PHYSICAL REVIE% A VOLUME 19, NUMBER 4' APRIL 1979

Microscopic calculation of the nonlinear current fluctuations of a metallic resistor: Theproblem of heating in perturbation theory

A.-M. Tremblay~ and B. Patton~Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

P. C. MartinPhysics Department and Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138

P. F. Maldague~IBM Thomas J. 8'atson Research Laboratories, Yorktown Heights, ¹wYork 10598

(Received 18 September 1978)

The effect of a constant electric field 0 on the low-temperature Johnson noise of a metallic resistor iscalculated by nonequilibrium diagrammatic perturbation theory. The expansion, in powers of E, is shown todiverge, and the divergence is associated with accumulating Joule heat. An insight into how to treat a' steadystate properly when there is heating is provided by the Boltzmann equation. A vertex function thatcorresponds to a steady-state solution of this equation is identified. With this vertex function it is possible toevaluate not only one-particle operators, for which the Boltzmann equation is sufficient, but also troublesomecontributions to the current fluctuations and other two-body observables for which a many-body approach isrequired. The calculated corrections to the current correlations agree qualitatively, but not quantitatively,with simple phenomenological agruments. -In' particular, it is not possible to describe the leading effects ofthe electric field solely in terms of a variable local temperature. The model calculation illustrates and shedssome light on several problems that only appear when nonlinear deviations from thermal equilibrium must beconsidered: the complicated effects to which heat generation leads, the identification and treatment of theseeffects in perturbation theory, and the difficulties these effects pose for hypothetical generalizations of thefluctuation dissipation theorem, Some possible connections between the calculation and 1/f noise are alsobriefly explored. In particular, recent experiments that may confirm the conclusion of this calculation, thatsystems in which there is only impurity scattering do not exhibit 1/f noise, are noted.

I. INTRODUCTION

Ever since the work of Bernard and Callen'theorists have sought in vain for an explanationof nonequilibrium current or voltage fluctuationsthat was as deep and general as the explanationof equilibrium fluctuations in terms of the fluctua-tion-dissipation theorem. To nonlinear orderthere probably is no correspondingly generalexplanation; the details of specific models playa major role.

In this paper we report on a first-principlemicroscopic calculation of the nonlinear responseand fluctuations for a specific model. To makethe necessarily lengthy calculations as short andeasy as possible we study a system of metallicelectrons that interact only with dilute, static,and isotropically scattering impurities. This canbe a reasonable model for a metallic resistorat low'temperatures. We calculate the proper-ties of this system using a method developed sometime ago." The method has been widely appliedbut it has not, to our knowledge, previously beenused to evaluate nonlinear fluctuations.

One motivation for performing these calculationswas our desire to understand the intriguing ex-perimental phenomenon' (1/f noise) that non-

equilibrium current fluctuations are supposed toexhibit. Our system does not appear to exhibitthis phenomenon. This result seems to agreewith recent experimental findings of Hooge andVandamme' on dirty systems. These authors arguethat their negative results show that. the noisemust be due to phonons. For reasons discussedin the Conclusion, we consider this finding sug-gestive but not compelling. In any event, weagree with these authors that since most systemsexhibit 1/f noise, a counterexample may well beuseful in unravelling the riddle.

Although the calculation does not explain 1/fnoise and is not necessarily generalizable, itdoes provide some new, interesting, and unex-pected insights. For example, the reader maybe surprised, as we were, to discover that per-turbation theory yields divergent current fluctua-tions for the model. In retrospect this divergencecan be simply understood on physical grounds.In thermal equilibrium, for frequencies uo muchsmaller than both the collision frequency and theaverage thermal energy, the spectrum of cur-rent fluctuations is given by

gQ) g 02kT

1721 1979 The American Physical Society

1722 TREMBLAY, PATTON, MARTIN, AND MALDAG UE 19

A is the resistance, k Boltzmann's constant, andT the absolute temperature. We are interestedin current fluctuations about a nonzero averagecurrent driven by an applied constant electricfield. If we treat the entire electric field as anexternal perturbation, we cannot avoid the factthat, to second order in this field, the tempera-ture of the system will rise (Joule heating) duringa typical measurement. Indeed, if the resistanceis temperature independent, we should expectto replace the expression 2kT/R by

(2k/8) [T(t,)+ (t —t,)dT(t, )/dt] .

It is the second term that produces the divergence.our calculation shows how heat generation appearsin the lowest-nonlinear-order perturbation theory,suggests how it must occur in higher orders ofperturbation theory, and indicates why it can beneglected in linear response and what difficultiesit poses for generalized fluctuation-dissipationthe orems.

To obtain a finite result at long times, we mustexplicitly cool the system. We must include theinteraction with the environment that compen-sates the Joule heating and allows it to reach asteady state. The-actual cooling mechanism isneither general nor universal. We might expectto be able to balance locally the heat generatedby the electric field and the thermal energy fluxproduced by a thermal gradient. Although thisidea is qualitatively correct, we shall find thatthe cooling mechanism in a true nonequilibriumstationary state and a thermal gradient are notprecisely identical.

By eliminating the shortcomings of perturbationtheory and allowing the heat to flow away, weobtain, with certain restrictions, finite non-equilibrium current fluctuations. Our field-de-pendent corrections to Johnson noise do not re-solve the 1/f mystery. At best, they eliminatesome theories and offer some clues. We discussthese matters briefly in the Conclusion.

Our discussion is organized as follows: In Sec.II, using the Boltzmann equation, we study theproblem of heating and cooling for simple one-body observables. In Sec. III we show, guidedby the results of Sec. II, that after the diver-gences have been removed meaningful resultscan be obtained to leading order. In Sec. IV wesummarize the results and in Sec. V we commenton 1/f noise, how to go beyond our calculation,and nonequilibrium perturbation theory in general.

II. JOULE HEAT FOR ONE-BODY OBSERVABLES:STATIONARY DISTRIBUTION FUNCTION

Although it is unavoidable, Joule heat appearsin a slightly obscure manner when we calculate

&f ---- f f—+eE ~ V f+v V,f=-Bt P

(2 1)

where f is a, distribution function which dependson the momentum (p = m v) and the position x,is an energy-independent scattering time, and

(2.2)

Assuming the solution can be expanded in powersof the applied electric field, we write

with

f (o) +f (i) +f (2) +. . . (2.3)

f (0) —(I/[ 8(P /2m-)L) + 1]J (2.4)In Eq. (2.4) we have introduced the conventionalnotations p for the chemical potential and P for1/kT. Substituting Eq. (2.3) into Eq. (2.1), wefind in a uniform system, for times much longerthan ~, that

7eE. v f (o) (2.5)A

pf'"= —7(eE V )'f")t.4&

Equations (2.3)-(2.6), the "standard" pertur-bation solution' of the Boltzmann equation, shouldbe contrasted with Huberman's' "long-time ex-pansion. " The "standard" solution is valid onlyfor systems that remain close to equilibrium(6T/T « I).

The second-order change in the energy densitycan be calculated from

(2.6)

(2.7)

where 0 is the volume of the system, e =P'/2m—)), and N(0) is the single-spin density of statesfor a system whose Fermi wave vector is k~(we take k = 1)

N(0) =mk /27('. (2.8)

Substituting Eq. (2.6) in Eq. (2.7), we find that

& 8 = (v '/R) t, (2 9)where 8, the resistance of the system, ' can be

current fluctuations. To achieve a clearer under-standing let us first discuss the effects of heatingon one-body observables using the Boltzmannequation.

The results we shall establish will guide us ininterpreting and evaluating the infinite vertexcorrections that arise in calculating current fluc-tuations and shed light on some general aspectsof heat generation in perturbation theory.

The Boltzmann equation corresponding to ourmodel can be written in the form

MICROSCOPIC CALCULATION OF THE NONLINEAR CURRENT. . . 1723

expressed in terms of L, the length along whichthe voltage V =EJ is applied, andA, the cross-sectional area, as

produced by the electric field and thermal gradientcan be computed classically. For a uniform cur-rent" (V ' j =0) we have

8 = rnf, /ne'vA;n is the density.

n = a,'/3~'.

(2.10)

(2.11)

+ (VT)' -T (—V T) j + )(V'T,(2.19)

Clearly, Eq. (2.9) represents Joule heating.Since most experiments are performed in steady-state configurations, we seek stationary solutionsof the Boltzmann equation. Intuitively, we expectto be able to apply a "thermal gradient" to thesystem that will take advantage of the electronicthermal conductivity to carry away the heatgenerated by the electric field and lead it to anonequilibrium steady state. Let us obtain thestationary solution of Eq. (2.1) in the presenceof an electric field and a "thermal" gradient.Expanding in powers of the electric field and al-lowing f (') to be position dependent, we obtainfrom the Boltzmann equation (2.1)

f (t) 7eE .& f (o) (2.12)&q. vf (2) + 7-x[f (2) f (2)j &(eE.p )2f (o)

(2.13)

where q is the wave vector in the direction of thegradient which we arbitrarily choose to be per-pendicular to the electric field.

Equation (2.13) can be solved by expanding inspherical harmonics and keeping only the firsttwo terms 'Subs. tituting the solution in Eq. (2.7),we find that

58 = 2+(0)(eE/q)'n, (2.14)

~8=C,~T, (2.15)

U'/R=)(q'5T, Q, (2.16)

where C~ is the specific heat for our model, "C), = 0 (m'/2) (kT/E), )nk,

and K is the thermal conductivity, "C~ n~~'K= =——O'T =——,To .Am33e'

(2.17)

(2.18)

The last equality has been written in terms of theelectrical conductivity. o to recover the Wiede-mann- Franz law.

Equation (2.16) can be obtained by the followingarguments: Suppose that we include a thermalgradient to remove the heat generated by an elec-tric field. The change in the local energy density

which is identical to the result that can be heuris-tically derived from

where Q is the thermopower and j the currentdensity. In the configuration where current andtemperature gradient are perpendicular, we have

V'T ~ j =0. (2.20)6T, the difference between the maximum andminimum temperatures across the sample, satis-fies the condition

6T«T.The inequality

(2.21)

I)(V'TI» -(~T)'

dT(2.22)

follows because

(2.23)

and

KV'T - )(5T/I. ' (2.24)where L is the length of the sample along thethermal gradient. Equation (2.19) therefore re-duces to

]. dg ]. y'dt

(2.25)

Thus the energy density will be time indepen-dent if

U'/0 R = —)(V'T . (2.26)In Fourier space, this equation is identical withEq. (2.16).

It should be emphasized that the solution to Eq.(2.13) is not exactly like the solution we wouldobtain by applying a simple spatially dependenttemperature to the system and requiring that heatbe carried away by the thermal gradient. %eelaborate on this point in the Appendix.

The second-order effect of the electric field ongny one-body observable can be calculated from

(A) =2K(o)of de j I'"(e) (2.27)where I'"(e) depends on the observable, e.g. , forthe energy density I'"(e) =e. Note that every one-body observable will have a time-independentvalue when the solution to Eq. (2.13) is used in Eq.(2.27). In the case of the energy, that value is the

TREMBLAY, PATTON, MARTIN, AND MALDAGUE 19

t„-~'T(uT/est)', (2.29)where t„ is the heating time of the system that canbe determined from

T Tcg" (2.30)These results suggest the reason why, in linear-

response theory, we need not examine how theheat is dissipated when we make realistic calcu-lations. However it is done, as long as the heatis dissipated the linear response is the same.

By contrast, a quantity that depends on f i'~ willbehave quite differently in experiments describedby Eq. (2.6) and Eq. (2.13). Joule heat is oneobvious quantity that depends on f".A less ob-vious quantity is the current fluctuation to lowestnonlinear order (Z') in the applied field.

The Boltzmann equation can only be used tocalculate one-body observables. To determinethe current fluctuations, a Aeo-body observable,we must use a microscopic theory. As we notedin the Introduction a straightforward applicationof microscopic perturbation theory leads to in-finities reflecting Joule heating. These infinitiesin the current fluctuations can be identified with acertain part of a diagram. %e can find how toinclude the "cooling" mechanism that leads to asteady state by establishing a formal equivalencebetween the equation for the troublesome partof that diagram (a vertex equation), and the per-turbative solutions to the Boltzmann equation dis-cussed here.

III. NONEQUILIBRIUM CURRENT FLUCTUATIONS

A. Model

1. Harniltonian

The resistor model' described by the Boltzmannequation (2.1) consists of free electrons in a posi-tive background scattering off random one-bodyimpurity potentials. The Hamiltonian is given by

one expected from simple macroscopic argu-ments.

Although the two solutions to the Boltzmann.equation we have presented correspond to dif-ferent physical situations, the values of f i'~ [seeEqs. (2.5) and (2.12)] are the same. Since the re-sistance of the system, for example, can be de-termined by calculating the nonequilibrium cur-rent from f i'~, the value of the resistance de-termined in a steady-state experiment [corres-ponding to Eq. (2.12)] or in an experiment thatlasts for a "short time" [corresponding to Eq.(2.5)] are the same. " Here, "short time" means

(2.28)

H= Ho+ H~, (3.1)

Ho= epCpsCps ~p, s

H, = g Q u(-q)e'~'"'CP, CP, ~, ,p, qs x&

(3.2)

(3.3)

where u(q) is the spatial Fourier transform of thescattering potential, x; are the positions of therandom impurities, e&= (P'/2m) —p, and Cp,destroys a particle in a state of momentum p andspin s. The field operator 4, is given by

eg(x, t) = g Cpg(t)e (3.4)In the dilute limit (an impurity concentration n,of, say, less than lfo), it is possible to deter-mine the properties of this model' by averagingover the positions of the impurities. Indeed, oncethe averaging is carried out, the problem isequivalent to solving a field theory with an effec-tive interaction Hamiltonian given, in the inter-action representation, by

a, = —' g gc;...(t')c;. -„(t')p, p', q ss'

&& [-ilu(q)l ]C&(t)CP,-„,(t) . (3.5)

where lt(q)l' is the t matrix for scattering off thepotential.

Many of our results are independent of assump-tions (a) and (b).' The Coulomb interaction be-tween the electrons can also be neglected sincewe shall be concerned with space-independentperturbations which do not alter the density andtherefore do not require us to take the long-rangeCoulomb interaction into account immediately.Finally, this model is realistic only when thetemperature is low enough so that electron-phononcollisions are less important than impurity scat-tering. For simplicity, and in line with this re-striction we shall take the electrons to be a highly

We must also neglect all diagrams that can bedisconnected by cutting an arbitrary number ofinteraction lines. We shall assume that the scat-tering potential (a) has no bound states, and (b)is so short ranged that it leads only to s-wavescattering of the electrons at the Fermi surface.Note that we do not need to assume that the Bornapproximation is valid. "'" When it is not, wemay set

lu(q)l'- lt(q)l' (beyond Born approximation),

lt(q) l'- lt(0) l' (s-wave scattering), (3.6)lul'-=lt(0) I',

MICROSCOPIC CALCULATION OF THK NONLINEAR CURRENT. . .

(3.7)

(3 8)\

(3.9)

(3.10)

degenerate Fermi gas.We shall need the following propagators"'":

G' (l, 1 ) = — &~(1)~'(1')),G' (1, 1') = i &4'(1'H'(1)),

G (1, 1') = —i [&4'(1)4' (1'))

+ &+'(1')e(1))]e(f,—f, ),G"(1, 1') = i [&e(1)e'(1'))

+ &~'(1')~(1))]e(f, —f,),

G' (k, (u) = if((u)A (k, (u),

G~(k, (u) = —i[1 -f((u)]A(k, u)),

where

(3.16)

(3.17)

obtain

i[G"(k, &u) —G"(k, (u)] =A(k, (u)

1/~((u —c„)'+(1/2~)' '

(3.15)

where 1 and 1' stand for the space-time coordi-nates and 6(t) equals one for t &0 and zero fort&0.

2. Thermal equilibrium solution of the model

In the limit

k'~L» 1, (3.11)where l is the mean free path [defined in Eq.(3.13)], we can neglect all overlapping diagrams.The solution for the electron propagator' equation(Fig. 1) is then, in units in which 8 = 1,

f(&u) = (e + 1) ' . (3.18)

H~t —— gt x&tA& x&$ d x

This completes the solution of the problem inthermal equilibrium.

3. Coupling to the external electromagnetic field

We wish to determine the effect of an electricfield described by the time-dependent transverse(V A = 0) vector potential in the interaction Ha, mil-tonian

G (i ~) f~"u=e " G-rx"~ "oo")p, (, i)A, (, t)A, (, f)d' .

&PSC(3.19)

= [G"(k, (u)]*. (3.12)

The collision time 7 and the mean free path / aredefined by

v '=—2vN(0)n, ~u~'; l —= @~7/m =v~T. (3.13)

We now make the assumption of thermal equili-brium by using the boundary condition" validin this case

The index i refers to the vector Cartesian com-ponents (with a summation convention). With thenotation of Schrieffer, "we have

j;(x, t)—= . g (kt(x, t)V, C, (x,t).Vl 2—[V;et(x, t)]e, (x, l))

p, (x, t)A, (x, t)SlCG' (k, (u) = —e ~G'(k, (u) . (3.14)

From Eqs. (3.12), (3.7)-(3.10), and (3.14), we=-jf (x, t) + p, (x, t)A; (x, t) .SIC (3.20)

r(r) r % rh r +

~r fr( r let ~l

rr( r Lrl r )rf r Lr

~X-~

rt r ~ rfrI r ~ rlr

+ ~ ~ ~ ~ ~ ~ ~

FIG. 1. Propagator equa-tion including all nonover-lapping self-energy dia-grams. The thick solidline is the "fully dressed"electron propagator withthe arrow pointing fromthe + coordinate towardthe 0 coordinate.

r+i/r f r 1

TREMBLAY, PATTON, MARTIN, AND MALDAGUE

The electric charge density is defined by

p, (x, t) -=—e g @~ (x, t)4, (x, t) . (3.21)C. Current autocorrelation function in thermal equilibrium:

Johnson noise

We have described the electric field by a vectorpotential and not a scalar potential for technicalconvenience.

B. Method of calculation

Let us define the propagator"

tGG(1, 1') =(T,4'(1)C ~ (1')), , (3.22)

IIG(tt )=t(f jv , G', (t; v)G,(rt)G(v)', (3.23)where A is a constant, U(~) an interaction po-tential, and C the KBL contour, rules for re-casting this expression in terms of the usualpropagators are useful. For this type of "series"multiplication we have

co

&G'(t, t') =2 d~ [G,'(t, 7)G", (7, t')

+ G, (t, T)G,'(~, t')]U(~), (3.24)

G" " ( t, )II(IfIjt~G"I"-I(t~),xG ( ~(~, t')U(T). (3.25)

These results can be generalized. Suppose that(symbolically), we have

&G:A G0G0G0 G() ~ (3.26)

We may then write)

6G' =W'[GGG"G",

+ GRGRG

. GA+ GRG'GA . ~ ~ GAGA+. . .+ GRGBGR. . . P(]

where T, represents the time-ordering operatoralong the Kadanoff-Baym-Langreth (KBL) con-tour. The usual rules of perturbation theory'"'"apply to nonlinear response if propagators aretime ordered along the KBL contour.

When perturbation theory" yields, for example,

Although the phenomenon of Johnson noise doesnot depend on the specific nature of the resistor,it is instructive to see how, without the fluctua-tion-dissipation theorem, Johnson noise is pre-dicted by the nonequilibrium formalism for ourmodel. The quantity measured in a Johnson noiseexperiment can ordinarily be expressed in termsof" /

(j' j' ) -=f e' "' "(j';(x, tjj,'(x'I')),(d0 A0&& dt dt' d x d'x' . (3.31)

The bracketed quantity on the right-hand sidecan be regarded as a time-ordered product alonga Kadanoff-Baym- Langreth' or Keldysh' contourwith t always later than t'. Since the time order-ing requires that t be later than t' we can writeEq. (3.31) as

d'k d(d e'k;k;~2 —(GPSS(IIGj . (2 )3 2 2

x G'(k, &tj)G'(k, &u+ ~,)QE . (3.32)The total time of the experiment V' and the volumeof the system A enter because the Green's functionsdepend only on the time- and space-coordinate dif-ference. The expression (3.3) corresponds to thediagram" of Fig. 2. Because the scattering isisotropic and no momentum is transferred at thecurrent vertex, all of the vertex correctionsvanish (i.e., loops of the type illustrated in Fig. 3vanish after integration over the direction of k).

Assuming particle-hole symmetry, we firstperform the c integration" and then the (d integra-tion, obtaining"

2 2

0

From linear response theory' we know that theresistance R of this model is given by Eq. (2.10).From the definition (3.31) we have

R(A) ~IGR{A)GR(A) QR(A), GR(A)0 0 0 0

(3.27)

(3.28)&q'. jj ) =l.' dtdt'e* G" -"(I'(t)I'(t')),

(3.34)The other rule we shall employ concerns parallel"multiplication. If G(k, )

&(t, t') = G(l(t, t')GG(t', t) (3.29)j (O, -cuo) j (O, cup)

(no integration implied) then we may write

&'(t, t') = G,'(t, t') G&(t', t) . (3.30)Equivalent rules may be obtained by the analyticalcontinuation procedures of Gor'kov and Eliash-berg or several other methods.

Q ( k v QJ + GJp )

FIG. 2. Diagram for the current fluctuations. Thesolid lines are dressed propagators. The vertex cor-rections have not been included because they vanish:see text.

19 MICROSCOPIC CALCULATION OF THE NONLINEAR CURRENT. . . 1727

k ~+o

j (O, cup)FIG. 3. Example of ver-

tex correction that vanishesfor our model.

=2R '[&u,/(e~o —1)] (&u,~«1)

- 2kT/R ((d, T «1,p~, «1).

(3.35)

where I is the macroscopically measured cur-rent. In the limit ~,«1/v, we deduce from Eqs.(3.34), (2.10), and (3.33)

D. Electric-field4ependent corrections to Johnson noise

In this section, we outline our calculation, em-phasizing the technical difficulties to which wereferred earlier. In particular, we examine thedivergent terms that arise in perturbation theory,and which will be treated more carefully in thefollowing section.

We want to find the current autocorrelationfunction to second order in the vector potentialA for the interaction Hamiltonian in Eq. (3.19).If we take the time integrals a,long the KBL con-tour, we can proceed exactly as in the equilibriumcase.'" Using Eqs. (3.19) and (3.20) and letting7;, inside the angular brackets denote a time-ordered product along the KBL contour, we im-mediately obtain

+ — d'x(r, j f(x)p, (x')j', (x)}A,(x')A, (x) +—,d'x (7,jf(x)jf(x')p, (x)}A„(x)A„(x}2

+ — —, d xd x(T,j~~(x)j~~(x')j~~(x)jf(x))A„(x)A, (x) . (3.36)

Here x stands for the four coordinates (spatialand temporal). The time integral extends alongthe KBL contour and the spatial integrals overthe entire volume of the infinite sample. Thefirst term in Eq. (3.36) comes from the diamag-netic contribution to j, and j;, the second andthird ones come from a first-order expansion ofthe S matrix and a diamagnetic term (respectivelythat ofj; and that of j;), the fourth term comesfrom the last term of Eq. (3.19) in first-orderperturbation theory, and the last term comesfrom the first term of Eq. (3.19) in second-orderperturbation theory.

Since the external field is real, we choose

(3.37)

(f)A~~. ( )

(A -())t A g l))t)C- Bt C

(3.38)

The Fourier transform of Eq. (3.37) is

e '(' "e' A,„,(t) d x dt

= 5 (q)[A5(z —p)+A+5((g+ v)](2v)'. (3.39)

If we substitute Eq. (3.37) in Eq. (3.36) we findterms proportiona, l to A', A*, and AA*. Weshall calculate the latter terms which give the

desired time-independent contribution to the cor-rela, tion function. "

As in the case of Johnson noise we shall be in-terested in

x dt dt' d'x g'g' . . (3.40)From Eq. (3.36) we can enumerate the diagrams

we need to evaluate Eq. (3.40). They are sketchedin Fig. 4. It is understood that each of thesegraphs yields a11 of the "time orders" the rulesof Sec. IIIB require. As noted previously, onlythe graphs proportional to AA* have been kept.Parts (a)-(d) of Fig. 4 come, respectively, fromthe first, second, third, and fourth terms of Eq.(3.36) while parts (e)-(h) all come from the lastterm of Eq. (3.36); parts (i) and (j) define theshaded areas that appear in the graphs and repre-sent vertex corrections. The various other sym-bols are defined in the figure caption.

As they did in Eq. (3.32), the corrections to asingle current vertex [Fig. 4(i)] vanish for iso-tropic scattering when q= 0. However, in general,the corrections of Fig. 4(j) do not.

Gauge invariance imposes many constraints onthe calculation. In particular, the vertex cor-rections in Fig. 4(i) (see, for example, Ref. 30)

1728 TREMBLAY, PATTON, MARTIN, AND MALDAGUE 19

(- o+v)—

Cd+ V

td + Itlp

& (-v)td + V

(- td p+ v) ' ' ' C' (tdp )

CCI+Cdp

f (v)td ~ Cd +V

A

(-Cdp) -~ ~&e o «(cdp v)

td + td p

A" A( v) (-v)

Cd+Id O

A

o- )

+(v ~-v)(a)

tdA

(-tdp+ v)e ~ ~ (e (Cdp)

td + tdp-V . Cd +CdpA ~(v~ „)

(-v)

(b)

(- o) A.& ~ ~ ~ (cdtl- v) ( C )Cd + tdo ~ td +Ido- V

A +(v~-v)(v)

Cd-Cdp

(- tdp)- & ~ (tdo) (d)

+(v ~-v)A A

( v) (-v)

as well as those of Fig. 4(j)" must be includedwhen the propagator in Fig. 2 is employed. Also,since there can be no response to a pure longi-tudinal vector potential at zero frequency, andsince the response must be even in v because ofthe v ——v symmetry, the leading contributionfor the diagrams of Fig. 4 will be proportionalto v'.

We proceed to evaluate the diagrams of Fig. 4.The only two-vertex diagram we must evaluateis shown in Fig. 4(a). When the vertex in Fig.4(i) is inserted for the shaded area, Fig. 4(a)clearly represents a density-density correlationfunction and vanishes at q= 0."

The three-vertex diagrams are those of Figs.4(b), 4(c), and 4(d). For a system with particle-hole symmetry these diagrams cancel" in pairs.We can prove this fact by using the identities

A&(-v)

A0 (-v)

Cd-V

(-tdp)(e)

(Cdp)

G'(Cd, ~) = -G"(-Cd, —e),G'(Id, C) = —G'(-Id, —C),

(3.41)

(3.42)

Cd+ tdo td+ tdo- VAII

(v)

Cd

(-tdo)

A" A)( v) )(-v). Cd+ V

td+ Cdo

( v) .~v. (-v)

(-cdo) (Cdtl)

(-tdp) (Cdp)

A" A

( v) (-v)

(v}

Cd+tdp~

td+Ido-V

(v) +(v~-v)A" A

( v)g (-v)g~ td+V .I

(- Cdp) (tdo)

td +td +(v~ -v)

Cd - tdp

(-tdp) (tdp)Cd ~-- I td

Cd-V yA" A +(v~ v)( v) (-v)

Cd - tdp

(-Cdp) (Cdp)I I td

+(v ~-v)A A

( v) (&v)

I ~ II II I

X X + ~ ~ ~ (t)I I

I I

and changing a few integration variables. Thesame method can also be used to show that eachfour-vertex diagram is exactly equal to its "up-side down" counterpart.

Before evaluating the finite corrections thatthe diagrams in Figs. 4(e), 4(f), and 4(g) yield,let us consider the troublesome diagrams inFig. 4(h). As in the other diagrams in Fig. 4,the vertex corrections in Fig. 4(i) vanish here.lt is the vertex corrections of Fig. 4(j) that leadto problems. Figure 5 shows schematically allnonvanishing time orderings of the diagram ob-tained by inserting the first term in Fig. 4(j) intothe first diagram in Fig. 4(h). (Each of the re-maining time orderings demanded by the rulesin Sec. IIIB has a loop that consists entirely ofadvanced or entirely of retarded propagators andvanishes). Using the identity

I II

I I

I I II I I

II

XI

I

X + ~ ~ - (j)I I

FIG. 4. Diagrams for the current autocorrelationfunction to second order in the applied field. The shadedareas are defined by Fig. 4(h) and 4(i). All diagramsmust be time ordered with the rules of Sec. III B. Onlythe frequency labels have been marked on the propaga-tors. The frequency in parentheses is the frequencythat comes out at a vertex (momentum never comesout of the v.ertices in these figures).

represents a j~ vertex [see Eq. (3.20)j;~ & ' ' represents a p, A vertex [see Eq. (3.20)];

represents a j .A vertex [see Eq. (3.19)l;represents a peA. vertex [see Eq. (3.19)].

Note that for each diagram there is another one with(& -&). Additional diagrams obtained by this simplevariable change must be included in the final result.Note that the vertex corrections in I'ig. 4(i) vanish for aj or j ~ A vertex.

G (Cd, e) =G"(td, c)*, (3.43)

]X v'+ 1/v' td,'+ 1/~' (3.44)

To include the remaining vertex corrections inFig. 4(j), we multiply the result of Eci. (3.44) bythe vertex shown in Fig. 6. Although we areinterested in the case &, = +,= ~, for the moment

and the fact that the product of G' and G' is real,we see that the diagram in Fig. 5(b) is real andthat thediagrams in Figs. 5(a) and 5(c) are complexconjugates. The sum of all these diagramS is realand its value is

2e' N(0) ~k' Cd, —v9 5 0-v ] e Mo ]

MICROSCOPIC CALCULATION OF THE NONLINEAR CURRENT. . . 1729

A" ( v) A(-v)

)G"( GA

+V

A (v) A(-v)JL

GRjG ) GA

G

A ( v) A(-v)

Gai G"i G»+v

G

FIG. 5. Only nonvanish-ing contributions to one ofthe diagrams in Fig. 4(h).

let the frequencies ~, and z, remain arbitrary.%e can write the solution to the vertex equation

I' = 1+n, iui' N(&) de1 1

(d~ —E —z 2T Q2 —f +2 27

(3.45)

using (3.13), in the form

r = (o), —u), —i/T)/(o), —u),) . (3.46)It is infinite when cu, = ~, = el Before discussingthis infinity further let us calculate the remainingdiagrams in Fig. 4(h) formally. For this purpose,we add the (v —-))) term to Eq. (3.44), multiplythe result by I', and double it to account for the"upside down" diagrams. The result is

c'm z 9 e ' o "& —1 e & o'"& —1 e o —1 (v'+1/7')(&u'+1/7') (3.4V)

%e shall come back to this equation.For the diagrams of Figs. 4(e), 4(f), and 4(g),

we obtain after a calculation whose tediousnesscannot be overemphasized

(j j ) = v'ni —+—k~$1 1 4 2e N(0)

field (v- 0), we have

(dp- V up+ V(+0) o o(QJ -v& ~ + 8(uJ +o&pp e p i e p

2(dpp

= 3(csch' —,'Po)o)( —,'Po)o coth —,'P(oo —1) . (3.49)(O)o —V) &dog ((d p-V) ] Q(d p

1X 2 i 2 o q 2AA*+(V~ —V) ~v2+ ly v p+ ly v

(3.48)

Note that (a) this correlation function, as wellas Eq. (3.47), vanish when w=~. To verify thatthis behavior is correct, observe that with non-interacting propagators the calculation would con-tain products of the form G'(e, e)G'(&u+ e„e)which vanish due to the 6 functions in the spectralweights. (b) Equations (3.47) and (3.48) also givea vanishing contribution when p =~ (7 = 0) or &so= ~as expected. (c) In the limit of a constant electric

To relate Eq. (3.48) to directly measured quan-tities, let us consider a special case. In the ex-Perimentally relevant regime, Puro«1 andpv«1, the approximation

lim B(o)o) = 18QJ p~ p

is valid. In terms of the constant external elec-tric field E, '4

(3.50)

2@~~+= c E (3.51)in the limit vv«1, coos«1, Eq. (3.48) reduces to

(j „j~ ) - &Q —+ — 4 I),„N(0) 'roE—(3.52)Using Eqs. (2.10), (2.11), and (3.34), we can writeEq. (3.52) as

GA

QJl II+ X

NpI

GR

FIG. 6. Nonvanishingsections in Fig. 4(h).

GA

g

~lIx

QJp II

GA

Ml

Olgi

GR

+ ~ ~ ~

contributions to the vertex cor-

dt' —te' p' ' ItIt' „, =—&I

= 0.156—

eEl ' 2'&

(3.53)

TREMBLAY, PATTON, MARTIN, AND MAL'DAGUE

where I is the length of the sample in the direc-tion of the electric field and V =El. is the appliedexternal voltage.

Disregarding the infinite contribution from Eq.(3.47), the total current autocorrelation functionincluding Johnson noise is given by Eqs. (3.35) and(3.53) as

(II)-.= „1+0156 u, +2kT eEl '

I

(3.54)

the dots standing for higher-order terms in apower series in (eEl/kT)'. This parameter canbe simply interpreted as the ratio of the energygained between two collisions to the average ther-mal energy kT: it must be small for the termcalculated in this section to be the dominant cor-rection to Johnson noise. " Note also that inthermal equilibrium and when Qpp7 «1, the cur-rent and voltage autocorrelation functions satisfyto a good approximation the relation

with &T, the small temperature increase, de-termined by

dT d5' ]. t/'6T = —,—~t=-d8 dt Cp (3.57)

where C~ is the specific heat. (For sufficientlysmall V', OT will be small even after a relativelylong time. ) Using Eq. (2.17), the free-electronspecific heat appropriate for our impurity model,we can calculate 5T in Eq. (3.57) and substitutein Eq. (3.56) obtaining

( )2kT E/~~1 5t

(p Q gT] g y) (3.58)

lf we combine Eqs. (3.45), (3.47), (3.49), (3.50),(3.51), (2.10), (2.11), and (3.34), we can expressthe contribution of the infinite diagrams in Fig.4(h) in the form

(VV) = jt (jj)This relation is not valid for Eq. (3.54) how-

ever. When nonlinear fluctuations are importantthe nonlinear response cannot be neglected. In-deed, the nonlinear fluctuations and the nonlinearresponse are both related to various analyticpieces of the same imaginary frequency four-particle vertex. '"

The significance of the infinite vertex correc-tions can be understood by considering how John-son noise is modified when the Joule heat pro-duced by the weak currents does not escape. Aftersome time, the thermal equilibrium noise shouldincrease by an amount

5(II)„=(2/jt )k5T, (3.56)

Clearly then, with"

9 5t 5t7t' 7 7 (3.60)

the infinite diagrams correspond to the modifica-tion of Johnson noise that results when the tem-perature continues to increase linearly with timeforever. In the calculation of this section weassumed implicitly that the experiment lasted foran infinite time when we took the Fourier trans-form in Eq. (3.40). The plausibility of the identi-fication in Eq. (3.60) may perhaps be enhancedby rewriting the vertex (3.45) in the limit cu,= =seas2

I = ].+I'= 1+ 1+ 1+ 1+ 1+ (3.61)

p- ux —xi +s ++sx(3.62)

where s is a spin index and x; is the position ofthe impurities interacting with the conductionelectrons through the potential u(x-x;).

To calculate the total energy ckange producedby an electric field (described as usual, by avector potential), we must evaluate the graphsin Fig. '7, which correspond to

and arguing that if I" counts the number of col-lisions that occur during the time of the experi-ment, then it is indeed approximately equal to5t/T where 5t is the time of the experiment (in-finite in Sec. IV), and v is the average time be-tween collisions.

Naturally, Eq. (3.58) is not valid when 5t islarge since the specific heat Eq. (2.17) is a func-tion of temperature and thus will change withtime. [See also Eqs. (5.5) and (5.6) andaccompany-ing remarks. ] We shall see in Sec, IV how thei~finite quantities can be avoided and the heuristicarguments given above put on a firmer basis.

E. Equivalence between the Joule divergent vertex equationand a Boltzmann equation for the one-body distributionfunction: Calculation of heating effects in the current

fluctuations

To relate the calculation of heating effects onone-body and two-body correlation functions, itis helpful to calculate the second-order effectof the electric field on the energy density. Weshall find that this calculation has infinite vertexcorrections similar to those we encountered whenwe calculated the current fluctuations.

We take for the energy density operator"

$(x) = p ——. + 4, (x) —.+ @,(x)V eA ~ 1 7 eAC SPZ 2 C

acr(e~zt)' r (3.59) 8 (q = 0, v' = 0) = f 8 (x, t) d'C dt . (3.63)

MICROSCOPIC CALCULATION OF THE NONLINEAR CURRENT. . .

('v)

2AA

-v)

(V)

2AA

-v)

(0)4 (8+v + (v~-v)

(0)

—mA"

wctu

wg 4 &t ++V(0) ~~

V)~2AA

)W,) o)%l[2AA

~ ~ ge(0)

A~ -++(v~-v)-v)

A

(v—mA

(-V)

+(v~-v)(t)

(c)

(o)Q;

ill W4(-v)

—.-+ A(v)

Qf m '~A(v)

it lU +V(0)

&-+A(- v)

+(v ~-v)

{0i tt)+VQJ+V

(0)—~A(-v)

+(v ~-v) + (v~-v)+ (v ~-v)(e)

ttl

h[

Cd

v)

2 AA"

)

Note that we can evaluate the diagrams as if wewere calculating the time average of a quantityindependent of time, i.e., no frequency emergesfrom the b vertex. Vfe know, on the other hand,that the total energy must be a linear function oftime when a constant electric field is applied.That is why the infinite-vertex corrections (pro-portional to the "infinite" time of the experiment)enter. I et us show this fact more explicitly.

First, note that the three-vertex diagrams inFig. 7 (in contrast to those in Fig. 4) may con-tribute; the additional factor of (s =k'/2m —)i)associated with the kinetic energy vertex changesthe parity of the integrands.

It is tempting to infer from this argument thatthe diagrams with two corners vanish. However,this conclusion is fallacious. %ith the additionalfactor of e from the energy vertex, the integralsextend far inside the Fermi surface and particle-hole symmetry is violated.

Because the 4 associated with an energy vertexis always later on the KBL contour than the cor-responding 4, the rules of Sec. IIIB for "series"multiplication can be used to enumerate the varioustime Orders.

The most interesting graphs, those that giverise to Joule heat, are illustrated in Figs. 7(i)-

V(l). Writing the contribution of Fig. 7(j) in theform"

E, = —2i' 2@A* 2, —e GGd'k'

x)~

s'QQI"Q'g',

(3.64)

we can show, by performing an integration byparts in the first integral, that 6,. equals mingsthe v-0 limit of the expression correspondingto Fig. 7(i)." Adding both terms we find that theonly nonvanishing time orders are those illus-trated in Fig. 8.

Evaluating them, we obtain"

8, +~, =, ', 2~(0),

nial

~&* e'k~ V3m v+1 72 (3.65)

which, with the help of Eqs. (2.10), (2.11), and(3.51), can be written in the form

o,. + e,. = (V s/R) rVI'' (3 66)Note that the vertex I' appearing in Eqs. (3.65)and (3.66) is the complex conjugate of the oneappearing in Fig. 6 with cu, =~,=e.

The diagrams in Fig. V(k) and 7(1) cancel. Theleading contribution to the diagrams of Fig. 7

FIG. 7. Diagrams for the change in total energy caused by an appliedexternal. electrric field. -„- represents a,.4",

kinetic energy vertex (c); ' — —'- represents a potential energy vertex. The potential energy vertices are associatedwith a, factor n~ ~u~ ~(i)(-i) f dak' [G(k', a)]/(2z)~. The frequency &u is unchanged at the vertex. The other symbols aredefined in Fig. 4. The shaded areas represent vertex corrections.

TREMBLAY, PATTON, MARTIN, AND MALDAGUE

~G

—-+A

GR

GR~G

~) RgA

Gc

GRR

G

G

GAGA

is thus given by Eq. (3.66)

FIG. 8. Nonvanishing contributions to the diagrams ofFigs. 7(i) and 7(j).

—'~(v)I

!)( +

I

CU —~A(-v)

A—'~(-s)I

X tg +I QJ- VI

Ql ~ ~A(V)

, (FIG. 9. Diagrammatic representation of the equation

for the vertex defined in Eq. (3.73). The time ordersof the inhomogeneous terms are the same as those ofthe corresponding part of Fig. 5.

Figs. 7(i) and 7(j), with a net outgoing momentumq at the energy vertex, can be written in the form

b- (V /R)f v I . (3.67) h(q, 0)= 2N(0) div &uf i'~gQ . (3.74)

%e may determine the value of the vertexheuristically by noting that the overall factor of1 in Eq. (3.66) arises from the a,ssumed timeindependence of S. Consistent with this assump-tion, we may write

$(q= 0, p' = 0)/K = h(q= 0, K) = (U '/R)TI'. (3.68)

In the limit v7 «1 the vertex I then obeys theequation (see Fig. 9)

evp~ ., si(q, ro) = v, EE'+0(rrl) )'3 8&di dk I"(q, u)T 4' —Vp ' q+1/7' (3.76)

But it is well known that for a resistor

~(t) = (~'/R)t.

We see that Eqs. (3.68) and (3.69) agree ifI"= r/7,

(3.69)

(3.70)

From this equation and the definition (3.73), wededuce that

2 2 g2f'iq'V f +7 &(f 2 -f )=7 ~ E

3 ~co(3.76)

A'(t) =e(t)[Ae '"'+A*e'"] -e(t)[A+A*]

corresponding to the electric field

&'(t) =8(t)&(t)

(3.71)

(3.72)

[where E(t) is given by Eq. (3.38)], we can showthat the vertex equation we obtain is equivalentto Eq. (2. 6) for the perturbed distribution function.Since we are interested in stationary rather thantime-dependent solutions, we present insteada proof of the equivalence between the vertexequation with a finite momentum flowing throughand Eq. (2.13) for the distribution function. Ifwe define

f '~(u&, k 'q, q) =-I'(q, &u)/(v~ 'q-i/r), (3.73)then the analytical expression corresponding to

in qualitative agreement with Eq. (3.60)." Thus,even when the nonlinear response to a harmonicperturbation seems to occur at zero frequency(and be time independent), observable quantities(e.g. , the energy in this section and the currentautocorrelation function in Sec. IIID), can dependon the total time of the experimentt The similaritybetween Eqs. (3.70) and (3.60) suggests that thedivergence that occurs in two-body and one-bodycorrelation functions has the same origin. Indeed,using the vector potential

which should be compared with Eq. (2.13). Thevertex equation (3.75) with a nonzevo momentumtransfer q is equivalent to the Boltzmann equation(~ &3) The solution of Eq. (3.75) to leading orderin (ql)' is

r (q, (g) = — ~ —,[1+0 (ql) '] .ere' s'fg ) 8

(3.77)

x. . . [1+O(ql)'] .(do+ 1/T (3.78)

FIG. 10. The first diagram in Fig. 4(h) can be calcula-ted by multiplying the piece of diagram illustrated hereby the vertex of Fig. 9 and various constant factors. Allthe integrals except the one over ~ have been performedin the result quoted for this diagram in Eq. (3.78),

The vertex that was infinite before is now finiteand it can be used to calculate the Joule divergentdiagrams. For this purpose it is useful to ob-serve that the diagram element that appears inFig. 10 has the value

2/2I s~ = 27I'N(0)if(co + (do)~ Sm'

19 MICRO SCOPIC CALCULATION OF THE NONLINEAR CURRENT. . . 1733

Using Eqs. (3.77) and (3.78), we obtain from allthe diagrams in Fig 4. (h)

that in equilibrium the current correlation functionis given by the combination G G' shown in Fig. 2,or since

2kT m' 5T, eE/ '// 9 T kT G = —i[1 -f((o)](1/7 ) [G (k, (u)G (k, (u)], (3.82)

where

B(~.)((o,v')'+1 ' (3.79)

by the product of (-i)[1 f(-~)]v ' and the graphi-cal elements shown in Fig. 10. The nonequili-brium current fluctuations contribution in Fig.4(h) is obtained from the sa, me elements withFig. 9 replacing (-i) [1 f(ur—)j~ '

kT(3.80)

is the temperature increase consistent with Eq.(2.16).

The difference between a pure thermal gradientand this stationary solution for the vertex equation(distribution function) is discussed in the Ap-pendix.

The terms of order (q/)' in Eqs. (3.77) and(3.78), give rise, in Eq. (3.79), to (eE//kT)' cor-rections, smaller than the 5T,/T terms We.have not evaluated these corrections. Indeed, theygive rise to a technical problem we have not re-solved. It appears that the value of these termsdepends upon how the momentum q is distributedamong the current vertices. For example, if wecalculate the terms of order (q/)' in Eq. (3.78) for

(j(-,'q(1+ o.'), /) j(2q(1 o.), /')),

n being an arbitrary real number, we find, in thelimit ~,~ «1

e'k'0(q/)'= 2m'(0) ~~ if(&u+ u),)

x [(v~q)'/15] v'[(- 7 —o.')/2] . (3.81)An unambiguous result should not depend on o..A similar difficulty appears to plague Eq. (3.77).If we introduce a parameter P to describe thedivision of the momentum between the currentvertices exhibited in Fig. 9, the value of the termsof order (q/)' depends on P.

Finally, let us observe that (a) the change in thecurrent correlation function of a uniform systemis a factor of m'/9 larger than we would guess byinserting the temperature change calculated fromJoule heating [see Sec. IIID and Eqs. (3.57) and(3.60)]. This conclusion agrees with Huberman's'observation that a uniform electric field does notsimply raise the temperature of a degenerateelectron gas. (b) The vertex in Fig. 9 describesthe change in the one-particle distribution functionproduced by the electric field. This identificationmay be substantiated diagrammatically by noting

IV. SUMMARY OF THE kESULTS FOR THE CURRENTFLUCTUATIONS

e' o' ' ItIt' dt' —t2kT m' &T, eE/ B(~,)I/ 9 T kT ((nor} +1 '

(4 1)

where B(~,) incorporates the quantum correc-tions [see Eq. (3.49)]

lim B(+ )= 10, lim B(eo)=0.QQJ ~ p gQ) ~ac

In Eq. (4.1), 5T„ the amplitude of the Fouriercomponent of the "temperature gradient" in thesystem, satisfies Eq. (2.16), T is the tempera-ture, R is the resistance [Eq. (2.10)], k Boltz-

(4.2)

We have ascribed to Joule heating the divergentaddition to the current fluctuations we found byperturbation theory to second order in the electricfield. More specifically we have shown that thediverging vertex reflects the secular growth intime of the E' correction to the one-body distribu-tion function when there is no "cooling" mecha-nism. A finite, time-independent solution of thevertex equation (or equivalently the Boltzmannequation) can be obtained by including an additional"self-consistent" cooling gradient tra~verse"to the electric field. With this stationary vertex,we obtain finite current fluctuations.

The stationary distribution function cannot bedescribed precisely by a spatially dependent localtemperature [see Eq. (3.79) and the Appendix].As a result, the shifted values of most observ-ables cannot be obtained from simple macroscopicarguments and the usual definition of temperature.The energy density is a notable exception.

Our final result for the steady-state current-current correlation function of an electron gaswith impurities (which is a good model for a,metallic resistor at low temperatures) is

TREMBLAY, PATTON, MARTIN, AND MALDAGUE

mann's constant, I the mean free path [Eq. (3.13)],and e the electron charge. The first term of Eq.(4.1) is the equilibrium result (Johnson noise), thesecond term, of order 5T,/T, represents thefact that the "temperature" of the system is onthe average slightly higher when an electric fieldis present [Eq. (3.79)], and the third term repre-sents'corrections that have been calculated inSec. IIID [see Eq. (3.53)]. These corrections in-volve the expansion parameter (eEl/kT)', thesquare of the ratio of the energy gained betweencollisions to the average thermal energy. " Asnoted in Sec. III E, additional ambiguous cor-rections of the same ordeg arise from the dia-gram of Fig. 4(h) after it is made finite. Theseambiguous corrections may reflect the fact thatthe corrections to the boundary dependent hydro-dynamic terms are themselves sensitive to theboundaries. However, by setting q- L~' in Eq.(3.80) we find that

3 L eFt(4.3)

which indicates that the ambiguous correctionsare probably unobservable. We shall thereforenot study them further.

Sufficient conditions for the validity of Eq. (4.1)are listed below. They fall into several cate-gories. Presumably several of the conditionsare not necessary.

A. Validity of perturbation theory

8. Validity of the impurity scat tering model

(i) The expansion parameters (eEl/kT)' and5T,/T must be small. The latter expansionparameter allowed us to use the "standard" per-turbative solution of the Boltzmann equation for theinfinite vertex [see the remark below Eq. (2.6) andEqs. (2.13), (3.73), and (3.75)]. Note that Eq.(4.1) describes the nonlinear current fluctuationsalong the applied constant electric field. Whenthe corrections to Johnson noise are important,the distinction between corrections to voltage andcurrent fluctuations is also important; they arenot, in general, simply related [see remarkfollowing Eq. (3.55)].

I.i,terally, our model is restricted to impurityscattering potentials which have no bound stateand are very short ranged [see Eq. (3.6)]. Thecondition on the range enters through the assump-tion of s-wave, isotropic scattering, If the scat-tering is anisotropic, vertex corrections whichaccount for forward scattering will alter thevalue of 7„, the collision transport time. How-ever, in terms of T„our conclusions are probablyunchanged. '

C. Validity of hydrodynamic assumptions

(iv) The conditions for the Boltzmann equationshould be satisfied. In particular, the samplemust be large enough for the inequality I ~/l»lto hold [see remark before Eq. (3.77)] (L, - q ').

(v) The frequency may have to be larger thanthe rate at which the system exchanges energywith the substrate or other parts of the apparatus.Since, in general, these couplings involve longrelaxation times, they can only, affect correlationsat very low frequencies. If J ~ is the shortestsample dimension, it is inconceivable that thosecouplings could modify our conclusions at fre-quencies higher than the thermalization frequency(&/L, )'/3T

Some time ago, Hooge and Hoppenbrouwers"reported observing high-frequency correctionsto Johnson noise that scaled with the square ofthe applied voltage. Since 5T, in Eq. (4.1) is re-lated to the electric field via Eq. (4.3), we sur-mise that they were simply looking at the firstterm of Eq. (4.1) (heating). (See Note added inproof. ) The more recent experiments of theEindhoven group' mentioned in the Introductionand in Sec. V maybe more relevant to our calcu-lation.

V. CONCLUSION

In this final section we comment on the relationbetween our calculation and 1/f noise, indicatehow to extend our calculation, and reflect on whatwe have learned about nonequilibrium perturbationtheory.

{ii) Impurity scattering must be dominant.Typically this condition holds at low temperatureswhere phonon scattering is negligible (see Sec.IIIA).

(iii) The impurities must be dilute. This con-dition on their concentration may be expressedin the form k~jt»1, where k~ is the Fermi wavevector and l the mean free path [see Eq. (3.11)].

A. Could our model exhibit 1/f noise?

Our calculation does not give 1/f noise. Thisresult may be confirmed by recent experiments'which show that as impurity scattering becomesdominant, the amplitude of 1/f noise decreases.However, the systems investigated experimentallyare Ohmic metal semiconductor point contacts

MICROSCOPIC CALCULATION OF THE NONLINEAR CURRENT. . .

whose modeling by free electrons scattering offimpurities is not entirely unambiguous. If ourmodel does not apply to the systems studied byHooge and Vandamme' and exhibits 1/f noise,should we have found it'P

It might be argued that 1/f noise can only occurin a regime in which perturbation theory fails.After all, although 1/f noise is observed to beproportional to E', for sufficiently small fre-quencies it must become larger than the (fre-quency-independent) term of zero order in theelectric field (Johnson noise). In addition, theterm of order E' is supposed to be larger thanhigher-order terms since no dependence uponE', E', etc. has been reported. We do not believethat these arguments make a calculation to orderE' senseless. If our model contained 1/f noise,we believe the lowest-order term in perturbationtheory should at least show an anomaly in thefrequency dependence. This does not mean thatit would not be valuable to treat our model with-out making expansions in the electric field. " 4'We doubt, however, that when 5T,/T&1 the re-sults would differ qualitatively from those we havefound.

Some investigations of the stationarity of the1/f phenomenon have suggested that I/f fluctua-tions imply that there is no stationary state. Notal1. experiments agree. " Since our calculationdoes not yield 1/f noise, there is little reasonfor expecting it to give additional credence tononstationarity. Indeed, at least in perturbationtheory, there is little doubt that our model has astationary state.

Moreover, the absence of 1/f noise cannot beascribed either to the omission of terms inverselyproportional to the volume, or to a disregard ofvarying collision times. A slightly different modelwith a distribution of scattering times would notbehave differently.

1. Factor of 1/N

Apart from the term of order 5T,/T, the currentfluctuations we have obtained in Eq. (4.1) are pro-portional to 1/N, where N is the number of chargecarriers. This factor is present in the pheno-menological description of 1/f noise' and might beexpected from general statistical arguments. Inour calculation, if we suppose that the nonlinearresistance is given by

/

R(E) =8 1+5 i +.AT) (5.1)

where 5 is a numerical factor, then, apart fromnumerical factors, we can write

(5.2)

Using V =El, 0 =XI and Eq. (2.10) we find that

(5 3)

where N=nQ is the number of charge carriers.

2. Natthiessen's rule

We cannot obtain 1/f noise from our model byincluding many decay times for the electrons be-cause our system obeys Matthiessen's rule. " Thisrule, which can be deduced from the propagatorequation in the dilute limit k~l ++ 1 implies, forexample, that if there are two types of scatteringimpurities described by the lifetimes T, and 7,,our result applies if we set 7 =7g +72 . Thiscomposition law should be contrasted with theaddition of the current autocorrelation functionsdue to 7, alone and T, alone. The latter composi-tion law, which is sometiines used to explain I/fnoise, ' may be valid for distinguishable, inde-pendent sources of noise, but it is certainly notvalid for impurity scattering in metal films, evenin the "classical limit" Pcyo«1.

To summarize, we do not believe that our modelor slight variants of it exhibit 1/f noise but ourarguments, which are based on perturbationtheory, are not rigorous. We cannot exclude thepossibility that 1/f noise arises entirely fromnonanalytical contributions that perturbation theoryomits.

B. Physical processes neglected in our calculation whichcould lead to 1jf noise

The only microscopic time in our model is thescattering time T. Longer times could be included,for example, by taking into account the substrate-sample interaction more realistically" and/orincluding the effect of hydrodynamic modes.

Since our calculation was performed for thebulk, " it does not account for all couplings tohydrodynamic modes that can conceivably occur. "However, it should be noted that our model cannotgive rise to 1/f noise via the mechanism of Vossand Clarke. " Our model has a temperature-in-dependent resistance and their proposal is basedon the temperature dependence of the resistance.Despite this fact, our model does involve a coup-ling to the energy density. " Although the re-sistance is temperature independent, the currentfluctuations are not. Note also that the first termin Eq. (4.1) is size dependent and thus in somesense is sensitive to the manner in which Jouleheat is dissipated. This agrees indirectly with

&v36 TREMBLAY, PATTON, MARTIN, AND MALDAGUK 19

some experiments. 55

Obviously many other processes and moresophisticated models could be investigated. Forexample, we might consider phonons"'" that dis-appear at the boundary of the system, liberatingJoule heat to the surroundings. Whether or notsuch models would be qualitatively different fromours is a matter on which we can only speculate.

C. Nonequilibrium perturbation theory

Our calculation shows that the manifestationsof heating must be understood before realisticcalculations on many-body systems beyond linearorder can be performed. It also sheds some lighton the general structure of high-order many-bodyresponse theory.

l. Adiabatic s~itchingWe have found that, in nonlinear order, when

an external perturbation is applied for a finitetime, secular terms that describe heating grow.Thus if the perturbation is applied adiabatically,divergences can occur. The secular terms thatoccur when the field is adiabatically applied canbe eliminated by treating the transfer of "heat"to the outside, simultaneously and "self-con-sistently. " In the case we studied, this self-consistency is achieved by including gradientterms. Stationarity could also be achieved byintroducing phonons which can disappear at theboundaries or are constrained to remain inequilibrium. "'" Once a satisfactory "cooling"mechanism has been included a calculation of theeffect of applying an external (electric) field adia-batically will not diverge.

2. Fluctuation-dissipation theorems

The basic idea behind recent efforts" " to de-rive generalizations of the Quctuation-dissipationtheorem resembles the idea behind the work ofBernard and Callen' even though the methods thathave been employed have evolved considerably. "The present work illustrates some of the dif-ficulties that are bound to plague such efforts.

Even if our calculations were performed using"equilibrium" diagrams so that what happens inthe nonequilibrium state is in some remote senseassociated with the behavior of the equilibriumstate, we wish to emphasize two difficulties (whichhave not to our knowledge been appreciated pre-viously) and which must plague any attempt torelate the two-current correlation function forsteady-state nonequilibrium experiments and thefour-current equilibrium correlation function:(a) Some of the "time-orders" of the four-currentcorrelation functions that are needed to calculatethe two-current nonequilibrium correlation func-

3. Structure of the perturbation expansion: dynamicsand heating

The successive terms of a perturbation ex-pansion in powers of an applied external fieldcertainly represent for any physical system (e.g. ,an atom) corrections to the dynamics of the sys-tem induced by the field. For a many-body sys-tem, however, there are also corrections to thestatistics of the system that occur naturally asterms that depend on powers of the applied field,multiplied by the time. Loosely speaking, wecan say that for statistical systems, there is anew expansion parameter in the theory given by5T/T, the ratio of the "temperature" change in-duced by the field to the starting equilibriumtemperature.

The results derived in Ref. 60 suggest stronglythat these terms of order 5T/T occur as follows:If the temperature T appears in the response ofan observable to a given order in powers of theapplied external field, then an infinitesimal risein the temperature caused by the field will mani-fest itself only in the next-higher-order terms.This is consistent with the behavior of the cor-rections to the current fluctuations and to the re-sistance in our model. "

We also surmise that if the temperature changein the system is appreciable, it must manifestitself through higher-and-higher-order termsin the perturbation expansion. (Note that Joulepoles are rigorously excluded only in the zero-and linear-order terms of the expansion. ) Thisshould be clear from the fact that Eq. (3.58) forthe nonequilibrium current fluctuations yields,roughly speaking,

2kT(eEl)' 1 (lit) (5.4)

where 5t is the length of time (+T) during whichthe field has acted. Equation (5.4) cannot be validfor arbitrarily large 6t since we know that

tion cannot really be measured in. the equilibriumstate since they diverge as q- 0. (b) Any genera-lized fluctuation-dissipation theorem must reflectthe restrictions implied by the fact that the non-equilibrium state depends, to nonlinear order, onthe way the system is cooled. "

It is important to realize that, in our calcula-tion, the equilibrium density matrix played therole of an initial condition: it cannot act as athermostat preventing the system from heating.

The fact that in our model, when no cooling ispresent, the current fluctuations are divergent

. to order E' while the resistance is finite" toorder E' further illustrates the difference betweenresponse and fluctuations to nonlinear order.

19 MICROSCOPIC CALCULATION OF THE NONLINEAR CURRENT. . .

5(a).,- (2u/R)(r, r—), (5.5)where T& can be obtained by integrating Eq. (3.57),using the value (2.17) for the specific heat:

( )2lrT -i 2 Zl~

aT) 7 (5 6)

Clearly, Eq. (5.6) reduces to Eq. (5.4) when 5tis small enough. We expect that some of thehigher-order terms in the perturbation expansionshould correspond to the terms obtained from ex-panding the square root in Eq. (5.6). If this istrue, it suggests that the perturbation series willbreak down for a 5t (or a temperature rise) givenby the radius of convergence of the Taylor seriesfor the square root: [(Tz —T)/T 0.4]. (This isclearly only a rough estimate. Other correlationfunctions will in general have a functional de-pendence on the temperature very different fromthat of the current fluctuations. ) This also sug-gests that a perturbation theory to order E'might not be meaningful if the. heating effect ofthe electric field makes the temperature grow toan effective value T'»T. In such a case, manyhigher-order terms in the perturbation expansionmust be taken into account and it is not obviousthat these would only lead to an effective tem-perature T'.'

We summarize the most important remarks ofthis section by writing the perturbation expansionfor the current response to an applied fieldschematically

VI= —+ AV'+B 5TV i+0(V')+ ~ . (5.7)=8 ' BTV is the applied external potential and V/R theusual linear response. The first term in theparentheses (AV') represents the change in thedynamics of the system induced by the field; thesecond term, the change in the temperature (sta-tistics). The magnitude is also of order V' sincethe leading term in 6T is of order V'.

4. Justifying 1ineur-response theory

There has been some criticism of linear-re-sponse theory. Equation (5.7) clearly' shows thatthe complete theory does involve terms which,as a function of time, rapidly become larger thanthe linear-order terms. For example, we canestimate that at room temperature, the secondterm in the parentheses of Eq. (5.7) becomesdominant after less than 1 sec, for an appliedexternal field of 2.5x 10 ' V/cm, a mean freepath of order 10 ' cm and a collision time of order10 "sec [see Eq. (2.29)].

One second is rpuch larger than the durationtime of most experiments. However, even for

ACKNOWLEDGMENTS

We would like to acknowledge valuable discus-sions with many colleagues including VinayAmbegaokar, John Clarke, Paul Horn, MarkHuberman, Rolf Landauer, Karel van Vliet, andRichard Voss. The work was supported in partby the NSF under Grant Nos. DMR-77-10210and DMR-02988A03. Fellowship support to A. -M.Tremblay from the National Research Councilof Canada and the Ministhre de I'Education duQuebec is also gratefully acknowledged.

APPENDIX

In this Appendix, we show that the stationarydistribution function found from Eq. (2.13) or(3.77) and (3.73) cannot be exactly described bya local temperature. When the steady state canbe described by a Fermi distribution functioncharacterized by a local, spatially varying tem-perature we find that for times much larger thanv, but much less than the thermalization time,the perturbative solution to the Boltzmann equa-tion satisfies

f ——7V'V f (A1)A

f (2) ~l r (~v, Q )2f (o)t (A2)~ 4m

Substituting Eq. (A2) into Eq. (2.7) we obtain

~8g,at = KV' T+ —V'Twhere z is given by Eq; (2.18). This supports theinterpretation of the solution Eq. (A2) in termsof the classical notion of a temperature gradient. "

On the other hand, in the presence of an electricfield, Eq. (2.6) gave us

steady-state experiments that last a long time,the use of linear-response theory can be justifiedby allowing for the cooling mechanism that exactlycancels the secular heating terms in Eq. (5.7).If the cancellation were not exact the systemcould not be in a steady state. Our calculationgives an example of how such a cancellationarises. Equation (5.7) then shows that as longas the field is small enough so the terms of theform A. V' in Eq. (5.7) are negligible only thelinear terms remain. Often the cooling mechanismleads to additional linear effects but these areusually easily treated. In the case of a thermalgradient, for example, the cooling mechanismwould introduce a, thermoelectric term in Eq. (5.7).'0

Note added in Proof Age. neralization of thishypothesis was verified by T. G. M. Kleinpenning[Physica B 84, 353 (1976)].

TREIQBLAY, PATTON, MARTIN, AND MALDAGUE 19

& ~.~OUie ——1. (A6)

This result should be compared with Eq. (2.26).Equation (A6) determines the temperature fieldand thus the solution (A2). Using this solution andEqs. (2.6) and (2.27) we find for an arbitrary ob-servable, to leading order in kT/E~,

(A)...,. ez 'T ( "„,s'f, ( ))(A)„„, q 6T ( se'x d'g e —1""c

This ratio differs in general from one observableto another. " Consequently, when 6T is fixed byEq (A6), . for most observables we will obtain

(AB)

where C is a time-independent constant. Thisproves that a distribution function described by a

We might prefer to set up a stationary state byadjusting the temperature field so that the re-sultant total energy is time independent

(A6)

From Eqs. (A3) and (A4) we see that when Eq.(2.21) .is satisfied

local temperature cannot lead to a stationary so-lution in the presence of an electric field.

The correct stationary solution Eqs. (3.73) arid(3.77) is energy conserving, and thus can be in-terpreted in terms of Eq. (2.26) but its detailedenergy and momentum dependence is differentfrom the simple ansatz corresponding to a Fermidistribution with a spatially dependent temperature.All of this is in agreement with. the results ofHuberman' quoted at the end of Sec. IIIE.

Note that Eq (A.7) can also be derived from dia-grams by taking the ratio of the responses of aone-body observable to an electric field (to secondorder) and to a temperature gradient. " Since theobservable on which the "electric" and "thermal"vertices are "closed" is arbitrary, it is not sur-prising that the effect of both perturbations willnot be the same unless their .energy and momentumdependence is the same. " Thus, in this context,Eq. {A7) also says that the energy and momentumdependence of a "simple" temperature perturbationcannot exactly cancel the effect of an electric fieldperturbation to second order.

Finally note that although a stationary one-par-ticle distribution function can make all the observ-ables in Eq. (2.27). time independent, we knowof no proof that the two-particle correlation func-tions must be stationary whenever the one-particledistribution function is. This is nevertheless whatwe have found for the special case considered here.

*New address: LASSP, Clark Hall, Cornell University,Ithaca, N. Y. 14853.

)New address: Physics Dept. , Ohio State University,Columbus, Ohio 43210.

fNew address: Physics Dept. , Brande&s University,Waltham, Mass. 02154.

~W. Bernard and H. B. Callen, Rev. Mod. Phys. 31,1017 (1959).

J. Schwinger, J. Math. Phys. 2, 407 (1961); L. V.Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964). fSov.Phys. JETP 20, 1018 (1965)i; H. A. Craig, J. Math.Phys. 9, 605 (1968).

3D. C. Langreth, in l975 NATO Advanced Study Instituteon Linear and Non-Linear Electron Transport in Solids,Antwerp, Belgium, edited by J. T. Devreese and V. E.van Doren (Plenum, New York, 1976), p. 3.

C. N. Hooge, Physica B 83, 14 (1976) and referencestherein for a review.

5F. N. Hooge and L. K. J. Vandamme, Phys. Lett. A 66,315 (1978).

~Because our frequency integrals extend from -~ to+~, 2A ~kT is the correct result. The conventionalvalue 4A~kT applies when the frequency integral ex-tends from 0 to ~.

~M. L. Huberman (private communication); and Ph.D.

thesis (Cornell University, 1975) (unpublished).A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzyaloshinski,Methods of Quantum Iield Theory in Statistical Physics(Prentice-Hall, Englewood Cliffs, N. J., 1963), p. 327and following.

~P. M, Morse and H. I"eshba, ch, Methods of TheoreticalPhysics (McGraw-Hill, New York, 1953}.N. W. Ashcroft and N. D. Mermin, Solid State Physics(Holt, Reinhart, and Winston, New York, 1976), p. 47.

~~Reference 10, p. 255. This result can be checked ex-plicitly for the model we use.

~2See Ref. 10, p. 261.~3When the resistance is temperature dependent, there

should probably be slight differences of order 6T/Tbetween the two situations.

~4V. Ambegaokar, in SuPerconductivity, edited by R. D.Parks (Marcel Dekker, New York, 1969), p. 313.

~S. Fujita, J. Phys. Soc. Jpn. Suppl. 26, 238 (1969).~6We suppose that the propagators are diagonal in the

spin indices which we omit henceforth.~YThe method we briefly review is well explained in the

article by Langreth (Ref. 3).~8L. P. Kadanoff and G. Baym, Quantum Statistical

Mechanics (Benjamin, Menlo Park, 1962), pp. 5, 6.Note that their definitions sometimes differ from ours.

MICRO S CO PIC CALCULATION OF THE NONLINEAR CURRENT. . . 1739

J. R. Schrieffer, The Theory of Superconductivity(Benjamin, Reading, Mass. , 1964). See p. 207 forthe notation used here.A. L. Fetter and J.D. Walecka, Quantum Theory ofMany Particle Systems (McGraw-Hill, New York,1971).

2~For conciseness, the spatial coordinates have beenomitted.

2I.. P. Gor'kov and G. M. Eliashberg, Zh. Eksp. Teor.Fiz. 54, 612 (1968) [Sov. Phys. JETP 27, 328 (1968)].

23See Ref. 24, Chap. I for more details.A.-M. Tremblay, Ph.D. thesis (Massachusetts Instituteof Technology, 1978) (unpublished).F. Reif, Fundamentals of Statistical and ThermalPhysics (McGraw-Hill, New York, 1965), pp. 583-594.

26Diagrams composed of two distinct pieces, each ofwhich is connected to a different current Ineasurement,correspond to (j(t')) (j(t)), and thus will be ne-glected. Only the linked diagrams contribute to thecurrent fluctuations.

2~This would not be allowed if we were calculating theretarded or advanced correlation functions (Ref. 8,pp. 312 and 341). The subtractions, needed to performthe ~ integration first, cancel in Eq. (3.33) becausethe correlation function is proportional to the differ-ence between the retarded and advanced functions andthe subtractions needed for these are identical. SeeRef. 24, Chap. IV, Appendix B for more details.

28I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals,Series, and Products (Academic, New York, 1965),p. 306, integral 3.315.2.See Ref. 24, Chap. IV, Appendix D.

30See Ref. 19, pp. 224-240.3~See Ref. 24, Chap. IV, Appendix C.

D. Forster, hydrodynamic Fluctuations, BrokenSymmetry, and Correlation Functions (Benjamin,Beading, Mass. , 1975), Eq. (4.44a); J. M. Luttinger,Phys. Bev. 135, A1505 (1964).

33This cancellation is analogous to Furry's theorem.See Ref. 24, Chap. IV, Sec. 4.

34The effective constant applied electric field shouldequal the average of the square of the low-frequency(v 0) field. This can be made more precise; seeRef. 24, Cha, p. IV, Appendix D.

35The same parameter appears in Huberman's {Ref. 7)long-time expansion for the Boltzmann equation.

3 We shall comment. later on the factor of 9/z .3~h(g) =P ~ [@, (x} (i8/ Bt )@~ (x) +H.c.] can also be used

in principle but great care must be exercised. SeeRef. 24, Chap. IV, Appendix E.

8This expression is written in terms of the KBL propa-gators defined in Eq. (3.22).

39In other words, there is a Ward identity (cf. Ref. 19).relating both types of diagrams.The integral involving e in the numerator can be evalu-

. ated by contour integration only after performing afew manipulations to recast it in a form well behavedat jnflnlty

+The identification in Eq. (3.70) contains an ambiguousfactor of 2 which depends on the heuristic arguments.

42The direction of the heat flow should not affect thefirst term of Eq. {4.1). However, if the gradient wasin the direction of the current measurement, it couldbe coupled to the hydrodynamic modes. See the follow-mg footnote.

3By analogy with Sec.III E we should also allow for a. finitemomentum transfer q in the finite graphs of Sec. III D.However, unless the coupling to the hydrodynamicmodes transverse to the current measurement issingular, this will only produce corrections of order(ql)~ (eEl /kT)2.

44F. N. Hooge and A. M. H. Hoppenbrouwers, Physica45, 386 (1969).

45Some steps towards a nonperturbative approach haveapparently been taken by Qh. M. Kogan and A. Ya.Shul'man, Zh. Eksp. Teor. Fiz. 56, 862 (1969) [Sov.Phys. JETP 29, 467 (1969)] and by the. authors in Refs.46 and 47. We were unaware of Refs. 45-47 when thecalculation was performed

46S. V. Gantsevich, V. L. Gurevich, and R. Katilyus,Zh. Eksp. Teor. Fiz. 57, 503 (1969) [Sov. Phys. JETP30, 276 (197O)].

4~S. S. Rozhkov and P. M. Tomchuk, Zh. Eksp. Teor.Fiz. 72, 248 (1977) [Sov. Phys. JETP 45, 130 (1977)].

48M. Stoisiek and D. Wolf, J.Appl. Phys. 47, 362 (1976);W. J.Moore, J.Appl. Phys. 45, 1896 (1974); R. A.Dell, M. Epstein, and C. R. Kannewurf, J.Appl. Phys.44, 472 (1973); F. N. Hooge and A. M. H. Hoppen-brouwers, Physica 42, 33 (1969); J.J. Brophy, Phys.Rev. 116, 827 (1968).

"See Ref. 1O, p. 323.50For a theoretical review with a good bibliography see

A. van der Ziel, in Proceedings of the Symposium on1/f Fluctuations Tokyo, July 1977, edited by H. Fukuyoand T. Musha (Tokyo Institute of Technology, NagatsutaMidori-ku, Yokohama, 1977) and also Ref. 5. See alsothe introductory remarks of P. H. Handel, in l975NATO Advanced Study Institute on Linear and Non-Linear Electron Transport in Solids, Antwerp, BeI.—gium, edited by J. T. Devreese and V. E. van Doren(Plenum, New York, 1976), p. 515,

~~See S. H. Liu, Phys. Rev. B 16, 4218 (1977) for a re-cent theoretical attempt to incorporate substrate-sample interactions in the temperature fluctuationmodel of Voss and Clarke (Ref. 53). J. Clarke andY. Hsjang, Phys. Rev. Lett. 34, 1217 (1975) found ex-perimental evidence for the importance of the sub-strate in some cases. See also T.G. M. KleinpenningPhysica B 84, 353 (1976).

52The density dependence of the resistance, for example,could produce a coupling to the density mode.R. F. Voss and J. Clarke, Phys. Rev. B 13, 556 (1976).For the calculation of Fig. 4(h) in Sec. III E we neededq finite {cf.Ref. 43).

5~J. Clarke and M. B.Ketchen, Phys. Bev. B17, 114 (1978).Becentexperimentsby J.W. Eberhard and P. M. Horn[Phys. Bev. B 18, 6681 (1978)]suggest that the substrateis important for at least one of the "two types" (typeA) of 1/f noise they identified. However, theseauthors, as well as many others [for example, F. N.Hooge, Phys. Lett, A 29, 139 (1969)] suggest that, atleast in some systems, the surfaces are unimportant.See also T. G. M. Kleinpenning, Physica B 84, 353(1976).

5~G. N. Bochkov and Yu. E. Kuzovlev, Zh. Eksp. Teor.Fiz. 72, 238 {1977)[Sov. Phys. JETP 45, 125 (1977)].

5~R. L. Stratonovich, Zh. Eksp. Teor. Fiz. 58, 1612(1970) [Sov. Phys. JETP 31, 864 (1970)].

5 G. F. Efremov, Zh. Eksp. Teor. Fiz. 55, 2322 (1969)[Sov. Phys. JETP 28, 1232 (1969)].

TREMBLAY, PATTON, MARTIN, AND MALDAGUE 19

+While the Joule divergent diagrams are greatly in-fluenced by the cooling mechanism, the finite diagramsin Fig. 4 are not (cf. Ref. :43).

60See Hef. 24, Chap. IV, Appendix F.@We also note that the results that can be computed

with Eq. (A2) are exactly the same as those obtaineddiagrammatically using the prescription of L, P. Kad-anoff and P. C. Martin, Ann. Phys. 24, 419(1963),App-endix C, to def ine the inter action Hamilton ian corres-ponding to a temperature gradient. The latter method

can also be extended to two-body observabies (Ref. 24,Chap. IV, Appendix E).A similar expression can be derived for multiparticleobservables.

63At times larger than the thermalization time, the re-sult 58 = C&6T for a temperature perturbation is ob-viously different from Eq. (A4) for an electric fieldperturbation. We are concerned here with a cancella-tion between electric and thermal perturbations at"short times" when Eq. (A2) is valid.