PHYSICAL REVIE& B VOLUME 48, NUMBER 11 15 SEPTEMBER 1993-I

Time-dependent transport through a mesoscopic structure

Ned S. WingreenNEC Research Institute, 4 Independence 8'ay, Princeton, ¹wJersey 08540

Antti-Pekka Jauho*NORDITA, Blegdamsuej 17, DK-2100 Copenhagen, Denmark

Yigal MeirDepartment of Physics, Uniuersity of California, Santa Barbara, California 93106

{Received 17 June 1993)

A general formulation is presented of the nonlinear, time-dependent current through a small interact-ing region, where electron energies are changed by time-dependent voltages. An exact solution is ob-tained for the noninteracting case when the elastic coupling to the leads is independent of energy. Tem-poral phase coherence in a double-barrier tunneling structure produces "ringing" in the response of thecurrent to a voltage pulse, which can be observed experimentally in the dc current by varying the pulselength in a train of voltage pulses. The nonlinear current due to an ac bias also shows complex timedependence.

The importance of the spatial coherence of electronicwave functions is one of the hallmarks of the mesoscopicregime. A panoply of mesoscale phenomena includingweak localization and weak antilocalization, ' Aharonov-Bohm oscillations, and universal conductance Quctua-tions all rely on the phase coherence of electrons in smallstructures. Since the traditional experimental probe ofthese effects is steady-state transport, the role of temporalphase coherence is generally subsumed under the effectsof spatial coherence. However, recent experimental pro-gress has opened the door to direct measurement ofelectronic phase coherence in time.

Interest in the time domain has sprung both from de-vice applications of double-barrier resonant-tunnelingstructures, and from possible current standard applica-tions of single-electron tunneling circuits. While the in-teractions among electrons are mean-field-like in thethree-dimensional resonant-tunneling structures, thelow-dimensional confinement of electrons in the single-electron circuits results in strong many-body correlations.A general theory for temporal coherence in these struc-tures must therefore rigorously account for interactions.

In this paper, we first present a general formulation ofthe nonlinear, time-dependent current through a small in-teracting region coupled to two noninteracting leads. Ageneral result, Eq. (6), expresses the time-dependentcurrent Aowing into the interacting region from one leadin terms of local Green functions. This expression is thenused to explore temporal coherence in the response of amesoscopic system to time-dependent external driving.We focus on a fully nonlinear, exactly solvableexample —noninteracting electrons traversing a double-barrier tunneling structure with energy-independent cou-pling to the leads. The temporal coherence of this systemis evident in the "ringing'* of the current in response to arectangular pulse of the bias. This ringing can be ob-served experimentally in the dc current by varying thepulse length in a train of voltage pulses applied to thestructure. Similarly, the current Aowing in response to

an ac bias displays complex time dependence, which isreAected in oscillations of the dc current vs driving fre-quency.

The formalism and Hamiltonian are similar to thoseemployed for the steady-state current in Ref. 8,

X ek.(t)ck.ck.+H;.t[I .l [d.'I t)k, aE.L, R

+ g [Vk „(t)ck d„+H.c.],k, aCL, R

where ck (ck ) creates (destroys) an electron withmomentum k in channel a in either the left (L) or theright (R) lead, and Id„J and [d„I form a complete,orthonormal set of single-electron creation and annihila-tion operators in the interacting region. However, timedependence due to external driving is now explicitly in-cluded in the energies of states in the interacting regionand in the leads, in the hopping matrix elements, and inthe interactions themselves. A central assumption is thatinteractions between electrons in the leads and betweenelectrons in the leads and in the interacting region can beneglected. Geometrically, the leads must therefore rapid-ly broaden into large metallic contacts in which interac-tions are strongly screened. Since transport experimentson mesoscale structures typically satisfy this requirement,the model is directly relevant to experiment.

The Keldysh approach to calculating the current Bow-ing into and out of the interacting region treats the con-tacts as systems separately in equilibrium in the distantpast, possibly with different chemical potentials. Physi-cally, applying a time-dependent bias (electrostatic-potential difference) changes the energies of states in theleads via ek (t), without changing their occupations, aswe will discuss. This preserves the coherent evolution ofphase in the leads, %ka(t) o- exp[ i jdt'Ek (t'—) j, andproduces interference in tunneling between the leads andthe interacting region, due to their different time-

0163-1829/93/48(11)/8487(4)/$06. 00 1993 The American Physical Society

8488 NED S. WINGREEN, ANTTI-PEKKA JAUHO, AND YIGAL MEIR

dependent energies. The energies and other parametersin (1) depend on the detailed geometry and on the self-consistent response of charge in the contacts to externaldriving. We assume these parameters are known andproceed to calculate the current through the interactingregion.

The time-dependent current from the left lead into theinteracting region is

JL(t)= Re g Vk „(t)G„k (t, t)' k, aEL

(2)

where in (2) we have just rewritten the expectation of thecurrent matrix element in terms of the Keldysh Greenfunction G„k (t, t'):i(—ck (t')d„(t)). Since the Hamil-tonian describing the leads is noninteracting, one has theDyson equation

G„ka(t, t')= g fdt, Vka ~(t, )

XI G„" (t, t, )g„'.„.(3)+ Gn, rn ( t t 1 )gka, ka ( t 1

where G„(t,t')=i(d (t')d„(t)), and G„" (t, t')i8(t ——t')( [d (t'), d„(t)] ) is the retarded Green

function. The time-dependent Green functions in theleads for the uncoupled system, which appear in (3), aregiven by

gka, ka(t t )=if(eka)e"P '

"topeka(

1)(4)

gk k (t, t')=i8(t' —t)exp —i dt, ek (t))where ek is the energy of the state k, with occupationf (ek ), when the system was prepared in the distant past,and the advanced Green function is related to the retard-ed Green function by g'(t, t')=[g "(t', t)]*. It is con-

I

[I ( et', t)] „=2' g p (e)V „(e,t)V* (e, t')aEL

Xexp i f dt, b, (e, t&), (5)

where p (e) is the density of states in channel a and theenergy, ek (t), of each state in the leads is separated intoa constant part ek =e and a time-dependent part b, (e, t)Rewriting the current in terms of the elastic coupling,and using matrix notation (denoted by an underbar) forthe level indices in the interacting region, we find

f dt' f Im Tr[e"' '~I (e t', t)fg 2m

X[G (t t')+fL(e)G "(t, t')]] .

(6)

JL(t) =—

An analogous expression applies for the current Aowingin from the right lead.

Equation (6) is the central formal result of this work.The strong resemblance to the steady-state result in Ref.8 means that the time-dependent problem is not concep-tually harder than the time-independent one. For a rangeof mesoscopic systems, a rigorous formula for thesteady-state current is a useful tool, and we believe (6)will be a useful tool to study dynamics. While interactingquantum transport is addressable with (6), we use it hereto explore dynamics in an exactly solvable, noninteract-ing system corresponding to a quantum-well structure.

In general, if interactions are neglected, the retardedand advanced Green functions are given by standardDyson equations, ' and the Keldysh Green function in(6) is related to them via

venient in (2) to turn the sum over momentum states k inthe leads into an integral over energies, and to define theelastic coupling between the leads and the states in the in-teracting region via

G (t, t )=i f dt, f dt G"(t, t, ) g f e ' ' fL/R(e)r ' (e, t~, t~) 6'(tgL, R

(7)

A tractable, noninteracting example of particular interestcorresponds to a double-barrier tunneling structure con-taining a single resonant level, eo(t)=co+6, (t), withenergy-independent coupling to two leads,

r«, (t', t)=rL/Rexp i, dtl~L/R(t

In (8), the time dependence in the leads is restricted to arigid shift of all states in the left (right) lead by b, L~R&(t)Within this framework, application of a voltage bias cor-responds to a shift of energies in one lead with respect tothe other, and generally a shift of the resonant level aswell. For energy-independent coupling to the leads, theretarded Green function for the resonant level is indepen-dent of energy shifts b,L/R(t) in the leads, and is givenby" G"(t, t')=exp[ —I (t —t')/2)g "(t,t'), where g "(t,t')is the retarded Green function for the uncoupled level,

g "(t, t') = i 8(t t')exp i d—t, eo(t, )——

I

leads.The resulting expression for the retarded Green func-

tion of the resonant level, G "(t, t'), can be used in (7) togenerate the Keldysh Green function, G (t, t'), whichcontains all the information concerning nonequilibriumoccupations'. Physical quantities such as the occupancyof the level and the currents through the barriers canthen be expressed directly in terms of G (t, t'), or, moreconveniently via (7), in terms of G "(t,t') and the occupa-tion function in the leads. The occupancy of the level,n(t)= iG (t, t), and th—e current, JL/R from (6), aregiven by

rL/R f fL/R(e)l &L/R(e, t)l, (lwG 6'

2

L, R 2m

8 1JL /R ( t ) rL /R n ( t ) +— d EfL /R ( e )

fi 7T

and where I = I L +I R is the total elastic coupling to the

TIME-DEPENDENT TRANSPORT THROUGH A MESOSCOPIC. . . 8489

where we have defined ~=& I «[fL(~) f~—(~ ] (13)

t&Lq&(~, &)= f dt, exp ie(r —t, )+i dt2bl~~(t, )

oo 1

XG "(r, r, ) . (12)

In the time-independent case, Al z~(e, t) is the Fouriertransform of the retarded Cireen function, and one findsthe usual result for the resonant-tunneling current,

I

While arbitrary time dependence of both the bias andlevel energy can be addressed via Eqs. (9)—(12), we havechosen to consider the response to a rectangular pulseand to an ac bias as examples of experimental relevance.From Eq. (12), one finds for a rectangular pulse of dura-tion s starting at t =0,

i (6 Ep+/I /2)(t s) i (6 E'p 6+51 /g +/I /2)se —eo+ i I /2 —( b, —b,l q~ ) [ 1 —e (1 0 liR )]w~, ~(e, r) =

(e co+i—I /2)(e eo —b, +b—.l &++i I /2) (14)

—10

0.1—

0L 06

Expression (14) for Ai~z(e, t) applies for times, r &s,after the pulse has ended. For times during the pulse,0 & t &s, one should replace s by t, which follows fromcausality. In Fig. 1(a), the current fiowing though thebarriers is plotted for a voltage pulse of durations =3'/I' (dashed curve). Before the pulse, the chemicalpotentials pL &z and the level energy eo are equal so thecurrent is zero. l3uring the pulse, energies in the left leadare increased by AL =10I and the energy of the reso-nant level is raised by 5=5I, appropriate for a sym-metric structure [inset, Fig. 1(b)]. (The choice b,z=0 is arbitrary since only the relative shifts ofenergy are significant. ) Initially the current througheach barrier grows proportional to t ln(1/t), ' and then

t

oscillates with a pulse-amplitude-dependent periodAtL )~ =2m A/I (pL ~~ +b I ~~ ) —(co+5 ) I [= 1.26 in Fig.1(a)]. For the case shown in Fig. 1(a), the occupancyremains fixed at n =0.5 by symmetry, and so the currentsthrough the two barriers must be equal. The "ringing" ofthe current rejects the movement of the sidebands ofIm[AL&z(e, t)] through the left and right Fermi ener-

gies. ' The decay time for the current oscillations is theresonance lifetime A/I, i.e., the time required for thestates comprising the resonance to fall out of phase.

There are two important expenmenta1 caveats. First,the time-dependent current Aowing in the contacts will

include capacitive contributions from the accumulationand depletion layers on either side of the tunneling bar-riers. ' However, there can be no net accumulation ofcharge over long times, so consequently, by time averag-

ing the current measured in the contacts, one also mea-sures the time average of (11). The "ringing" can be ob-

served in the time-averaged current by applying a seriesof pulses such as that of Fig. 1(a), and then varying thepulse duration. ' In Fig. 1(b), the derivative of the dccurrent with respect to pulse length is plotted, norma1-

ized by the repeat time ~ between pulses. For pulse

'0

g 0.50.4

30

20

2 3 4pulse length s

FIG. 1. (a) Time-dependent current J(t) through a sym-metric double-barrier tunneling structure in response to a rec-tangular bias pulse. All energies are in units of the elastic cou-pling to the leads I, the current is in units of eI /A, and alltimes are in units of A/I . Initially, the chemical potentials pLand p& and the resonant-level energy ep are all zero. At t =0, abias pulse (dashed curve) suddenly increases energies in the leftlead by hL = 10 and increases the resonant-level energy by 6= 5(see inset). At t =3, before the current has settled to a newsteady value, the pulse ends and the current decays back to zero.By symmetry, the occupancy of the resonant level is always 0.5and the currents through the two barriers are equal. The tem-perature in all figures is k&T=0. 1I . (b) Derivative of the in-tegrated dc current Jd, with respect to pulse duration s, normal-ized by the interva1 between pulses ~. For pulse durations muchlonger than the resonance lifetime R/I, the derivative is just thesteady-state current at the bias voltage, but for shorter pulsesthe "ringing" response of the current is evident.

0. 1

0.24—I

1 2 3 4 5 6time t

10

—0

0.22

0.2

0. 18

3 4period 27r/u

FIG. 2. (a) Time-dependent current J(t) through a sym-rnetric double-barrier tunneling structure for an ac bias of fre-quency co=2I /A (dashed curve). The ac driving amplitude isAL =10 about pL =10 in the left lead, 6=5 about op=5 for thelevel, and 6& =pz =0 in the right lead (see inset). By symmetryn =0.5 and the currents through the two barriers are equal. (b)Time-averaged current Jd, as a function of the ac oscillationperiod 2~/m. The dc amphtudes are the same as those in (a).

8490 NED S. WINGREEN, ANTTI-PEKKA JAUHO, AND YIGAL MEIR 48

lengths s of the order of the resonance lifetime fi/1, thederivative of the dc current mimics closely the time-dependent current following the pulse, and, likewise,asymptotes to the steady-state current at the new voltage.

The second caveat is that the treatment of time depen-dence in the leads assumes that charge in the accumula-tion layers has time to follow the external bias. The fre-quencies in the driving signal must therefore be smallerthan the plasma frequency and also small enough that the

I

capacitive current is a small fraction of the incident Aux

of electrons, which assures that the change of occupationof each state in the leads is negligible. For the experi-ment of Brown et al. , this implies an upper limit of —10THz, already a large frequency by experimental stan-dards, which limits the rise time of a pulse.

Instead of applying a bias pulse, it may be experimen-tally more practical to apply an ac bias to the tunnelingstructure. From Eq. (12), one finds for an ac potential,

~L /RAL /g (E, t) ='exp i — sin(cot) g Jk

%co k = —oo

exp( ik cot )

e—eo —kAco+i I /2(15)

wllele AL/R (t) =5L/R cos(co, t), 5(t) =6 cos(cot), and J„is the kth-order Bessel function. In Fig. 2(a), the currentis plotted for an ac potential of frequency co=21 /A'.

While the current has the same period, 2m. /co, the com-plex time dependence inside each period is similar to the"ringing" response to a pulse shown in Fig. 1. Oneconsequence of this complex harmonic structure is thatfor temperatures k~ T & A'co the time-averaged current Jz,oscillates as a function of period 2m/co, as shown in Fig.2(b). An oscillation occurs whenever a photon-assistedresonant-tunneling peak aligns with one chemical poten-tial, so+ kAco =pz&~, giving oscillations periodic in 1/co.

In conclusion, a general formula (6) for the time-dependent current through an interacting mesoscopic re-gion has been obtained in terms of local Green functions.While for the interacting case only approximate Green-function solutions may be available, for a single nonin-teracting level with energy-independent coupling to twoleads an exact solution has been obtained [(9)—(12)]. Thiscase corresponds experimentally to resonant tunneling

through a double-barrier structure. We find that tern-poral coherence of electrons tunneling through the reso-nant level leads to "ringing" of the current in response toan abrupt change of bias, and similarly complex timedependence in response to an ac bias. This "ringing" canbe observed experimentally in the dc current by varyingthe pulse length in a train of voltage pulses. We hopethat time dependence will provide a new window oncoherent quantum transport and will lead to significantnew insights in the future.

We thank S. J. Allen, M. Kastner, and L. Kouwenho-ven for valuable discussions. One of us (N.S.W. ) grate-fully acknolwedges the NORDITA mesoscopic programfor hospitality during early stages of this work. Work atUCSB was supported by NSF Grant No. NSF-DMR90-01502 and by the NSF Science and TechnologyCenter for Quantized Electronic Structures, Errant No.DMR 91-20007.

Also at MIC, Technical University of Denmark, DK-2800Lyngby, Denmark.

For a review, see P. A. Lee and T. V. Ramakrishnan, Rev.Mod. Phys. 57, 287 (1985).

For a series of review articles, see Mesoscopic Phenomena inSolids, edited by B. L. Altshuler, P. A. Lee, and R. A. Webb(Elsevier, Amsterdam, 1991).

J. F. Whitaker et al. , Appl. Phys. Lett. 53, 385 (1988); E. R.Brown et al. , ibid. 58, 2291 (1991).

4L. P. Kouwenhoven et al. , Phys. Rev. Lett. 67, 1626 (1991);fora recent review, see Single Charge Tunneling, edited by H.Grabert, J. M. Martinis, and M. H. Devoret (Plenum, NewYork, 1991).

5P. S. S. Guimaraes et al. , Phys. Rev. Lett. 70, 3792 (1993).A linear-response formula for the time-dependent noninteract-

ing current was recently derived by Y. Fu and S. C. Dudley,Phys. Rev. Lett. 70, 65 (1993), and a general formula, fortime-independent leads, was obtained by J. H. Davies et al. ,Phys. Rev. B 47, 4603 (1993).

7The time-averaged current in response to an ac bias has beenanalyzed previously for this model: D. Sokolovski, Phys.Rev. B 37, 4201 (1988); P. Johansson, ibid. 4I, 9892 (1990).Johansson also considers the time-dependent current in aphenomenological approach.

Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992).9L. V. Keldysh, Zh. Eksp. Tear. Fiz. 47, 1515 (1964) [Sov. Phys.

JETP 20, 1018 (1965)]; C. Caroli et al. , J. Phys. C 4, 916(1971); S. Hershfield, J. H. Davies, and J. W. Wilkins, Phys.Rev. Lett. 67, 3720 (1991).G. D. Mahan, Many-Particle Physics (Plenum, New York,1990).

N. S. Wingreen, K. W. Jacobsen, and J. W. Wilkins, Phys.Rev. B 40, 11 834 (1989); D. C. Langreth and P. Nordlander,ibid. 43, 2541 (1991)~

The initial change in current (11) is due to a change in

AI /&(e, t). The change 6AL/&(e, t) has an overall amplitudeproportional to t, and a long tail (~1/e) out to energies

~ e~ -A/t, which leads to a growth of the current ast f"„,de/e-t ln(1/t).In a previous calculation of time-dependent resonant-tunneling currents, the phase coherence which produces thesesidebands is explicitly absent, and so only exponential relaxa-tion is found following an abrupt change of bias [L. Y. Chenand C. S. Ting, Phys. Rev. Lett. 64, 3159 (1990);Phys. Rev. B43, 2097 (1991)].K. L. Jensen and F. A. Buot, Phys. Rev. Lett. 66, 1078 (1991).Leo Kouwenhoven (private communication).