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Noise Thermal Impedance of a Diffusive Wire B. Reulet1,2 and D.E. Prober1 1Departments of Applied Physics and Physics, Yale University, New Haven CT 06520-8284, USA 2Laboratoire de Physique des Solides, UMR8502, bât 510, Université Paris-Sud 91405 Orsay, France 5 (Dated: February 2, 2008) 0 0 The current noise density S2 of a conductor in equilibrium, the Johnson noise, is determined 2 by its temperature T: S2 = 4kBTG with G the conductance. The sample’s noise temperature n TN = S2/(4kBG) generalizes T for a system out of equilibrium. We introduce the ”noise thermal a impedance” of a sample as the amplitude of the oscillation of TN when heated by an oscillating J power. For a macroscopic sample, it is the usual thermal impedance. We show for a diffusive wire 7 how this (complex) frequency-dependent quantity gives access to the electron-phonon interaction 1 time in a long wire and to the diffusion time in a shorter one, and how its real part may also give access to the electron-electron inelastic time. These times are not simply accessible from the ] frequency dependenceof S2 itself. l l a PACSnumbers: 72.70.+m,05.40.-a,73.23.-b,07.57.Kp h - s e For decades, measurements of the electrical response is closely related to the recently proposed and explored m of condensed matter systems have provided powerful in- third moment of the current noise[7, 8]. . sights into the physics at the macro- and micro-scale. t We consider a conductor biased by a time-dependent a Measurements of the conductance G = dI/dV (with I voltage V(t) = V +δV cosωt. For simplicity we treat m dc the current when biased by a voltage V), and of the eδV eV ,k T. The meansquarecurrentfluctuations dc B - noise (the variance of the current fluctuations) are ex- ≪ d aremeasuredthroughthe spectraldensityof the current amples of the many successful approaches[1]. Yet, for n noise, S , integrated over a frequency band ∆Ω around 2 o good conductors, the frequency dependence of the con- thefrequencyΩ. Inequilibrium(V(t)=0),S =4k TG 2 B c ductance and the noise is determined only by charge with G the electrical conductance, taken to be indepen- [ screening by the electron fluid, due to the long range dent of T and V[9]. S is averaged over a time τ such 1 interaction of the electrons. For normal metals (non- that ω τ−1 ∆Ω,2Ω, to give S (t). Experimemntally, v superconductors), the physics associated with inelastic ≪ m ≪ 2 this could be implemented by coupling the sample noise 7 processes,energyexchange,dephasing,ordiffusiontimes, through a bandpass filter centered at frequency Ω to a 9 is usually accessible only in the quantum corrections[2] 3 bolometer with a response time τm [10]. We treat low- 1 or fromtunneling measurementson specific materials[3]. frequency noise, ~Ω eV ,k T, so our conclusions do dc B 0 Superconductorsareanexceptionsincethegap,andnear not depend on Ω, ∆Ω≪or τ . Under the time-dependent m 5 TC the resistance, is sensitive to the distribution of ex- bias V(t), S (t) is amplitude modulated at frequency ω 0 citations. Thus, time- or frequency-domain transport 2 (see inset, Fig. 2). We define the sample’s noise tem- t/ measurements provide direct access to the time scales of perature T (t) = S (t)/(4k G), and the instantaneous a N 2 B m microscopic processes, such as electron-phonon inelastic Joule power P = I(t)V(t) = GV2(t) dissipated in the J relaxation[4] , diffusion removal of energy [5], or quasi- sample. Fromtheir(complex)componentsδTω andδPω - particle recombination[6]. N J d atfrequency ω, one defines the (complex)response func- n tion (ω)=δTω/δPω. measures how much the noise o In this letter we develop the theory for a novel ap- R N J R temperature oscillates when the system is heated by an c proach to directly measure dynamic processes of elec- v: trons in a normal metal. For an electron system in oscillating power. R has units of a thermal resistance, K/W. For a macroscopic sample, T is the sample tem- i equilibrium (V = 0), the temperature is reflected in the N X perature, and is simply the thermal impedance be- Fermi-Dirac distribution of state occupancy and can be R r tweenthesampleanditsenvironment. Thus,wewillcall determined from the Johnson-Nyquist noise. If the oc- a (ω) the ”noise thermal impedance” (NTI) of the sam- cupancy is perturbed in a charge-neutral fashion, its re- R ple at frequency ω. It exhibits a frequency dependence laxation is governed by the microscopic processes that on the scale of the inverse thermal relaxation time. For we wish to access. One can determine the relaxation a thin film or wire at low temperature, as we consider of the electron temperature (and more generally, excita- later, this thermalization time is determined by energy tions that are charge-neutral)from the time-dependence removalprocessesexperiencedbytheelectrons(electron- of the magnitude of the noise (measured at frequencies hole relaxation)[11]. much higher than the inverse relaxation time) when the systemisdrivenbyanacvoltage. Ourideais,inessence, In the following we calculate (ω) for a diffusive wire R to use the driven noise to determine the dynamics. This of length L between two normalmetal reservoirs(see in- 2 set of Fig. 1), in several limiting cases: i) long wires This occurs on a time scale set by the diffusion time τ . D Le−e Le−ph L; ii) wires of intermediate length OnecanstilldefinealocaltemperatureTe(x,t)sincethe ≪ ≪ Le−e L Le−ph; and iii) short wires L Le−e. electronsequilibratewitheachotherlocally. Teispeaked ≪ ≪ ≪ Here Le−ph stands for the electron-phonon interaction along the wire, given by: length and Le−e for the energy relaxation length due to ∂T ∂ ∂T electron-electron (e-e) interaction. These lengths, much C (x,t) e =P (t)+ G (x,t) e (3) e J WF longer than the mean free path, are related to the corre- ∂t ∂x(cid:18) ∂x (cid:19) sponding times by, e.g., L2e−e =Dτe−e with D the diffu- with the boundary conditions: T (0,t) = T (1,t) = T, sion coefficient. We show that (ω) gives access to the e e R with T the temperature of the contacts. x denotes the electron-phonon relaxation time in i), and that it gives position along the wire in units of L: 0 x 1. G the diffusion time τD = L2/D for ii) and iii). The two is the electron thermal conductance, rela≤ted t≤o the eWlecF- latter cases differ significantly if one measures the real tricalconductanceGthroughtheWiedemann-Franzlaw: and imaginary parts of (ω). Using this difference one R G = GT with =(π2/3)(k /e)2theLorentznum- can probe the electron-electroninteraction time. Finally WF L e L B ber. Eq. (3) is linear in T2 and we compute: we discuss how (ω) is related to the third cumulant of e R the noise and its environmental corrections. T2(x,t)=T2(x)+2Re[A(x,ω)exp(iωt)] (4) e 0 Ageneralpropertiesof (ω)is that, atlow frequency, R themagnitudeofthenoisefollowsadiabaticallythevolt- whereT issolutionofthedccase,T2(x)=T2(1+αx(1 0 0 − age variations, such that: R(ω = 0) = dTN/dPJ. The x)) with α = (3/π2)(eVdc)2/(kBT)2, and A the ac solu- Joule power has a component at frequency ω, δPω = tion of the V = 0 case (usually called the ”weak heat- J dc 2GVdcδV,soonehasR(0)=(dS2/dV)/(8kBG2Vdc)[12]. ing” limit), for which T0 =T. We find: i) We first consider a long wire, L Le−ph. The electrons give the energy they acquire f≫rom the electric δPωT coshq(x 1/2) A(x,ω)= J 1 − (5) field to the phonons. We refer to this regime as phonon- G q2 (cid:18) − coshq/2 (cid:19) WF cooled. For a wire made of a thin film, the phonons of the film and substrate are well coupled and repre- withq =√iωτD. ForasmallacexcitationandVdc finite, sent the thermal bath[4]. Phonon emission occurs uni- the ac response of the electron temperature is given by: formly in the wire, except near the ends, on a length δTeω(x) = A(x,ω)/T0(x). TN is the average of Te along Le−ph where the hot electrons can leave the sample the wire. For eVdc ≪kBT, T0(x)≃T and we obtain: without emitting a phonon. Such finite length effects (ω) q 2tanh(q/2) are negligible for L Le−ph, so we consider the elec- R =12 − (6) ≫ (0) q3 tron temperature T (t) = T (t) to be position indepen- e N R dent. In the absence of ac excitation, Te(T,Vdc) is such and (0) = G−1 /12. We do not have an analytical that the phonon cooling power Pe−ph(Te,T) equals the expreRssion for W(Fω) for all V , but numerical calcula- Joule power GV2. The electron-phonon thermal con- R dc dc tions show that the dependence of (ω)/ (0) on V is ductance Ge−ph =dPe−ph/dTe has been studied with dc extremely weak. Curves for differeRnt valRues of V dcare heating[13]. For ac excitation we have: dc indistinguishable on a linear plot. Real and imaginary Ce(t)dTe =PJ(t) Pe−ph(Te,T) (1) p1.artAstohfiRgh(ωfr)eaqsueancfyu,ncωtτion of ω1,τDRea[re(ωpl)o]tdteedcaiynsFliikge. dt − D ω−3/2 whereas Im[ (ω)] dec≫ays like ωR−1. The magni- where the phonons remain at temperature T [11]; Ce = tude (ω) is plotRted on Fig. 2. The frequency for γTe is the electron heat capacity. The electron tempera- which|R (ω|)2 = 1/2, i.e., the bandwidth of this ”ther- ture oscillates: Te(t)=Te(Vdc)+Re[δTeωexp(iωt)], and: mal” re|Rspons|e, is ∼10τD−1. iii) We now consider the case of elastic transport, R(ω)= δδPTeJωω = (1+Gi−eω−1τpeh−ph) (2) tLhe≪elLece−treo.nTshtirsavisetlhaeloinndgetpheendweinret-ewleitchtorountreexgpimereie,nscinincge inelastic collisions. There is no local temperature, but with τe−ph =Ce/Ge−ph the electron-phonon time at Te. one can define a local noise temperature: (ω) is the electron-phonon thermal impedance at tem- RperatureT (T,V ). MeasurementsofT (t)foravoltage +∞ e dc N T (x,t)= f(x,E,t)(1 f(x,E,t))dE/k (7) step have recently been undertaken [14]. N Z−∞ − B ii) We now turn to the case of intermediate length, Le−e L Le−ph. This is the hot electron, diffusion- where f(x,E,t) stands for the local energy distribution ≪ ≪ cooled regime. The energy stored in the sample relaxes function in the wire. If f is a Fermi function at tem- because energetic electrons leave the sample and are re- perature T, Eq. (7) gives T = T. The wire’s noise N placed by thermalized ones coming from the reservoirs. temperature T (t) is the average of T (x,t) along the N N 3 1.0 1 0.8 Re 8 L 6 0.6 4 ) R(0 0.4 0) ) 0.2 R( 2 i(t )/ R(w 0.0 )/0.1 R(w 8 |R|, hot el. -0.2 6 |R|, indep. el. -0.4 Im hot electrons 4 Re[R], hot el. independent el. Re[R], indep. el. 0 10 20 30 40 50 2 tw 2 4 6 8 2 4 6 8 2 D 1 10 100 tw D FIG. 1: Real and imaginary parts of R(ω) as a function of ωτD, in the hot electron, diffusion cooled regime ii) (solid FIG.2: Magnitude (thicklines) and real parts(thinlines) of lines)andindependentelectronregimeiii)(dashedlines). The R(ω) as a function of ωτD, in the hot electrons, diffusion- values of R(0) differ by ∼ 10% for eVdc ≪ kBT. Inset: the cooled regime ii) (solid lines) and independent electrons geometry considered: a wire of length L between two thick regime iii) (dashed lines). Inset: Current noise amplitude- normal metal contacts. modulated bythe time-dependentbias V(t). wire. The distribution function f(x,E,t) obeys the 1D diffusion equation[15]: Re[ (ω)] crosses zero and is negative above ωτ 31. D R ∼ ∂f(x,E,t) D ∂2f(x,E,t) At this frequency, in the hot electron, diffusion-cooled = (8) regime ii) Re[ (ω)]/ (0) has only dropped to 0.1. ∂t L2 ∂x2 R R ∼ Thisremarkabledifferenceisdefinitelymeasurable. Note The effect of the external voltage appears only in the that there is no principle preventing Re[ ] from being boundaryconditions: f(0,E,t)=fF(E)andf(1,E,t)= negative. ThisoccurswhentheaveragedisRtributionfunc- fF(E +eV(t)) with fF(E) the Fermi distribution function in the center of the wire oscillates outof phase with tion at temperature T. Solving Eq.(8) for f to first the excitation voltage. order in δV we obtain the ω component of the time- dependent noise temperature profile: Tω(x) (1 The case of intermediate electron-electron time, L x)sinh(qx)/sinhq with q =√iωτ . We deNduce:∝ − Le−e, is beyond the scope of this article. But this coul∼d D be consideredby adding e-e relaxationto the right hand R(ω) =6sinhq−q (9) side of Eq. (8). This eq., in the limit τe−e → 0 leads to (0) q2sinhq the heatdiffusion equation (3) [16]. It wouldbe of inter- R esttocalculatehowtheexistenceofthezeroofRe[ (ω)] FGo−r1e/V1d0c.9≪. TkhBisTdioffneershfarsomR(c0a)se=ii)(bπy2/o1n0l8y)G−W11F0%∼. irnelacxaasetioiini)ischeannerggeys-dweipthenfidneintte, eth-eesptoresintgiotnh.ofStihnecRezeer-oe WF ∼ Real and imaginary parts of (ω) as a function of ωτD should be voltage- and temperature-dependent. R are plotted on Fig. 1. For ωτ 1, Re[ (ω)] decays D exponentially whereas Im[ (ω)] ≫decays likRe ω−1. The In this last section, we consider the relation of our R results to other kinds of noise measurements, and other magnitude (ω) is plotted on Fig. 2. The frequency for which |R(ω)2|=1/2 is 9τ−1. possible applications. As a first example of its relation, |R | ∼ D we can contrastthe NTI we have calculated to the noise We now compare the three cases. (ω) has a R underacvoltageexcitationconsideredpreviously,theso- Lorentzian shape, Eq. (2), for case i), and in ii) has called”photonassistednoise”[17,18,19,20]. Thelatter a frequency dependence that is very similar. The roll- off frequency, τ−1 , of (ω) for phonon cooling i) is refers to the effect of an ac voltage on the time-averaged temperature-depee−npdhent, sRince τe−ph ∝T−p, whereas the Tnohiisse.diffIterhsassigfenaifitucarenstlaytfreoVmdct=henN~TωI,(wwihthichnminetaesguerre)s. roll-offfrequencyfordiffusioncoolingii)isrelatedonlyto thetimedependenceofthenoiseaveragedonatimescale the diffusion time and is thus temperature-independent. τ , revealing the dynamics of the energy exchanges. Ameasurementoftheroll-offfrequencyof (ω)vs. tem- m R perature(ordcvoltage)incasei)givesadirectmeasure- Our result also elucidates the importance of correla- ment of τe−ph(Te). The shape of the magnitude (ω) tions in the scattering matrix formalism,whichhas been |R | in cases ii) and iii) is similar although case iii) exhibits very powerful in treating noise properties of coherent a kink (see Fig. 2). However the real part of (ω) is systems[1]. (ω) could be treated within this formal- R R quite different. For the independent electron regime iii), ism, starting from its definition in terms of the classical 4 fluctuating current at frequency Ω, i(Ω) [21]: Last, we consider a generalization of our result. (ω) R is a response to an external excitation. At equilibrium i(Ω)i(ω Ω)δV−ω (forV =0),itmustberelatedtosomecorrelationfunc- (ω)= h − i (10) dc R 8kBG2Vdc δVω 2 tion through the fluctuation-dissipation theorem. The h| | i latter gives for a macroscopic system [10] : δT2(ω) = ThefrequencydependenceofR(ω)onthescaleofτD−1for 4kBRe[G−th1(ω)]T2. A similar expression shhould reliate cases ii) and iii) comes only from correlations within the δT2(ω) to (ω). Since T S , δT2(ω) is related scatteringmatrixatdifferentenergies,onthescaleofthe h N i R N ∝ 2 h N i to S , the fourth cumulant of noise [24]. Thus, we con- 4 Thouless energy ~/τ . As a consequence, provides a D R jecture that is related to S4. directprobeofthecorrelations,whicharenotconsidered R WethankW.Belzig,M.Bu¨ttiker,M.Devoret,L.Levi- in usual calculations of S . 2 tov,S.Pilgram,M.Reese,P.Samuelsson,D.Santavicca, Our calculation sheds new light on the environmental R.Schoelkopf,M.ShenandA.Shytovforfruitfuldiscus- effects on the third cumulant of noise. These have been sions. This work was supported by NSF DMR-0407082. considered recently[8, 22], specifically for tunnel junc- tions. The expression of (ω) in Eq.(10) makes it ap- R pear as a third order correlation, like the third cumulant of the noise S (ω ,ω ) = i(ω )i(ω ω )i( ω ) , 3 1 2 1 2 1 2 h − − i except the sample-generated current fluctuation at fre- [1] Forareviewonnoise,seeY.M.BlanterandM.Bu¨ttiker, quency ω2 has been replaced by the external applied Phys. Rep.336, 1 (2000). − current(orvoltage). Asamatteroffact,ithasbeencal- [2] B.L.AltshulerandA.G.Aronov,inElectron-electron in- culated that the external current noise, by modulating teractions in disordered systems, A.L. Efros and M. Pol- lak eds, North Holland (1985). the noise emitted by the sample, does contribute to S 3 [3] H. Pothier et al., Phys.Rev.Lett. 79, 3490 (1997). [22]. This mechanism has been explicitly demonstrated [4] E.M.Gershenzonetal.,Sov.Phys.JETP70,505(1990). inexperimentbyapplyinganacvoltagetoatunneljunc- [5] P.J. Burke et al., Appl.Phys.Lett. 68, 3344 (1996). tion and detecting S [8]. This extrinsic contribution to 3 [6] C.M. Wilson, L. Frunzio and D.E. Prober, Phys. Rev. tSe3disby∝tRhedωenSv2einrvo(nωm)eRn(t−;wωe)wsuipthpoSs2eenhv(eωre)~thωe noeisVe,ekmBiTt-. [7] SL.etPti.lg8r7a,m0,67K0.0E4. (N2a0g0a1e)v, aannddrMef.s.Btu¨htetriekienr., Phys.Rev.B Our approach for the NTI can also be u≪sed for the 70, 045304 (2004). calculationofthe environmentaleffects onS foradiffu- [8] B. Reulet, J. Senzier and D.E. Prober, Phys. Rev. Lett. 3 91, 196601 (2003). sive wire. Inparticular,we predict that the contribution [9] For ωτRC ≪1 (with τRC the RC time), the admittance of the environmental noise to S vanishes at frequencies 3 is thedcconductance G. much larger than τD−1. The intrinsic contributions to S3 [10] P.L. Richards,J. Appl.Phys. 76, 1 (1994). ofadiffusivewirealsodecaysforfrequencies>τ−1,even [11] Wewillconsiderthinenoughfilmsatlowenoughtemper- D for voltage bias[7]. We believe this also may be under- ature so we can neglect the Kapitza resistance between stoodfromthebehaviorof (ω). Certainlythemeasure- thephononsofthewireandthephononsofthesubstrate. ment of (ω) is simpler thRan that of S . [12] AtzerovoltageVdc =0,theJoulepoweroscillatesat2ω. 3 R Similarly thenoise, whichis anevenfunction ofvoltage, ThefrequencyscaleofR(ω)incasesii)andiii)issetby is proportional to δV2 and oscillates at 2ω. Thus one the escape time of the electron-hole excitations from the simplyhastoreplaceω by2ω inourdefinitiontoextend wire. We believe this statement applies qualitatively to it to thecase of zero dcbias. other systems. Indeed, in chaotic cavities, (ω) should [13] M. Roukeset al.,Phys. Rev.Lett. 55, 422 (1985). R also decay on the scale of the inverse dwell time (as S [14] M. Shen et al.,unpublished. 3 does [23]); in a quasi-ballistic wire, it likely decays on [15] Thiseq.stillholdsinthepresenceofanacexcitation, at the scale of the inverse transit time L/v with v the least to first order in δV (A. Shytov,privatecomm.). F F [16] K.E. Nagaev, Phys. Rev.B52, 4740 (1995). Fermi velocity. The use ofour method for a carbonnan- [17] G.B. Lesovik and L.S. Levitov,Phys.Rev.Lett. 72, 538 otube may provide an example of its applicability. For (1994). most single wall nanotubes, it is not known if the con- [18] R.J. Schoelkopfet al.,Phys.Rev.Lett. 80, 2437 (1998). ductance results fromscatteringthat is equalfor allfour [19] A.V. Shytov,cond-mat/0312013, unpublished. quantum channels, or fromsome of these channels being [20] M.H.PedersenandM.Bu¨ttiker,Phys.Rev.B58,12993 blocked,andtheothersopen. Thetimescaledetermined (1998). from (ω), might distinguish these two cases. The noise [21] In Eq. (10), the issue of ordering the current operators R has been disregarded, for thesake of simplicity. relaxation time scale of (ω) for a normal metal wire R [22] C.W.J. Beenakker, M. Kindermann and Yu V.Nazarov, between superconducting reservoirs should also be stud- Phys. Rev.Lett. 90, 176802 (2003). ied. We suspectthe relaxationtime forthis caseis much [23] K.E. Nagaev, S. Pilgram and M. Bu¨ttiker, Phys. Rev. longer than τD, because electron-hole excitations of en- Lett. 92, 176804 (2004). ergy smaller than the superconducting gap of the reser- [24] M.KindermannandS.Pilgram,Phys.Rev.B69,155334 voirs cannot escape the wire. (2004)