PHYSICAL REVIEW B VOLUME 40, NUMBER 3 15 JULY 1989-II

Landauer's conductance formula and its generalization to finite voltages

Philip F. Bagwell and Terry P. OrlandoDepartment of Electrical Engineering and Computer Science, Massachusetts Institute of Technology,

Cambridge, 'Massachusetts 02139(Received 16 December 1988; revised manuscript received 14 March 1989)

We recast the expression for tunneling current to show that the energy averaging due to a finite

voltage is statistically independent from the energy averaging due to finite temperature. The energy

averaging takes the mathematical form of convolutions. Convolving the standard Landauer con-ductance formula in energy with a voltage windowing function averages the conductance over an

energy range equal to the applied voltage, just as the independent convolution with the Fermi-Diracprobability density function averages the conductance over an energy range equal to the tempera-ture. We illustrate the effects of voltage broadening versus ordinary thermal broadening of the con-ductance using a quasi-one-dimensional ballistic conductor as a model system, as in the recent ex-

periments of van Wees et al. [Phys. Rev. Lett. 60, 848 (1988)] and Wharam et al. [J. Phys. C 21,L209 (1988)].

I. INTRODUCTION

Since Landauer's original work' relating electricalconductance to the transmission probability through a re-gion of elastic scatterers between two temperature baths,his formula has been extended to the case of multiplequantum channels, ' extended to include inelasticscattering within the sample, applied to many differentexperimental geometries, ' and derived from linear-response theory. ' A review of the two types of Lan-dauer formulas, discussing the different measurementconditions under which each applies, is given in Refs. 6,8, 9, and 10. References 6, 8, and 10 review the higher-dimensional analogs of these two types of Landauer for-mulas. The extension of Landauer's formula to finitevoltages has also been investigated. "'

The main goal of this paper is to express the effects ofenergy averaging due to finite voltage and finite tempera-ture on the electrical conductance mathematically interms of convolutions. This objective is important be-cause convolutions are a very natural way to incorporaterandom processes leading to a broadening of energy lev-els, as is the case with the energy level broadening due toelastic and inelastic scattering in a normal dirty metal ~

'

We show in Sec. II that to generalize the zero-voltage andzero-temperature Landauer conductance formula to finitevoltage and finite temperature, one merely convolves it inenergy with both a voltage windowing function and athermal smearing function, respectively. These two con-volutions show that the effect of finite temperature is toaverage the conductance over a region kz T near the Fer-mi level, while one effect of finite voltage is to average theconductance over an energy range e V near the Fermi lev-el, as previously mentioned in Refs. 11 and 12. By prov-ing that the difference in Fermi functions characterizingelectrical conduction can be written as a convolution of

two separate functions, one function depending only onthe applied voltage and the other depending only on tem-perature, we show in this paper that voltage broadeningand temperature broadening are statistically independent.We apply this convolution broadening method to aquasi-one-dimensional ballistic conductor to understandthe qualitative similarities and differences between volt-age broadening and ordinary thermal broadening of theconductance. By emphasizing that electrical conductionnecessarily occurs at finite voltages, we gain additionalinsight into the physical mechanism responsible for thequantum contact resistance in one-dimensional ballisticconductors. In Sec. III we examine Landauer's formulain the classical diffusive limit in one dimension, showingthat it reduces to the standard Drude conductance result.We show that one also obtains the quantum contact resis-tance from the Drude-Sommerfeld conductance formulain the limit that the mean free path grows to its upperbound of the device size.

A subsidiary goal of this paper is to examine the rela-tionship of Landauer's conductance formula to previoustreatments of electron tunneling at finite voltages. In thetunneling limit, by which we mean many parallel quan-tum channels with no scattering between them, we showin Sec. IV that the finite-voltage Landauer formulareduces to many well-known treatments of electron tun-neling at finite voltages. ' ' ' However, even in thiswell-known tunneling limit, we can gain physical insightinto the shape of the current density versus Fermi energyin different spatial dimensions by noting that, because thetunneling Hamiltonian is separable, we can obtain thetunneling current density in two spatial dimensions byconvolving the one-dimensional tunneling current withthe one-dimensional free-electron density of states, re-peating this process to obtain the three-dimensional tun-neling current density.

1456 1989 The American Physical Society

LANDAUER*S CONDUCTANCE FORMULA AND ITS. . . 1457

II. LANDAUER'S FORMULA IN ONE DIMENSION

Our model for a quantum conductor in one dimensionis a potential U(x, V) between two thermal reservoirs asin Fig. 1, where U depends both on position x and the ap-plied voltage V. The potential U(x, V) depends on theapplied voltage because electrons reAected from a scatter-ing obstacle pile up on one side of the obstacle, as em-phasized in Ref. 1, leading to a density gradient andnonuniform electric field concentrated around the obsta-cle. These density and potential gradients in turninAuence the tunneling potential so that U(x, V) must bedetermined self-consistently, and our treatment totallyneglects such effects. We emphasize that the self-consistent determination of U(x, V) is the central issue inany specific calculation of tunneling currents. An ap-proximate method to self-consistently determine the tun-neling potential at finite voltages, but which still assumesa uniform electric field, is given in Ref. 11. In this paperwe will seek results which will not depend strongly on theexact shape of the tunneling potential, as long as the elec-tron tunnels through some U(x, V).

Electrons deep inside the two reservoirs are assumed tohave a Fermi-Dirac distribution shown in Fig. 1. We as-sume the effect of an applied voltage between the tworeservoirs is to create an imbalance in the electrochemicalpotentials deep inside the reservoirs equal to the appliedvoltage. This results in a current'

I(p, V, T) =e fU+(E)T(E, V)N+(E)

X [f(E —(p+eV), T)

f (E —p, T—)]dE,where U+(E) is the group velocity for electrons movingin the positive x direction, N+(E) the density of statesfor electrons of both spins moving in the positive x direc-tion in a one-dimensional free-electron gas, f is the Fermifunction, T the temperature of both reservoirs, and

T(E, V) the transmission coeKcient through the potentialU(x, V). By enforcing time reversal symmetry andcurrent conservation during tunneling, it is possible toshow that the current transmission probability throughthe potential U(x, V) is the same from left to right asfrom right to left, i.e., T(E, V)=T~, (E, V)=T,&(E, V).While Eq. (1) is intuitive, it is also possible to derive it us-ing the transfer Hamiltonian method and Fermi's golden

12, 14

The product of the group velocity and the electrondensity of states in one dimension is a constant given by

0+(E)N+(E) =dk

1 1 1

~I dE/dkl

The Fermi function in Eq. (1) can itself be expressed as acomposite probability function

f (E —p, T)= [1 0(E ——p)] — (E, T)dE

where (3 denotes a convolution in energy, 0 is the unitstep function, and

d= IO(E —p) —8[E—(p+eV)]I — (E, T)dE

is the thermal smearing function or Fermi-Dirac proba-bility density function. The convolution (3 of two func-tion A (E) and B (E) has its usual meaning,

A (E)B (E)= f A (E E')B(E')d—E'

= f A (E')B(E E')dE' .— (5)

The difference of Fermi functions in Eq. (1) can thereforebe written as

f(E (p+e—V), T) f (E —p, T)—

p. +eVOO| ) P).~

ee00 jp. .

left Ideal

reservoir conductor

U(x v)

Scattering

potential

'Ideal

conductor

e~e+oeoee

Right

reservoir

(6)

Equation (6) shows that the difference of Fermi functionsin Eq. (1) can be expressed as the convolution in energy oftwo independent functions, one of which depends only onthe applied voltage and the other which depends only onthe temperature. Equation (6) therefore separates theeffect of finite voltage on the electrical current from theeffect of finite temperature.

Our expression for the current now becomes

Xp

FICx. 1. Model for a one-dimensional quantum conductor:An elastic scattering potential U(x, V) located between twothermal reservoirs. p& and p& are the electrochemical potentialsin the left and right ideal conductors, respectively. An appliedvoltage V creates an imbalance between the electrochemical po-tentials p~ =p+ e V in the left reservoir and p, =p in the rightreservoir. Note that plop& and p,&p2.

I(E, V, T)= T(E, V)s W(E, V)s — (E, T)

2pT(E, V) . (7)

Equation (7), where W(E, V)=[9(E+eV)—8(E)], is ourinterpretation of the finite-voltage Landauer formula inone dimension in terms of convolutions. T(E, V) is theenergy averaged transmission coefficient. After carryingout the convolutions we set E =p. We display Eq. (7)

1458 PHILIP F. BAGWELL AND TERRY P. ORLANDO 40

graphically in Fig. 2. At this point we ignore the last twoconvolutions shown in the figure. If both e V and kz T aresmall compared with any structure in the transmissioncoefficient T(E, V), Eq. (7) reduces to the well-known re-sult

e VI (E)= T(E)V =mA' R,

(8)

W (E,Vj

—)E (E,Tj

I (E,V,T) = —'5 T(E,V) s S-eV 0 =E

g lo

lcm'

-E

FIG. 2. Eft'ect of finite temperature and finite voltage on thetunneling current displayed graphically as independent convolu-tions. Energy averaging due to finite voltages is included in thevoltage window 8'(E, V), while the energy averaging due tofinite temperature results from the thermal smearing function—df (E, T)/dE. The two convolutions with the one-dimensional free-electron density of states relate tunnelingcurrent density in three dimensions to the one-dimensional tun-neling current.

which is Landauer's conductance formula in one dimen-sion. Here R, is the total resistance of the electron mov-ing from one contact to the other. This formula was also

. obtained explicitly in Ref. 12 in a discussion of tunnelingin one dimension between two thermal reservoirs.

Equation (7) has a simple physical interpretation: Theconvolution with the voltage window W(E, V)= [0(E +e V) —0(E)] implies that the electron beam nearthe Fermi surface contributing to conduction is notmonochromatic; it has a width in energy of eV. The box-like shape of W(E, V) as shown in Fig. 2 argues that theelectron beam near the Fermi surface has a uniform prob-ability to have any energy between the electrochemicalpotential in the right contact and that energy plus eV.Equivalently, the average tunneling electron near the Fer-mi energy has a variance about its mean energy ofAE=eV due to the applied voltage. ' The convolutionwith df IdE is ju—st the standard thermal broadening ofthe conductance. Just as thermal broadening of the con-ductance makes it necessary to average the zero-temperature conductance over an energy range kz T nearthe Fermi level, finite voltages require averaging the con-ductance over an energy scale eV near the Fermi level.The separate convolutions imply that thermal broadeningand voltage broadening are statistically independent.Note also that the phase coherence time ~v=6/eV intro-duced by the nonmonochromatic electron beam at finitevoltages is analogous to the better-known phase coher-ence time at finite temperatures ~z-=A/k~T, both times

being set approximately by the variances of W(E, V) and—df (E, T)/dE, respectively.

Equation (8) predicts that a conductor having perfecttransmission T(E)=1 has a contact resistance ofR, =h/2e . This result seems contrary to one's intuitionfrom the Drude formula as well as from Landauer's origi-nal formula, ' which gives the device resistance Rd as

h R(E)2e' T(E) (9)

where n (EF )=N (EF )e V is the carrier density nearthe Fermi level. As the Fermi level is raised, the Fermivelocity increases but the carrier density decreases.Equation (2) tells us that, in one dimension, these two fac-tors exactly cancel each other for all energies. The con-tact resistance in Eq. (8) is then a consequence of theseemingly fortuitous cancellation in Eq. (2). We wish topropose a heuristic explanation for Eq. (8), which does

Here R (E) is the reflection coefficient so thatT(E)+R (E)=1. Biittiker discusses the assumptionsnecessary to obtain Eq. (9). Both the Drude formula aswell as Landauer's original formula, Eq. (9), assert that alength of perfect conductor has zero resistance. ' It isnow understood ' that whether one obtains Eq. (8) or (9)in an actual experiment depends on how the resistance ismeasured. Connecting two weakly coupled voltageprobes to the ideal conductors in Fig. 1, which measurethe electrochemical potentials p, and p2 as defined in Ref.18, one obtains the original Landauer formula

Vd=(pi —p2)/e =IRd=I(h/2e )(R/T),

which is Eq. (9). Landauer's original formula for Rd cor-responds to the standard four-probe geometry formeasuring resistance, in which the voltmeter is onlyweakly coupled to the sample one wishes to measure andno current Aows into the voltmeter. If, on the otherhand, one wishes to obtain the resistance using a two-probe geometry, which measures the electrochemical po-tentials IM+ e V =

p& and p =p„one obtains

V, = V=(p& —p„)/e =IR, =I(h/2e )(1/T)

which is Eq. (8). In this case the voltmeter is the reser-voir, which is strongly coupled to the system one wishesto measure. Due to the incoherence introduced by thereservoirs in Fig. 1, the total resistance R, is simply thesum of the device resistance Rd plus the contact resis-tance R„ i.e., R, =Rd+R, . That R, is in fact a contactresistance associated with the introduction of the tworeservoirs in Fig. 1 was explained by Imry' in terms ofthe efFusion of a gas through a small hole, and byButtiker ' who emphasized that carrier motion from onereservoir to the other was essential to obtain the contactresistance.

The contact resistance implied by Eq. (8) follows fromthe argument in Eq. (1) that the current is simply a prod-uct of the carrier charge, velocity, and density. For smalltemperatures and voltages this gives

2

I=evF n (Ez;)=evF+[N+(Ez)eV]= V,

LANDAUER'S CONDUCTANCE FORMULA AND ITS. . . 1459

not rely on Eq. (2), and which emphasizes the way inwhich current is drawn out at, a contact.

To gain further insight into the contact resistance andmake a plausibility argument for Eq. (8), introduce thetimes

I= = T(E)V and I= = V~, +~d vrA ~A

(10)

T' he time ~d corresponds to the original Landauer formu-la Rd while the time ~, corresponds to the contact resis-tance R, . If the conductor is in the classical diffusivelimit, the time ~d is proportional to the time for a classi-cal particle to diffuse across the conductor. '

We can now make a plausibility argument for Eq. (8).We consider a single quantum ballistic channel havingunity transmission so that the resistance Rd is negligible,i.e., ~d —+0. That the conductance of the perfect wire byitself is infinite one can understand simply from theDrude formula, in which an impulse function electricfield excites a current which flows forever without decay-ing. Our emphasis must be on understanding the resis-tance R„which we do by arguing for the time ~, as fol-lows: Consider the case of zero temperature. Far awayfrom the interface between the device and the reservoir,the right contact must be described by a Fermi functionhaving a single current-carrying state at the electrochem-ical potential p„. As the right contact rapidly draws outan electron at energy p„another electron can occupythat energy level. It cannot, however, do so instantane-ously. When the average tunneling electron near the Fer-mi surface reaches the right contact, it must dissipate onaverage an amount of energy AE =e V in that contact.Even if inelastic scattering processes occur on very rapidtime scales in the contact, there is a limit imposed by theuncertainly principle for the rate at which energy can bedissipated into the thermal bath, namely 1/hr=b, E/fi.The time h~ to dissipate an amount of energy AE =e Vinto the contact should be the same order of magnitudeas the time ~, to equilibrate with the measurement reser-voir. This equilibration step will limit the current Howbecause, if the electron is out of equilibrium with themeasurement reservoir, the reservoir cannot detect theelectron. Therefore, only one electron of charge e canAow across the device every A/eV sec. Imposing thisrate-limiting process in the contacts, we argue that theelectrical current even in this ballistic case is

which define the effective time of an electron in the device7 d and the contact ~, . One obtains for these times

2~Pi 2~A' 1 —T(E)—'Ty and 'Td =.V ' ' .V T(E)

be transported into the right contact even in a perfectlyballistic device. Since the rate-limiting step for thecurrent occurs in the contact, not in the device itself, theconductance formula (7) is independent of the length ofthe device as T(E )~ 1. The contact resistance is alsoplausible from the perspective that the voltage sources,which inhuence only the electrochemical potential deepinside the reservoirs, have no way of knowing that anelectron has, in fact, transited the device until it comesinto equilibrium with a Fermi function inside the reser-voir. While the above argument is only heuristic, we be-lieve it can be placed on firmer mathematical grounds byanalogy with the onset of resistance in narrow supercon-ductors via "phase slips. "

The factor of T(E) which enters into Eq. (8) can alsobe understood from an energy dissipation viewpoint:Only those electrons which in fact transit the device dissi-pate energy and wind up contributing to the resistance.We also note that the analog of Eq. (9) at finite voltageshas been suggested in Ref. 11 to beI/Vd =(2e /A)(T/R ), where T is simply the energyaveraged transmission coefficient defined in Eq. (7) andT+R =1.

Consider now a quasi-one-dimensional ballistic wiredescribed by the separable potential U (x,y, V)= U, (x, V)+ U (y), where U (y) is a confining potentialwhich quantizes the electron energies perpendicular tothe direction of transport. The separable Hamiltonianimplies that there is no scattering between the differentsubbands created by the confining potential U~(y). Equa-tion (7) therefore generalizes, as we show in Sec. IV, tosimply a sum over all the possible paths for electrons totransmit between the left and right contacts:

I(E, V, T)= g T, (E, V)e W(E, V)e — (E, T)l

where the sum is over all occupied subbands. For aballistic quasi-one-dimensional wire with U (x, V) =0, wehave

T;(E, V)=()(E —E;)

at moderate voltages. Here E, is the subband energy forquantized motion perpendicular to the direction of trans-port. The limiting case of this formula at zero tempera-ture is when the thermal smearing function becomes a 5function so that

I(E, V, T=O)= +8(E E, )e W(E, V) . —

e e e(12)

We can carry out the convolution to obtain

8(E E, )e W(E, V)—which is the same order of magnitude as Eq. (8). Thiscontact resistance corresponds to energy dissipation inthe reservoirs, since no energy can be dissipated by elasticscattering in the sample. For the electron to dissipate anamount of energy e V in the reservoirs requires some time~, =2M/eV, which limits the rate at which current can

0, E(E;—eV

E —(E; —eV), E; —eV&E &E; .

eV, E)E;We sketch the current in a ballistic quasi-one-

1460 PHILIP F. BAQWELL AND TERRY P. ORLANDO

T=QT-Q

V "=0

V =0.5mV

III. CLASSICAL DIFFUSIVE LIMITOF LANDAUER'S FORMULA

""""T=0.6K V-"-0p

C3 ~-

Pr

oE~-eV

0.5meV

I

Ee-eV Eq

p. (Energy)Ei

FIG. 3. Ballistic contact resistance in a quasi-one-dimensional quantum wire illustrating the effects of finite volt-ages and finite temperature. The conductance is shown versusthe electrochemical potential p in the right contact. The finitewidth of the steps in conductance is inversely proportional tothe phase coherence time introduced by finite voltage or finitetemperature.

dimensional wire with several subbands versus the elec-trochemical potential p in the right contact in Fig. 3.The conductance 6 is defined as the current divided byvoltage, even at finite voltages. These steps in the ballis-tic conductance have been observed ' and explained us-ing formalisms similar to the one in this paper as quan-tum contact resistances. The conductance at finite volt-ages, given by the dashed line in Fig. 3, shows initially alinear dependence versus Fermi energy instead of a stepat zero temperature. We also show the effect of finitetemperature on the conductance in Fig. 3 for comparison.Both finite temperature and finite voltage smear out thesharp steps in the zero-temperature conductance. Themaximum slope of these steps in the conductance versusFermi energy occurs at conductance values half-way be-tween the plateaus, and is given by

We are interested in the case with there are n scatterersrandomly distributed in one dimension between tworeservoirs as in Fig. 4. We consider the diffusion of clas-sical particles down the chain of scatterers. EvaluatingLandauer's formula for this case requires evaluating thetransmission probability of a classical particle down achain of n identical scatterers. This can be treated as aclassical random walk down the n scattering sites. Theclassical transmission probability T„down the total chainof n scatterers can be obtained in terms of a recursion re-lation

1 Tn —1

1 —RiR„where T„+R„=1 . (19)

The solution of this recursion relation, which can bemotivated from physical arguments involving the addi-tion of classical resistors in series, is

P1

1

(R, /T, )n+1 (20)

21

~A (R, /T, )n +1 (21)

If we chose the scattering centers to be different, withscatterer a having a transmission coefficient T„etc., weobtain for the resistance R &D after solving a recursion re-lation similar to Eq. (19)

where Ti is the probability of a classical particle beingtransmitted by a single scatterer. This result has been ob-tained in Ref. 2, and the special case with T=R =

—,'

treated in Ref. 23 using a different method. Substitutingthis classical transmission coefficient into Landauer'sconductance formula in one dimension, Eq. (8), gives thediffusive conductance

dG e 1 e ~Tfor V=O

dp M 4k~T mA(17)

Ra Rb Rc'+ + '+. . +1, (22)

at finite temperature and

dG e 1 e &yfor T=O

dp ~A eV ~Pi

which is just the classical sum of the original R /T Lan-dauer resistance from each scatterer plus the contactresistance.

If we choose T, =R, =—,' for each scatterer, corre-

sponding to the assumption of isotropic scattering in the

at finite voltages. The maximum slope of the steps inconductance versus Fermi energy is limited by the phasecoherence times ~T and ~z, and is proportional to thephase coherence time. In a field effect transistor wherethe chemical potential is proportional to the gate biasvoltage V, and if the chemical potential were controlledby such a gate, one could interpretg =dI/dV =eVdG/dp as the transconductance of thedevice. For such a structure, the transconductancewould have a finite maximum value even in the ballistictransport regime. When the applied voltage is muchgreater than the temperature, this maximum transcon-ductance in the quasi-one-dimensional ballistic wire be-comes just g =2e /h.

1 2 545 n

L =(n+I ) g

FIG. 4. n randomly positioned scatterers between twothermal reservoirs in one dimension. The two reservoirs arealso regarded as scatterers so that the mean separation I be-tween scattering centers is L =(n +1)l, where L is the samplelength.

LANDAUER'S CONDUCTANCE FORMULA AND ITS. . . 1461

conductor, each elastic collision with a scatterer com-pletely randomizes the particle's momentum direction.Our form for the classical diffusive conductance, Eq. (22),suggests that we interpret the contact resistance on theright-hand side as stating that the thermal reservoirscompletely randomize the momentum direction of theelectron when it enters the reservoir. The reservoir cap-tures all particles incident upon it, then immediately ran-domizes their momentum direction due to rapid scatter-ing inside the reservoir. This assumption is included inour derivation of the conductance formula (8) by assum-ing the reservoirs are described by a Fermi distribution.

We now wish to compare Eq. (21) to the standardDrude-Sommerfeld conductivity expression in one di-mension,

2 e0 1D e N(EF )D(Ez ) =e ~ vp&«=2 I, (23)

2e' lt

L mAL(24)

In order for the diffusive conductance from Landauer'sformula, Eq. (21), to equal the Drude conductance in Eq.(24), we must conclude that

where l«=vF~« is the transport mean free path at theFermi level and D (Ez ) =Uz~« is the diffusion coefficient.To obtain the conductance in one dimension, we merelydivide by the length of' the sample:

the arrangement of the n scatterers between two thermalreservoirs as shown in Fig. 4. The reservoirs are regardedas being two additional scatterers which absorb all car-riers incident upon them while simultaneously randomiz-ing the carrier s momentum direction, so they must be in-cluded in computing the mean free path /. From thegeometry of the scatterers we therefore have for a givenscattering configuration that L =(n + 1)l. The mean freepath in the average configuration of the scatterers istherefore L =2(n +1)l as in Eq. (26). We see again thatthe Drude conductance, given by Eq. (24), andLandauer's conductance formula, Eq. (8), in the classicaldiffusive limit are the same.

The Drude conductance formula and the Landauerconductance formula can also be made equivalent in theballistic limit of no scattering between the thermal reser-voirs (n =0), if we reinterpret the meaning of the trans-port mean free path which enters the Drude conductancein Eq. (24). The standard interpretation of the transportmean free path /„=vF~« is that ~„ is the scattering timein an infinite system. In this standard interpretation /«can increase without bound. If we now reinterpret l«asthe average distance for the conduction electron to ran-domize its momentum direction in a finite sample, we seethat the longest possible transport mean free path for anygiven sample is just the sample length. For the averagesample described by Eq. (24), this argument gives2l« =2l =L, in which case the ballistic conductance is

l«2

1

(R /iT, )n +1 (25)

2

Glr =~ (27)

To show that Eq. (25) is correct, consider the special caseof isotropic scattering in the conductor. HereT& =R

I=

—,', so that the transport mean free path is equalto the mean distance between the scatterers, i.e., l«=l.In that case Eq. (25) becomes

1

L n+1 (26)

In the limit that n becomes large, Eq. (26) implies thatthe mean free path is half the average spacing betweenscattering centers. We would obtain this result for themean free path in one dimension if we average the dis-tance to collide with a scatterer over a uniform distribu-tion of all possible starting positions for the randomwalk. This statistical average is consistent with the stan-dard assumption used in deriving the Drude conductivi-ty, in which we consider an electron executing a randomwalk inside an "average" conductor that is spatially uni-form. We can either keep the scatterers fixed and aver-age over all possible starting positions for the randomwalk, or we can keep the origin of the random walk fixedand average over all possible impurity locations. Eitherway we obtain that the mean free path which should ap-pear in the Drude formula, Eq. (24), is half the averagespacing between scatterers. Consequently, for large n, Eq.(26) is correct, so that the Drude conductance isequivalent to the diffusive limit of the Landauer conduc-tance. Moreover, even for small n, Eq. (26) can be shownto be valid by the following argument: Consider again

as before. The contact resistance can be thought of fromthis viewpoint as a consequence of the electron's trans-port mean free path being limited to at most the samplelength.

We can gain further insight into the Drude-Sommerfeld conductance by considering a quantum par-ticle moving in a set of potential barriers in one dimen-sion, with barrier a having a quantum-mechanicaltransmission probability T„etc. In computing the totaltransmission probability of a quantum particle down thischain of scatterers one cannot ignore the phase of theparticle. The total transmission coefficient down thechain will therefore be much more complicated than inEq. (22). Biittiker has worked out the general case forthe addition of quantum resistors in series. In the casewhere the particle inelastically scatters with probabilityone between each elastic scatterer, his results can be writ-ten very simply as

2

~ R1DR, +1 +T.

+ ~ ~ 0 + 1

Rb R,+1 + +1b t~c

(28)

The quantum particle picks up the contact resistanceeach time it inelastically scatters. Equation (28) was de-rived for zero applied voltage. In the case of a finite volt-age, we must average each transmission and reAectioncoefticient over the energy dissipated into the inelasticscatterer. This energy averaging will be of the order of

1462 PHILIP F. BAGWELL AND TERRY P. ORLANDO

the applied voltage divided by the number of additionalinelastic scatterings inside the sample. If the number ofinelastic scattering events increases with temperature,then the energy averaging due to voltage and temperaturewill be related, in contrast to the case where there is noinelastic scattering inside the device. Another interestingproperty of Eq. (28) is that the total transmissioncoefTicient decreases linearly with the length of the 1Dchain as expected if the motion is diffusive. It is generallybelieved that, for most types of disorder, the resistance ofa 1D chain without inelastic scattering increases ex-ponentially with the length of the chain. Therefore, itappears that inelastic scattering processes are necessaryto obtain diffusion in 1D. Equation (28}, if comparedwith the Drude conductance in Eq. (24), givesMatthiessen's rule it,

' =1,&

' +1;„', where L /2l,&=(R, /T, )n for the case of identical barriers and

L/2l;„=n +1.

IV. TUNNELING LIMIT OF LANDAUER'S FORMULAIN TWO AND THREE DIMENSIONS

We consider here the generalization of Landauer's con-ductance formula to include many quantum channels orsets of good quantum numbers in the ideal device leads asdescribed in Refs. 3 and 10. Define the current transmis-sion probability T;. from channel j in the left reservoir tochannel i in the right reservoir evaluated at a given ener-gy as

j,left(29)

Xf&(E —(p+eV))[1 f,(E p)]dE—. (30—)

The current from right to left is similarly

I,&=pe f v, (E)N, (E)T/;(E, V)f,. (E —p)

X[1 f,(E —(p+eV—))]dE . (31)

Here v+ =v and X+ =X =X/2.Current conservation along with time reversal syrnme-

try give the constraint tlj tjl implying that Tlj Tjl.The net current is therefore

f TJ(E, V)[f,(E (p+eV)) f,(E —p)]dE . — —1,J

(32)

Equation (32) reduces to the general result

when all other incoming currents are zero. Here t; is thecurrent transmission amplitude as defined in the scatter-ing matrix. ' The current from left to right, in the spiritof the derivation in Ref. 14, is therefore obtained as

I&, =g e f v/+ ( E)X~+(E)TJ (E, V)

I = g T,,(E, V)e W(E, V)@e df (E, T)

1,J

where W(E, V) is again the voltage window given by

W(E, V)=[8(E+eV) 8(—E)] .

(33)

Equation (33} shows that temperature broadening andvoltage broadening are statistically independent even inthis multichannel case. Equation (33) reduces at low tem-peratures and small voltages to the well-known result ' '

eV eV eV2M . . '' 2m' 'J '' 2vrR

17J 1,J(34)

where the sum in Eq. (34) also runs over the individualspin quantum numbers, thus differing by a factor of 2from Eq. (33). Equation (34) is analogous to the Lan-dauer formula proportional to T, Eq. (8) describing atwo-probe measurement. References 3, 6, and 10 give themultiple channel Landauer formula which describes afour-probe measurement, analogous to the T/R Lan-dauer formula in Eq. (9).

For a general T;, Eq. (34) allows the possibility thatelectrons can scatter between different quantum channels.We are interested in the tunneling limit of this formula,for which T;.= T;6; -. The current transmission probabil-ity takes this form only if the Harniltonian is separable as8„„~=8(x)+8' (y)+8, (z), thus justifying Eq. (13) inour earlier treatment of the ballistic quasi-one-dimensional wire. This tunneling limit corresponds tothe classical addition of resistors in parallel, i.e., there isno communication between the parallel resistors. Be-cause there is no scattering between the different con-ducting channels inside the sample, we only have to sumover one channel index in Eq. (34), namely,

eV eV2~&" =2.~&'1&J 1

(35)

Expressing the multichannel Landauer formula, Eq. (35),in terms of a current density then gives

I3D

y z

T(E —E (k )—E,(k, )) .

2M Ly Lz k k gplz

(36)

This limit also corresponds to the usual treatment of elec-tron tunneling, for which we assume a static potentialhaving spatial variation only along the x direction. ' ' 'To prove this, note first that the Harniltonian is separableso that E =E„+E +E,. Furthermore, the eigenfunc-tions along y and z are plane waves so that E» =A' k» /2mand E, =Pi k, /2m. The transverse channel index i can belabeled by i =(k, k, ). Translational invariance along the

y and z directions implies there is no scattering potentialto couple the transverse channels. We enforce conserva-tion of energy by writing

T,.(E„)=T(E E(k ) E,(k, ))—. —

LANDAUER'S CONDUCTANCE FORMULA AND ITS. . . 1463

Recognizing the one-dimensional identity

2 g~fNg(E )dELy k,

(37)

in the previous section, we note that the multiple-channelLandauer formula has been applied to consider voltageAuctuations in small diffusive conductors.

where N &'D'(E) is the one-dimensional free-electron densi-

ty of states including spin1 j2 1/2

(E)=— — 8(E)1D

the current density now becomes2

y f f ,'N';-(—E,) ,'N';"(-E, )spin

X T(E E—(k ) E,(k,—))de dE, .

(39)Equation (39) can be written as a convolution

J I3D

Z

2

T(E)S —,'Ni'D (E) —,'Ni'D'(E) . (40)

If we had started from a standard tunneling formal-ism, ' ' we would have obtained

J»(E, V, T) = T(E, V)e 8'(E, V)e — (E, T)dE

s 'N '"(E)s i N—'"(E)1D 2(41)

as the current density at finite voltage and finite tempera-ture. Equation (41) is shown graphically in Fig. 2. FromEq. (41) one can obtain any tunneling current at finitevoltage and temperature found in Refs. 12, 14, and 15. Ahelpful identity in obtaining these tunneling currents is

1 m1Nfree(E) 1Nfree(E) —1Nfree(E) g(E)1D 2 1D 2 2D 2 2 A2

(42)

Results analogous to Eqs. (42) and (40) for the density ofstates and electrical currents follow for any separable po-tential. ' Using Eqs. {42) and {40),we see that in the stan-dard type of quantum-mechanical tunneling calculationsin three dimensions, the quantum contact resistance nev-er appears. The reason is, of course, that in going over toa continuum of states we add an infinite number of con-ducting channels in parallel. For tunneling at finite volt-ages, Eq. (33) reduces to the standard tunneling currentdensity at finite voltages, Eq. (41), thus having the correctbehavior in the tunneling limit.

By analogy with our consideration of classical diffusion

V. CONCLUSIONS

Using the Landauer formalism we have examined theeffects of finite voltages on quantum transport, showingthat the finite voltage acts to average the electrical con-ductance over an energy range equal to the applied volt-age. This voltage broadening is similar to ordinarythermal broadening of the conductance, which averagesthe conductance over an energy range equal to the tem-perature. We showed that both thermal and voltagebroadening of the conductance can be expressed as in-dependent convolutions with known functions, proving,that voltage broadening and ordinary temperaturebroadening are statistically independent. Motivated bythe recent experiments of van %'ees et al. ' and %'haramet a/. , we examined thermal and voltage broadening ofthe conductance using a quasi-one-dimensional ballisticconductor as a model system. The finite width of therises in the quantized steps in conductance versus electro-chemical potential in these experiments is inversely pro-portional to the phase coherence time of the electron in-troduced by either finite voltage or finite temperature.The quantum contact resistances can be thought of eitheras a consequence of the rate limitation imposed by theuncertainty principle in the device leads or as a conse-quence that the mean free path is limited at most to thedevice size.

We also examined both the diffusive and tunneling lim-its of Landauer's conductance formula. In the classicaldiffusive limit the one-dimensional Landauer formulareduces to the Drude-Sommerfeld conductance in one di-mension. In the tunneling limit, the finite-voltage Lan-dauer formula is equivalent to many previous treatmentsof electron tunneling at finite voltages. In this tunnelinglimit, we have shown that simply by repeatedly convolv-ing the tunneling current in one dimension with the one-dimensional free electron density of states, one obtainsthe tunneling current density in two and three spatial di-mensions.

ACKNOWLEDGMENTS

We thank Dimitri A. Antoniadis, Henry I. Smith, andKhalid Ismail for useful discussions. We gratefully ac-knowledge financial support from U.S. Air Force ofScientific Research Contract No. AFOSR-88-0304.

R. Landauer, IBM J. Res. Dev. 1, 223 (1957).2Rolf Landauer, Philos. Mag. 21, 683 (1970).M. Biittiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev.

B 31, 6207 (1985).~Daniel S. Fischer and Patrick A. Lee, Phys. Rev. B 23, 6851

(1981).5M. Buttiker, Phys. Rev. B 33, 3020 (1986). See also M.

Biittiker, IBM J. Res. Dev. 32, 63 (1988).M. Biittiker, Phys. Rev. Lett. 57, 1761 (1986). See also M.

Biittiker, IBM J. Res. Dev. 32, 317 (1988). This last referencepoints out that Eq. (9) is not the most general four-proberesistance, even in the case of weak coupling to the voltage-measurement probes. A more general formula given byBiittiker considers possible wave-interference effects sn theprobes, as well as the possibility that the voltage probes cou-ple differently to left- and right-moving carriers. In a four-probe —measurement geometry it is even possible to obtainnegative resistances for either weakly or strongly coupled

1464 PHILIP F. BAGWELL AND TERRY P. ORLANDO

[0(E+eV)—6(E)], eV(p[9(E+p)—g(E)], eV~ p .

(43a)

(43b)

Rolf Landauer, Phys. Lett. 85A, 91 (1981).~8H. L. Engquist and P. W. Anderson, Phys. Rev. B 24, 1151

(1981).~9Use the Drude-Sommerfeld formula to describe the conduc-

voltage probes.S. Datta, M. Cahay, and M. McLennan, Phys. Rev. B 36, 5655

(1987).8A. Douglas Stone and Aaron Szafer, IBM J. Res. Dev. 32, 384

(1988).9Rolf Landauer, Z. Phys. 8 6S, 217 (1987).

Yoseph Imry, in Directions in Condensed Matter Physics, edit-ed by G. Grinstein and G. Mazenko (World Scientific, Singa-pore, 1986).D. Lenstra and R. T. M. Smokers, Phys. Rev. B 38, 6452(1988).

Walter A. Harrison, Solid State Theory (McGraw-Hill, NewYork, 1970). See Eqs. (3.23)—(3.25).Philip F. Bagwell, Dimitri A. Antoniadis, and Terry P. Orlan-do, in Advanced MOS Device Physics, edited by N. G. Ein-spruch and G. Gildenblat (Academic, New York, 1989). Seealso P. F. Bagwell and T. P. Orlando, Phys. Rev. B (to bepublished).

t~E. L. Wolf, Principles of Electron Tunneling Spectroscopy (Ox-ford University Press, New York, 1985). See Eqs. (2.5) and(2.6).

~5R. Tsu and L. Easaki, Appl. Phys. Lett. 22, 562 (1973).If the applied voltage is greater than the Fermi energy, then8'(E, V) must be cut off at the Fermi energy, namely,

tance of one electron of each spin in a device of length Ld by

2e 2D (E )~(E )V

where V is the applied voltage and D(EF)=v+~F is deter-mined by the Fermi velocity and momentum relaxation time.We later show that Eq. (44) is equivalent to Eq. (9) in the clas-sical diffusive limit. The time ~d is then given as

&(EF)«=D [X(EF)e VL„] (45)

describing an electron gas in one dimension of arbitrary den-sity n=X{E+)eV. Thus ~d is the time for one electron todiffuse across the device divided by the number of electrons inthe device.This argument makes one wonder if it would be possible to seeany indication of this frequency eV/h, which is half theJosephson frequency, in the current of a ballistic quasi-one-dimensional conductor as a function of time. Because of thespin degeneracy two electrons cross the conductor everyh /e V sec.B.J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Wil-liamson, L. P. Kouwenhoven, D. van der Marel, and C. T.Foxon, Phys. Rev. Lett. 60, 848 (1988).

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