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.2 Exact ManyBody Calculations, Spin Blockade . . . . . 12

3. The Kondo Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1 Kondo Effect in Metals. Impurity Model . . . . . . . . . . . . . 133.2 Kondo Effect in Dots . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Theory Tool: Greens Functions . . . . . . . . . . . . . . . . . 143.4 Unitary Limit at T = 0 : . . . . . . . . . . . . . . . . . . . . . . 153.5 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.6 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4. Coupled Quantum Dots . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.1 Superposition . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.2 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Theory Tool: Master Equations . . . . . . . . . . . . . . . . . . 194.3 Microwave Radiation . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Spontaneous Emission of Phonons . . . . . . . . . . . . . . . . . 20

4.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.5 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 214.6 Stationary Current . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.6.1 Experiments in Double Quantum Dots: SpontaneousEmission of Phonons . . . . . . . . . . . . . . . . . . . . 24

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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1. INTRODUCTION

1.1 Quantum Dots: Basic Classifications

1.1.1 Semiconductor Based Systems

Electron density easily controllable. Lower density larger Fermi wavelengthof electrons. Electrons can be confined in small structures. Very high mobility2DEG. Both lateral and vertical dots possible (pictures).

Vertical dots can have a high symmetry (circular pillar): analogy to realatoms but instead of 3 dimensions only 2 dimensions. Number of electrons inthose dots very well defined (1 to 10 or more).

Lateral dots have a low symmetry: can be useful for the study of quantumchaos. Easier to integrate into circuits.

Fig. 1.1: LEFT: Vertical quantum dot (picture from a re-view article by Kouwenhoven, Austing, and Tarucha[1]. RIGHT: lateral quantum dot (C. Marcus, Harvard;http://marcuslab.harvard.edu/research.html).

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1. Introduction 2

1.1.2 Metallic Systems

Disorder, mesoscopic fluctuations due to disorder.

1.1.3 Transport vs. Optics

Here we only deal with transport, although optics appears at some points(microwave radiation plus transport etc.)

1.1.4 Nonlinear Transport vs. Linear Transport

1.1.5 Single Dots vs. Coupled Dots/Dots in more complicated structures

1.2 Basic Phenomena

1.2.1 Orthodox and Classics: Coulomb Blockade (CB), CB Oscillations and CBstaircase

IV curve deviates from a straight line (Ohms law). Already known in the50ies (Gorter 1951) for granular metallic systems. Experiments (60ies/70ies)difficult because of varying grain sizes.

1.2.2 The Dot as a Many-Body System (Artificial Atom): Level Structure,Selection Rules, Spin Effects

Tarucha, Kouwenhoven. Periodic Table. Spin Blockade.

1.2.3 Kondo Effect: Many Body Theory at its best.

1.2.4 The Quantum World enters: Coherent Effects in Coupled Dots, ArtificialMolecules

1.2.5 Best Regards from Quantum Optics: Spontaneous Emission of Phonons

1.2.6 Not done here

Single Electron Box: Interesting , also to discuss dissipation. Superconduc-tivity. Statistical properties of Dots, fluctuations due to disorder. AharonovBohm Effect and Dots. Fano Effect in Dots Electron pumps, turnstiles, adi-abatic and nonadiabatic pumping of electrons ... Further literature + infos(hopefully in the very near future) on my homepage http://bursill.phy.umist.ac.uk/ .

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2. THE SINGLE ELECTRON

TRANSISTOR

2.1 Basic Properties

Device with two tunnel junctions in series. Island in between is called dot.Literature: chapter 3 in [2], also Beenakker 1991 [3].

Energy scales:a) internal: U := e2/2C charging energy . E: single particle level

spacing . kBT: Temperature.b) external : Vtransport := L R difference of chemical potential of left

and right lead. Vgate gate voltage leads to shift of single particle levels.Why must the temperature T be low to see charging effects: for kBT

U := e2/2C, Coulomb blockade can be overcome by thermal excitation suchthat charging effects disappear. Example: capacitance of small device with

size 100nm yields typical C 1016F corresponding to 10K.Why must the connection between the dot and the leads be weak (weak

tunnel barriers): Invoke uncertainty principle for total energy Et chargeshould leak out through resistance R very slowly so that charge on dot is welldefined and can be counted classically (in terms of integer multiples of theelementary electron charge e. Use Ett > and estimate

Et = U = e2/2C, t = RC R h/e2 25.8k. (2.1)

2.1.1 Charging Energy

Orthodox theory of Coulomb blockade due to Likarev et al., developed in the80ies.

In metallic systems, basically classical electrostatic phenomenom with thediscreteness of energy levels neglected.

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2. The Single Electron Transistor 4

In semiconductor based systems, discreteness of levels (level separation E

becomes important. In that case, interplay between resonant tunneling andCoulomb blockade.

We treat the case E = 0 first: Charge Q = Ne on dot can only changein integer steps. Electrostatic potential ext can be changed continuously.

CCLL

CCRR

CCGG

VVRR

VVGG

VVLL

Ne

Fig. 2.1: Scheme of single electron transistor (SET).

Potential D(Q) of dot (central region) with N electrons and charge Ne,where e is the electron charge:

D(Q) = NeC

+ ext

ext =VLCL + VRCR + VGCG

C, C := CL + CR + CG (2.2)

From this one obtains the total electrostatic energy U(N) (ground stateenergy of N electrons in the dot) as

U(N) =

Ne0

dQD(Q) =(N e)2

2CN eext. (2.3)

Tunneling process N N + 1: an electron tunnels into the dot.dot(N + 1) := U(N + 1) U(N) = (N + 1/2)e

2

C eext (2.4)

is called the electrochemical potential of a dot with N electrons: difference ofground state energies.

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Condition for current to flow from a left lead to the island and then to the

right lead: To go from the left lead into a dot with N electrons, the electronhas to provide the energy U(N + 1) U(N). Clearly, this is only possible ifL := eVL > U(N+ 1)U(N). To go from the dot to the right lead, electronleaves and has energy U(N + 1) U(N) which has to be above the Fermienergy of the right lead, otherwise Pauli blocked.

eVL > U(N + 1) U(N) > eVRL > dot(N + 1) > R. (2.5)

For fixed VL and VR, depending on the gate voltage VG one is either in the

Coulomb blockade regime (no current flows), or one has a finite current.

2.1.2 Linear Transport VL VR 0: Coulomb Blockade OscillationsREVIEW for this part : Chapter 5 of [4], and Beenakker 1991 [3].

Varying the gate voltage VG leads to Coulomb blockade oscillations. In-creasing the gate voltage will change the number of electrons one by one.

Small bias voltage VL VR: Energy degeneracy at

U(N + 1) U(N) = 0 ext = (N + 1/2)eC

. (2.6)

Conductance peaks at these values of ext. Of course we are not satisfied withthat explanation: want to know exact form of the peaks (widths, height, as afunction of temperature, magnetic field etc etc.).

2.1.3 Linear Transport: Conductance Formula

Always kBT h: no level broadening. Two regimes: a) kBT E classi-cal regime. b) kBT E resonant tunneling regime. Explicite derivationfor b) below: only one level participates in transport.

Two different temperaturedependent curves in a) and b).

2.1.4 Experiments: Coulomb Blockade Oscillations

First observation of single charge tunneling in microfabricated samples by Ful-ton and Dolan (1987 [5]).

Meirav, Kastner, Wind 1990. First interpretation as Wigner crystal.Show curves from Meir, Wingreen, Lee.

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2.1.5 NonLinear Transport: Stability Diagram. Coulomb Blockade Staircase

Consider again condition for current flowing through dot:

eVL > U(N + 1) U(N) > eVR, VL > VR (2.7)

Assume symmetric situation VL = VR = V /2. If V > 0 we must haveeV /2 > U(N + 1) U(N) > eV /2. If V < 0 we must have e|V|/2 >U(N+ 1)U(N) > e|V|/2 (current flows in opposite direction). Therefore,the condition for current to flow is

e|V|/2 > U(N + 1) U(N) > e|V|/2 (2.8)

e|V|C/2 > QG/e (N + 1/2) > e|V|C/2, QG := CGVG + V(CL CR)/2.Define y := eV C/2 and x := QG/e. We have the inequality |y| > xN1/2 >|y| or |x (N + 1/2)| < |y| as a condition for current to flow (shaded areasin the Figure). Outside the shaded areas, i.e. within the Coulomb blockadediamonds, no current flows due to Coulomb blockade. For such values of Vand VG, the dot has a fixed and stable number of N electrons.

2.2 Nonlinear transport theory: Anderson Impurity Model

Literature: See Meir, Wingreen, Lee 1991 [6]; Weinmann, Hausler, Kramer1996 [7].A model that already comprises a lot of the physics: charging energy U,

level spacing E, finite temperatures kBT, finite voltages Vtransport = (L R)/e.

Consider dot with two single particle states Ea and Eb with 0, 1, or 2spinless electrons, attached to left and right leads. Dot can be in either offour states 0,a,b, 2 with energy 0, Ea, Eb, and Ea + Eb + U, where U is the(Hubbard like) interaction energy. The external voltage is directly controlledby the gate voltage Vg, Eq. 2.2, and shifts the energy levels, a := Ea ,b := Eb

.

2.2.1 Rate Equations

Denote rates for internal transitions between dot states due to tunneling as0a (rate for internal transition from state a with 1 electron to state 0 withno electron), etc.

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Rate equation (master equation) for probabilities as a function of time:

pa = a0p0 (2a + 0a)pa + a2p2pb = b0p0 (2b + 0b)pb + b2p2p2 = 2apa + 2bpb (a2 + b2)p21 = p0 + pa + pb + p2. (2.9)

The rates are given by

a0 = LfL(a) + RfR(a), b0 = LfL(b) + RfR(b) (2.10)

0a = LfL(a) + RfR(a), 0b = LfL(b) + RfR(b)

2a = LfL(U + b) + RfR(U + b), 2b = LfL(U + a) + RfR(U + a)a2 = LfL(U + b) + RfR(U + b), b2 = LfL(U + a) + RfR(U + a).

Here,

fL/R() := f( L/R), f() =

exp

kBT

+ 1

1f() := 1 f(). (2.11)

The stationary current I through the left junction is obtained from the sta-tionary solution of the master equation (stationary values p0, pa, pb, p2)

I = e La0 + Lb0p0 + L2a L0apa+

L2b L0b

pb

La2 +

Lb2

p2

. (2.12)

It doesnt matter if we calculate the current through the left or the rightjunction: both currents are the same.

It is very easy to solve the system of linear equations Eq. (2.9). Theimportant ingredients are the transition rates Eq. (2.10).

2.2.2 Linear Transport Regime: Coulomb blockade oscillations and addition

energiesFor Vtransport := (LR)/e 0, the Coulomb blockade oscillations discussedabove are visible as two consecutive peaks at the chemical potentials

dot(N = 1) = a 0 = adot(N = 2) = a + b + U a = b + U. (2.13)

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0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3

current,electronnumber

U=1.0, T=0.01

L=0.04, R=0

Ea=1.0,Eb=1.1

currentelectron number

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3

current,electronnumber

U=1.0, T=0.01

L=0.4, R=0

Ea=1.0,Eb=1.1

currentelectron number

Fig. 2.2: Current and number of (spinless) electrons in the linear (left) and the

nonlinear (right) transport regime through a quantum dot, modeledas an Anderson impurity with two energy levels Ea and Eb. Thecurrent is in units ofeLR/(L + R). The external voltage onthe dot is directly controlled by the gate voltage Vg, Eq. 2.2, and shiftsthe energy levels, a := Ea , b := Eb . The transport (bias)voltage Vtransport := (L R)/e is kept fixed here. The repulsionenergy of two electrons is U.

The distance between the peaks is given by the addition energies E(N) :=(N)

(N

1) ,

E(N = 1) := dot(N = 1) dot(N = 0) = aE(N = 2) := dot(N = 2) dot(N = 1) = b a + U. (2.14)

2.2.3 Linear Transport Regime: Peak Shape, Limit b, UIn this limit, only 0 and a are involved and we have

pa = a0p0 0apa, 1 = p0 + pa (2.15)In the stationary case, this becomes

p0 = 0a

0a + a0, pa =

a0

0a + a0

I = eLa0p0 L0apa = e 1L + R

La00a L0aa0

= e LRL + R

{fL(a) fR(a)} . (2.16)

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In the linear transport regime, this expression can be further simplified: With

fL(a) fR(a) = f(a L) f(a R) = f(a L) f(a L + (L R))= (L R)(f(a L)) + O(L R)2, (2.17)

we find

I = e(L R) LRL + R

1

4kBTcoth2aL2kBT

+ O(L R)2. (2.18)The linear conductance G is defined as the limit L R := ,

G := lim(LR)0

Ie(L R) , (2.19)

so that we obtain

G =LR

L + R

1

4kBT coth2a2kBT

. (2.20)This describes a temperaturedependent resonance peak, when the gate volt-age Vg and with it , Eq. 2.2, and the energy a = Ea is swept through thechemical potential or, alternatively, a is kept fixed and is swept through

(see Fig. 2.3).

EXERCISE: Repeat and check the steps that lead to the conductance formulaEq. (2.20).

2.2.4 Nonlinear transport regime

Interpretation of the results from Fig. 2.2: The chemical potentials are

dot(N = 1) = a

0 = a

dot(N = 2) = a + b + U a = b + U. (2.21)for the transition N N + 1 between groundstates, and

dot(N = 1) = b 0 = bdot(N = 2) = a + b + U b = a + U. (2.22)

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0

0.01

0.02

0.03

0.04

0.05

0.06

0.4 0.6 0.8 1 1.2 1.4 1.6

current

U=1.0L=0.01, R=0

T=0.10T=0.10T=0.05T=0.05

E = 0.1

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3

current

Vtransport

U=1.0, T=0.01

Ea=1.0

R=0, =0

E=0.1

E=0.5

Fig. 2.3: LEFT: Current in the linear transport regime through a quantum

dot (as above). Comparison is made between the exact solution of themaster equation(solid lines) and the analytical form Eq. 2.20 whichworks well for kBT E, where E is the separation between thetwo levels. As above, the Coulomb repulsion is U = 1 here, andE = 0.5 for the two lower curves, where the agreement is perfect.For a smaller level separation, E = 0.1 and T = 0.05 (upper curve),the agreement with the analytical form (for T = 0.05) is not good.RIGHT: Coulomb staircase , i.e. currentvoltage characteristics ofthe same system for fixed voltage and variable transport voltageVtransport := L R.

for the transitions N N + 1 between ground and excited states. Increasingthe gate voltage increases and the following enter into the transport window[R, L]: first, dot(N = 1) one transport channel. Then,

dot(N = 1)

two channels, current even larger. For the next current peak one has to increase until dot(N = 2) (not

dot(N = 2)) enters the window: immediately two

transport channels (large current). Further increase of shifts dot(N = 2) outof the window only one channel, further increase (dot(N = 2) < R) leadsto zero current. Somewhat tricky when visualised in energy level picture.

2.2.5 Comparison to Experiments: Nonlinear Transport

In semiconductor dots: Johnson, Kouwenhoven, de Jong, van der Vaart, Har-mans, Foxon 1992 [8]. Weis, Haug, v. Klitzing, Ploog 1992, 1993 [9,10].

See double Coulomb staircase: not only dot ground states but also excitedstates.

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2.3 Symmetries, Many Body Effects, Spin

Beyond the constant interaction model. Anderson impurity model above:addition energy for one electron is the sum of (constant) Coulomb energy Uand single particle energy difference b a = E. Not much information onthe dot contained in the addition energy spectrum.

2.3.1 Symmetries

Fig. 2.4: a) Current through a circular dot (pillar). The number of electronsN in the dot is indicated. b) is a simple shell model picture, giving

rise to a periodic table of artificial elements . This picture is takenfrom a review article by Kouwenhoven, Austing, and Tarucha [1].

Literature: Kouwenhoven, Austing, Tarucha 2001 [1].2d circular symmetric dots: single particle states (orbitals) are 2d harmonic

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oscillator states |nl, eigenstates of single particle potential

V(r) =1

2m20r

2. (2.23)

Gives rise to a shell structure in the addition energies, similar to 3d real atomswhere electrons move in the spherically symmetric 1/r potential of the nucleus(states |nlm).

Magic numbers N = 2, 6, 12,...: more energy is needed to add an electronto a dot with 2, 6, 12,... electrons. This shell structureis due to degeneracies insingle particle energy spectrum (FockDarwin, perpendicular magnetic fieldcan be included): single particle effect.

Phenomenological model: three energy scales single particle energy, directCoulomb energy, and exchange energy for electrons with parallel spin. Thelatter is an extension to include manybody effects beyond the simple chargingmodel. Seems to work surprisingly well.

Periodic Table of artificial atoms can be created and controlled withinone and the same quantum dot. Application of a perpendicular magnetic fieldto resolve different energy scales.

2.3.2 Exact ManyBody Calculations, Spin Blockade

Literature: Weinmann et al. [11] . Spin effect in tunneling: tunneling of an

electron with total spin 1/2, spin projection 1/2 into a dot with manyelectron state with total initial spin Si, total initial spin projection Mi: finalmanyelectron state must have with one electron more (or less) and totalspin projection Mj increased (or decreased) by 1/2. Tunnel rate must containClebschGordan coefficient

Si, Mi; 12

,12|Sj, Mj (2.24)

If total spins of groundstates with successive electron numbers n n1 differby more than 1/2: Coulomb oscillation peak is suppressed (linear transport,

spinblockade type II). Not possible in one dimension (LiebMattis Theorem:spin of ground state is always 0 or 1/2).

Additional effects in nonlinear transport: negative differential conduc-tance possible.

Experiments: Huttel, Qin, Holleitner, Blick, Neumaier, Weinmann, Eberl,Kotthaus [12].

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3. THE KONDO EFFECT

3.1 Kondo Effect in Metals. Impurity Model

Kondo 1964: Calculation (second order perturbation theory) of resistanceR(T) of a metal with magnetic impurities (localised spins, example: magnetic

alloy CuMn), increases logarithmically for small T, R(T) = R0+R1 ln(kBT /D)+R2(T /Tp)

5, R1 < 0, R0 residual resistance due to impurites, term T5 phononcontribution resistance minimum, known since the 30ies.

Literature here: Yosida chapter 16 [13]Higher order perturbation theory: terms lnn(kBT /D) appear, can be summed

up as a geometric series with a result that diverges below the Kondo temper-ature TK. Haldane 1978: within the Anderson impurity model (one electronwith spin up or spin down on a level 0, broadened due to coupling toFermi sea, double occupancy prevented by Coulomb repulsion U)

TK = 12Uexp

0(0 + U)U

. (3.1)

Spin flip processes : consider 0 (Fermi energy of metal). Spin-up elec-tron can tunnel out of the impurity into the metal to occupy a virtual state(energetically forbidden, but allowed in QM due to energytime uncertaintyrelation). Electron then replaces by metal electron with spin down spinflip. Leads to extra Kondo resonance in the local impurity density of statesat the Fermi energy of the metal). This drastically changes (linear) transportproperties (which are determined by states at the Fermi surface). Importantproperty of that phenomenon: single parameter scaling, i.e.

R/R0 = f(T /TK), (3.2)

i.e. the whole temperature dependence is governed by one universal functionf with the material parameters only entering into the value for the Kondotemperature TK.

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3. The Kondo Effect 14

3.2 Kondo Effect in Dots

Glazman, Raikh 1988 [14]; Ng, Lee 1988 [15]: Consider Anderson impuritymodel as a very simplified model for a dot: one spin-degenerate single particlelevel 0 with strong Coulomb repulsion U:

H =

;kL,R

kckck +

0cc + Unn

+

;kL,R

Vkc

kc + H.c.

. (3.3)

Linear transport Fermi level of reservoirs L = R = with 0 < < +U

at very low temperatures: there is one electron on the dot. The conductanceG, however, is not zero (as one would expect due to the Coulomb blockade),but

G =2e2

h

4LR(L + R)2

f(T /TK) (3.4)

with f(0) = 1. Here,

L(R) := L(R)(), L(R)() := 2

kL(R)

|Vk|2( k) (3.5)

is the tunnel rate between the dot and the left (right) lead. In particular,for symmetric coupling to the leads, L = R, G = 2e

2/h corresponding tocomplete transmission.

What happens to the system at T = 0: new correlated many body groundstate with one electron on the impurity, forming a spin singlet with the elec-trons of the leads.

3.3 Theory Tool: Greens Functions

Literature: Book by Haug/Jauho [16]; Meir, Wingreen 1992 [17], 1994 [18].The stationary current I through the dot can be expressed with the help ofthe local density of states () on the dot:

I =e

d [fL() fR()] L()R()L() + R()

()

() := 1

ImGr(), Gr(t) = i(t)c(t)c(0) + c(0)c(t). (3.6)

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The last line defines the retarded Greens function Gr of the dot. It has to be

calculated in presence of both interaction (U) and the coupling to the leads(tunnel matrix elements Vk). Even for this simple model the Greens functionin general cant be calculated exactly.

3.4 Unitary Limit at T = 0 :

At zero temperature T = 0, it is possible to make an exact prediction basedfor the conductance when L R = and the Greens function has tobe evaluated at the Fermi energy . In fact, this case is analogous to thesituation of a magnetic impurity in a metal, and this analogy has been used by

Glazman/Raikh and Ng/Lee. Basically, at T = 0 and in equilibrium L R,there are not many possibilities for scattering. The self energy of the dotat the Fermi energy fulfills

Im( i) = (R + L)/2, (3.7)

i.e. is only determined by elastic scattering. As a result, it is possible to for-mulate a relation between the scattering properties of the Anderson impurityand its occupation number. This Friedel Sum Rule can be exploited [15,19] inthe regime 0 < < 0 + U with one electron on the impurity: It follows

G = 2e2h

4LR(L + R)2

sin2

2

nd

(3.8)

with nd = n+ n = 1. In particular, for symmetric coupling to the leads,L = R, G = 2e

2/h corresponding to complete transmission. Eq. (3.8) looksquite noninteracting but the underlying physics isnt: What happens to thesystem at T = 0, is the emergence of a new correlated many body ground statewith an odd number of electrons on the impurity, forming a spin singlet withthe electrons of the leads. This is the Fermi liquid fixed point of the Kondoproblem. At very low but finite temperatures the prediction for the deviationfrom this point was made by Nozieres back in the 70ies: in our context here,it reads

G =2e2

h

4LR(L + R)2

1

2T2

T2K

, (3.9)

where the Kondo temperature TK was defined above.

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On the other hand, approaching from high temperatures T TK, per-turbation theory for the Greens function yields logarithmic divergences of thetype ln(T /D) (D is a cutoff due to, e.g., the finite bandwidth of the conductionband). These logarithmic singularities get even worse in higher order pertur-bation theory. There is, however, a remedy to systematically deal with thisproblem: renormalization group theory . This allows to establish the connec-tion between the high temperature T TK and low temperature T TKregime, yielding

G =2e2

h

4LR(L + R)2

f(T /TK) (3.10)

with f(0) = 1.

3.5 Experiments

Since 1998, a lot. to name a few: GoldhaberGordon, Gores, Kastner, Shtrik-man, Mahalu, Meirav 1998 [20]; W. G. van der Wiel, S. De Francheschi, T.Fujisawa, J. M. Elzerman, S. Tarucha, and L. P. Kouwenhoven 2000 [21] unitary limit:

3.6 Further topics Kondo effect in an integer spin quantum dot: spin S = 1 of triplet

state also screened by series of cotunneling events. Spin S = 1 tripletcan occur due to Hunds rule (exchange energy wins over single particlespacing).

Experiment: Sasaki, De Franceschi, Elzerman, van der Wiel, Eto, Tarucha,Kouwenhoven 2000 [22]: vertical rectangular pillar, number of electronsN in dot known exactly. Conductance as a function of magnetic field Band gate voltage B: spintriplet to spinsinglet transition in the groundstate.

For absence of evenodd behaviour in Kondo effect see also Schmid, Weis,Eberl, v. Klitzing 2000 [23]

Theory: Eto, Nazarov 2000,...

Nonlinear transport and Kondo effect. Dephasing and Kondo effect.

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Fig. 3.1: LEFT: Unitary limit of the Kondo effect in quantum dots: exper-

imental results by van der Wiel et al [21]. Conductance G in unitsof e2/h as a function of the gate voltage for different temperatures(T = 15 mK for the black trace, up to T = 800 mK for the redtrace. Inset: logarithmic Tdependence ofG for one fixed gate votage.RIGHT: Conductance as a function of temperature T for three dif-ferent gate voltages Vg. Inset: scaling of the conductance to a singlecurve as a function ofT /TK, where TK is the Kondo temperature thatdepends on Vg.

Magnetic field and Kondo effect; AB ring and Kondo effect.

Many level dot and Kondo effect; Kondo effect in coupled dots: not spin,but pseudo spin (dot 1 or dot 2).

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4. COUPLED QUANTUM DOTS

4.1 Motivation

Why coupled quantum dots: motivation from quantum logic gates, exploitingtwo core quantum mechanical phenomena a) superposition b) entanglement.

4.1.1 Superposition

A (single) Qubit is a superposition (linear combination) | = a|0 + b|1 inthe 2d complex Hilbert space C2. Example: |0 = | , |1 = | (a spin 1/2).At the moment, many if not perhaps most of the activities on solid state basedqubits are based on the spin. Other example: |0 = |L, |1 = |R, an electronwith fixed spin in either a left (L) or a right (R) quantum dot (charge basedqubit).

4.1.2 EntanglementDeepest difference between classical physics and quantum physics. Mathemat-ically again due to the linear structure of QM (superposition principle). In QMwe describe a system composed of more than one particle by tensor productsof Hilbert spaces and not direct products: Example: Hilbert space of 2 Qubitstates for two spins 1/2 basis of 1 singlet and 3 triplet states

|S0 := 12

(| |)|T1 := |

|T0 :=12 (|+ |)

|T1 := |. (4.1)|T0 and |S0 are entangled: we cant say if the first spin is up or down (samefor second spin). |T1 and |T1 are not entangled. Gets more complicated ifwe have electrons with spin: indistinguishable particles.

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4. Coupled Quantum Dots 19

Quantum computing : basically, it is sufficient to have 1qubits and 2

qubits.Experiments on entanglement are most advanced in quantum optics , ion

traps etc. Entanglement of N = 2, N = 4 ions realized in ion traps (Sackettet al 2000) [24]. Keywords: nonclassical Schrodinger cat states, control ofentanglement. Aim: to do that in a solid state environment, where things canbe (hopefully) scaled up to large arrays of qubits.

4.2 Theory Tool: Master Equations

What has to go into a coupled quantum dot theory: 1. strong interactions 2.

coupling between the dots 3. coupling to the external world (leads) 4. couplingto other degrees of freedom such as bosonic excitations, photons, phonons.

Noone has succeeded to do all of them, although general framework doesexist: Greens function formalism. In the following, we concentrate on point4 (relation to decoherence). Treat coupling to the leads in lowest order per-turbation theory (T TK, no Kondo physics). Here only simplest case oftwo coupled quantum dots. Extension to many dots possible, see Wegewijs,Nazarov 1999 [25].

Literature here: Stoof, Nazarov 1996 [26]; Gurvitz, Prager 1996 [27]; Gurvitz1998 [28], Brandes [29,30].

Coulomb effects also dealt with in extremely simplified manner: Assumetwo dots coupled by a tunnel barrier. Coulomb charging energy U in smalldots typically U 1 meV Vtransport = (L R)/e: assuming there are Nelectrons in the left and M electrons in the right dot ground state |0 =|N, M. Additional transport electron is either on the left or on the right dot states |L = |N + 1, M,|R = |N, M + 1. Therefore, only three statesare involved! Nevertheless, including dissipation, microwave radiation etc stillcomplicated enough to be of interest.

Describe the space spanned by the two states L and R with the help of(pseudo) spin Pauli matrices: Hamiltonian for this subspace is

H0 = 2

z + Tcx, (4.2)

where z = |LL| |RR| and x = |LR| + |RL|. Here, = L R isthe difference of the ground state energies of |L and |R, and Tc describes thecoupling between the two dots.

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4.3 Microwave Radiation

Eigenvectors and energy eigenvalues of H0 are

| = 1N

[2Tc|L + ( )|R] , N :=

4|Tc|2 + ( )2

:= 12

, :=

2 + 4|Tc|2. (4.3)

Microwave experiment should be able to detect the squareroot form of : Ex-periment by Oosterkamp, Fujisawa, van der Wiel, Ishibashi, Hijman, Tarucha,Kouwenhoven [31].

4.4 Spontaneous Emission of Phonons

For metals: weak electronphonon coupling, electronphonon scattering ratesgo to zero when the temperature T 0. Reason: at T = 0, no occupied statesabove the Fermi level EF, all states below EF are occupied, scattering wouldbe only possible by absorption of phonons. Absorption is proportional to Bosedistribution nB() = [exp(/kBT) 1]1 0 for > 0 and T 0.

Quantum dots interact with the semiconductor substrate phonons. In cou-pled quantum dots (threestate basis as discussed above) there is no Fermi

distribution. For example, left dot can be higher in energy than right dot even at zero temperature there is spontaneous emission.

4.4.1 Model

We describe the coupling to phonons by a Hamiltonian

H =

2z + Tcx +

1

2zA +

Q

QaQaQ, A :=

Q

gQ

aQ + a

Q

,(4.4)

Here, Q are the frequencies of phonons, and the gQ denote interaction con-

stants. Although not exactly solvable, the model is quite well understood forclosed systems [32] (isolated dots with one additional electron). The couplingto external leads offers the possibility to study its nonequilibrium properties,such as the inelastic stationary current through the dots.

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Fig. 4.1: Left: Stationary current through a microwave irradiated doublequantum dot (zero transport voltage) for different microwave fre-quencies f as a function of the energy difference (= E here)in the experiment by Oosterkamp and co-workers [31]. Positive or

negative peaks occur whenever hf matches the energy difference,hf = = 2 + 4T2c , between bonding and anti-bonding state inthe double dot, cf. RIGHT: From the position of the current peaks,the relation ( E) = (hf)2 4T2c is tested for various interdotcoupling constants Tc (denoted as T in the picture). The pictures aretaken from [31].

4.5 Equations of Motion

We describe the dynamics of the double dot by a reduced (effective) statistical

operator (t). The coupling to the reservoirs is described by the additional

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Hamiltonian

Hres + HV, Hres =k

Lk ckck +

k

Rk dkdk (4.5)

HV =k

Vkc

ksL + Wkd

ksR + c.c.

, sL := |0L|, sR := |0R|.

(4.6)

We start from the Liouvillevon Neumann equation for the density operatorof the total system (DOT + PHONONS + RESERVOIRS),

d

dt(t) = i[H, (t)]. (4.7)The first step: transformation into the interaction picture with respect toH0 := HB + Hres, O(t) = e

iH0tOeiH0t for any operator O. Write the totalHamiltonian H = H+ V, and Eq. (4.7) becomes

d

dt(t) = i[V(t), (t)]. (4.8)

Iteration of Eq. (4.8),

d

dt (t) = i[V(t), (0)] t

0 dt

V(t), [V(t), (t)]

. (4.9)

We define the effective density operator for the double dot,

(t) = Trres,phonon(t). (4.10)

Now, second order Born and Markov approximation in the perturbation Vyields [26, 30]

tLL(t) = iTc [LR(t) RL(t)] + L [1 LL(t) RR(t)]

t

RR(t) = iTc [RL(t) LR(t)] RRR(t),

where the tunnel rates are

L = 2k

|Vk|2(Lk L) R = 2k

|Wk|2(Lk R). (4.11)

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and we allow for tunneling from a left reservoir at rate L into the left dot,

and from the right dot to the right reservoir at rate R.For the remaining equation for the offdiagonal element LR =

RL, one

has to choose between perturbation theory in gQ (weak coupling, PER), orin Tc in a polarontransformed frame (strong coupling, POL) . In general, noexact solution of the model is available: this is the case even for coupling toone bosonic mode only (gQ Q,Q0, Rabi Hamiltonian).

For the spinboson problem with R/L = 0, it is wellknown that POLis equivalent to a doublepath integral noninteracting blip approximation(NIBA) that works well for zero bias = 0 but for = 0 does not coincidewith PER at small couplings and low temperatures. In the following, we

compare both approaches for R/L = 0 and find nearly perfect agreement forvery large Tc, a regime that has been tested experimentally recently [33].The standard Born and Markov approximation with respect to A yields

d

dtPERLR (t) = [i p R/2] LR(t) + [iTc ] RR(t) [iTc +] LL(t).

Here, the rates are

p := 2T2c2

()coth(/2) , := Tc2

2()coth(/2) Tc

2()

() :=

Q|gQ|2( Q). (4.12)

where :=

2 + 4T2c is the energy difference of the hybridized levels, and = 1/kBT the inverse phonon equilibrium bath temperature. Note that besidethe offdiagonal decoherence rate p, there appear terms in the diagonalswhich below turn out to be important for the stationary current.

On the other hand, the polaron transformation [30] leads to an integralequation

POLLR (t) = t0

dtei(tt)

R2

C(t t)LR(t)

+ iTc(t) {C(t t)LL(t) C(t t)RR(t)}

,

where

C() := exp

0

d()

2[(1 cos t)coth(/2) + i sin t]

. (4.13)

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4.6 Stationary Current

Calculated as Istat = e2TcImLR(z = 0) from Laplace transforming the equa-tions of motion;

Istat =eLRG+

LG + (L + R)G+ LR . (4.14)

has to be used with either the perturbative result (small ep coupling) or thepolaron transformation (small interdot coupling),

G(PER) := 2TcImiTc

i

p

R/2(4.15)

or

G(POL)+ := 2TcIm

iTcC1 + (1/2)RC

, G(POL) := 2TcIm

iTcC1 + (1/2)RC

,(4.16)

where C :=0

dteitC(t). More details in [30,34].Most interesting regime at low temperatures: Experiment by Fujisawa et

al 1998 [33].

4.6.1 Experiments in Double Quantum Dots: Spontaneous Emission of Phonons

We now shortly describe a recent experimental realization of a twolevel systemin a semiconductor structure. This experiment has been performed in coupledartificialatoms, that is coupled quantum dots, by Fujisawa and coworkers atthe Technical University of Delft (Netherlands) in 1998.

The double quantum dot is realized in a 2DEG AlGaAsGaAs semiconduc-tor heterostructure, see Fig.(4.2). Focused ion beam implanted inplane gatesdefine a narrow channel of tunable width which connects source and drain (leftand right electron reservoir). On top of it, three Schottky gates define tunabletunnel barriers for electrons moving through the channel. By applying nega-tive voltages to the left, central, and right Schottky gate, two quantum dots

(left L and right R) are defined which are coupled to each other and to thesource and to the drain. The tunneling of electrons through the structure islarge enough to detect current but small enough to have a welldefined numberof electrons ( 15 and 25) on the left and the right dot, respectively. TheCoulomb charging energy ( 4 meV and 1 meV) for putting an additionalelectron onto the dots is the largest energy scale, see Fig.(4.2).

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Fig. 4.2: Left: Schematic diagram of a double gate single electron transistor

(double quantum dot) by Fujisawa and Tarucha (1997). The 2DEG islocated 100 nm below the surface of an AlGaAs/GaAs modulationdoped heterostructure with mobility 8 105 cm2 (Vs)1 and carrierconcentration 3 1011 cm2 at 1.6 K in the dark and ungated. Gafocused ion beam implanted inplane gates and Schottky gates definethe dot system. A double dot is formed by applying negative gatevoltages to the gates GL, GC, and GR. The structure can also beused for singledot experiments, where negative voltages are appliedto GL and GC only.

Right: Top view of the double quantum dot. Transport of electrons is through

the narrow channel that connects source and drain. The gates themselves havewidths of 40 nm. The two quantum dots contain approximately 15 (Left, L)and 25 (Right, R) electrons. The charging energies are 4 meV (L) and 1 meV(R), the energy spacing for single particle states in both dots is approximately0.5 meV (L) and 0.25 meV (R).

By tuning simultaneously the gate voltages of the left and the right gatewhile keeping the central gate voltage constant, the double dot switches be-tween the three states |0 = |NL, NR, |L = |NL + 1, NR, and |R =|NL, NR + 1 with only one additional electron either in the left or in theright dot (see the following section, where the model is explained in detail).The main experimental trick is to keep the system within these statesand to change only the energy difference = L R of the dots withoutchanging too much, e.g., the barrier transmissions. The measured averagespacing between singleparticle states ( 0.5 and 0.25 meV) is still a largeenergy scale compared to the scale on which is varied. The largest value of

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is determined by the sourcedrain voltage which is around 0.14 meV. The

main findings are the following:1. At a low temperature of 23 mK, the stationary tunnel current I as a

function of shows a peak at = 0 with a broad shoulder for > 0 thatoscillates on a scale of 20 30eV, see Fig.(4.3).

2. For larger temperatures T, the current increases stronger on the ab-sorption side < 0 than on the emission side. The data for larger T canbe reconstructed from the 23 mK data by multiplication with EinsteinBosefactors (the Planck radiation law) for emission and absorption.

3. The energy dependence of the current on the emission side is between1/ and 1/2. For larger distance of the left and right barrier (600 nm on a

surface gate sample instead of 380 nm for a focused ion beam sample), theperiod of the oscillations on the emission side appears to become shorter.Those who are interested in more details: T. Fujisawa, T. H. Oosterkamp,

W. G. van der Wiel, B. W. Broer, R. Aguado, S. Tarucha, and L. P. Kouwen-hoven, Science 282, 932 (1998).

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0

0.5

1

1.5

2

2.5

-100 -50 0 50 100 150 200

C

urrent(pA)

(eV)

R = L = 0.03 eV

Tc = 3.0 eV

g = 0.02

c =100 eV

d = 10 eV

PER, T=300 mKPOL, T=300 mKPER, T= 23 mKPOL, T= 23 mK

g=0

Fig. 4.3: Left: Current at temperature T = 23mK as a function of the energydifference = L R in the experiment by Fujisawa and coworkers.The total measured current is decomposed into an elastic and aninelastic component. If the difference between left and right dotenergies L and R is larger than the sourcedrainvoltage, tunnelingis no longer possible and the current drops to zero. The red circlemarks the region of spontaneous emission of phonons. Phonons arethe quanta of the lattice vibrations of the substrate (the whole semi-conductor structure), in the same way as photons are the quanta ofthe electromagnetic (light) field. The spontaneous emission is char-acterized by the large shoulder for > 0 with an oscillationlikestructure on top of it. Right: Stationary current through doubledot, calculated in perturbation theory (PER) or in the polaron ap-proach (POL).

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BIBLIOGRAPHY

[1] L. P. Kouwenhoven, D. G. Austing, and S. Tarucha, rep. Prog. Phys. 64,701 (2001).

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[3] C. W. J. Beenakker, Phys. Rev. B 44, 1646 (1991).

[4] (Ed.) H. Grabert and M. H. Devoret, Single Charge Tunneling, Vol. 294of NATO ASI Series B (Plenum Press, New York, 1991).

[5] T. A. Fulton and G. J. Dolan, Phys. Rev. Lett. 59, 109 (1987).

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[9] J. Weis, R. J. Haug, K. v. Klitzing, and K. Ploog, Phys. Rev. B 46, 12837(1992).

[10] J. Weis, R. J. Haug, K. v. Klitzing, and K. Ploog, Phys. Rev. Lett. 71,4019 (1993).

[11] H. P. Wei, L. W. Engel, and D. C. Tsui, Phys. Rev. B 50, 14609 (1994).[12] A. K Huttel, H. Qin, A. W. Holleitner, R. H. Blick, K. Neumaier, D. Wein-

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[13] K. Yosida, Theory of Magnetism, Vol. 122 of SolidState Series(Springer,Berlin, 1996).

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Bibliography 29

[14] L. I. Glazman and M. E. Raikh, JETP Lett. 47, 378 (1988).

[15] T. K. Ng and P. A. Lee, Phys. Rev. Lett. 61, 1768 (1988).

[16] H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics ofSemiconductors, Vol. 123 of Solid-State Sciences (Springer, Berlin, 1996).

[17] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992).

[18] N. S. Wingreen, Y. Meir, Phys. Rev. B 49, 11040 (1994).

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Ando (Springer-Verlag, Berlin, 1991), p. 53.

[20] D. Goldhaber-Gordon, J. Gores, M. A. Kastner, Hadas Shtrikman, D.Mahalu, and U. Meirav, Phys. Rev. Lett. 81, 5225 (1998).

[21] W. G. van der Wiel, S. De Francheschi, T. Fujisawa, J. M. Elzerman, S.Tarucha, and L. P. Kouwenhoven, Science 289, 2105 (2000).

[22] S. Sasaki, S. De Franceschi, J. M. Elzerman, W. G. van der Wiel, M. Eto,S. Tarucha, L. P. Kouwenhoven, Nature 405, 764 (2000).

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INDEX

addition energies, 8AlGaAsGaAs, 24Anderson Impurity Model, 6, 13

Born and Markov approximation, 22

charging energy, 3ClebschGordan coefficient, 12conductance, 9conductance peak, 9Coulomb blockade, 3Coulomb blockade diamonds, 6Coulomb Blockade Oscillations, 5Coulomb staircase, 10Coupled Quantum Dots, 18coupled quantum dots, 24current oscillations, 26

density operator, 22

electrochemical potential, 4Entanglement, 18exchange energy, 12

Fermi liquid, 15Friedel Sum Rule, 15

gate voltage, 25Greens Functions, 14

Hubbard, 6

inelastic current, 27

Kondo Effect, 13Kondo temperature, 13

Lateral dots, 1level spacing, 3Liouvillevon Neumann equation, 22logarithmic singularities, 16

master equation, 7, 19Microwave Radiation, 20

NIBA, 23Nozieres, 15

orthodox theory, 3

periodic table, 11Phonons, 20polaron transformation, 23

Quantum computing, 19

quantum optics, 19

renormalization group theory, 16retarded Greens function, 15

Schottky gates, 24shell structure, 12Spin Blockade, 12Spin flip processes, 13spin singlet, 14spinboson problem, 23

Spontaneous Emission, 20Superposition, 18

tunnel rates, 7TwoLevel System, 19

Vertical dots, 1

31

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