Photon-assisted shot noise in mesoscopic conductors

ARTICLE IN PRESS

Physica E 40 (2007) 123–132 www.elsevier.com/locate/physe

Photon-assisted shot noise in mesoscopic conductors Dmitry Bagretsa, Fabio Pistolesib, a

Forschungszentrum Karlsruhe, Institut fu¨r Nanotechnologie,76021 Karlsruhe, Germany Laboratoire de Physique et Mode´lisation des Milieux Condense´s, CNRS-UJF B.P. 166, F-38042 Grenoble, France

b

Available online 24 May 2007

Abstract We calculate the low-frequency current noise for AC biased mesoscopic chaotic cavities and diffusive wires. Contrary to what happens for the admittance, the frequency dispersion in noise is determined not by the electric response time (the ‘‘RC’’ time of the circuit), but by the time that electrons need to diffuse through the structure (dwell time or diffusion time). We find that the derivative of the photonassisted shot noise with respect to the external AC frequency displays a maximum at the Thouless energy scale of the conductor. This dispersion comes from the slow, charge-neutral fluctuations of the non-equilibrium electron distribution function inside the structure. Our theoretical predictions can be verified with present experimental technology. r 2007 Elsevier B.V. All rights reserved. PACS: 72.70.+m; 73.23.b; 73.50.Td; 73.50.Pz Keywords: Shot-noise; Quantum dots; Diffusive wires; Photon-assisted electron transport

1. Introduction In mesoscopic devices Coulomb interaction enforces charge neutrality on a time scale (tRC ) given by the product of the typical resistance and capacitance of the sample. The time scale (tD ) for non-interacting electron motion is instead fixed by either the dwell or diffusion time, depending on the transparency of the interfaces between the system and the external leads. The usual experimental situation is tRC 5tD : the slow diffusive motion of the electrons is converted to a fast flow by the electric field generated by the piling up of the charge during the motion of the electrons. As a consequence the typical response time 1 1 of the device is t1 ¼ t1 RC þ tD  tRC . This has been shown for the frequency dependence of both the admittance [1,2] and the noise [3–6] in mesoscopic chaotic cavities and diffusive wires. The inverse of the diffusion time appears instead as the relevant energy scale for the voltage or temperature dependence for both the conductance and the noise in superconducting/normal metals Corresponding author. Fax: +33 476 88 7983.

E-mail addresses: [email protected] (D. Bagrets), [email protected] (F. Pistolesi). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.05.006

hybrid systems [7–14]. The main difference is that in systems with broken gauge symmetry the energy dependence is due to interference of electronic waves that does not imply charge accumulation in the system. In normal metallic structure it is much more difficult to observe a dispersion of a physical quantity on the inverse diffusion time. To our knowledge, this has been predicted so far only for the third moment of current fluctuations [5] and for the finite frequency thermal noise response to an oscillating heating power [15]. An alternative, and less investigated possibility, is to study the low frequency current noise (S) as a function of the external AC bias frequency O. The noise through a quantum point contact was calculated long ago [16] and recently measured [17–19]. Since the quantum point contact is very short, electron diffusion does not introduce any additional time scale in the problem and the resulting frequency dispersion is simply linear [16]. More recently, the noise for an AC biased chaotic cavity has been considered [20,21] in the limit of small fields eV 5_O (V is the amplitude of the AC bias and e is the electron change). The authors of Ref. [20] found that in the noninteracting case the noise disperses on the inverse diffusion time scale. But, when the interaction is taken into account

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at mean field level, the noise disperses only on the t scale, that is the combination of the diffusion and the electric response time. For a diffusive wire the interaction forces the field to be constant inside the bulk (see also the more detailed discussion in the following), and, within this assumption, the noise induced by a short voltage pulse was investigated in Ref. [22]. Even if the dispersion of the noise is not considered in that paper, the authors find that the ratio of the diffusion time to the duration of the pulse may affect the observed noise. This implies that zero frequency noise can be sensitive to the diffusion time. More recently Shytov [23] for the same system and in the limit eV =_Ob1 found that the limiting values of the noise S for OtD b1 and OtD 51 do not coincide. These facts indicate that an external-frequency dependence of the noise on the scale of the diffusion time can be present, but at the moment there exist no explicit prediction for this dispersion. In this paper we derive the complete external frequency dependence of the photon-assisted noise in chaotic cavities and in diffusive wires. We find that in general a dispersion on the scale of the diffusion time is present and, in particular, that dS=dO shows a clear maximum for O  1=tD . We show that this dispersion comes from the slow, charge-neutral fluctuations of the non-equilibrium electron distribution function inside the structure. In the same frequency region the admittance does not disperse due to the enforcement of charge neutrality. A short account of these results has been given in Ref. [24]. The plan of the paper is the following. In Section 2 we give the definition of the photon-assisted shot noise. In Section 3 we discuss photon-assisted transport through a chaotic quantum dot; first through a simple phenomenological model (Section 3.1), then using the microscopic Keldysh technique for the current (Section 3.2) and the noise (Section 3.3). In Section 4 we consider photon-assisted transport in diffusive wires. We start the discussion by considering Maxwell relaxation of charge (Section 4.1) and general formula for noise (Section 4.2) and then show how to evaluate numerically the photo-assisted noise through the diffusive wire (Section 4.3). Section 5 gives our conclusions. 2. Photon-assisted shot noise We start from the preliminary definitions and consider the mesoscopic sample driven by AC voltage source V ðtÞ ¼ V 0 cos Ot. The average current over one period vanishes in this case. One can, however, detect the zerofrequency photon-assisted noise. The possible experimental realization of such measurement is shown in Fig. 1. From the theoretical point of view the noise is the current–current correlation function, which in case of an AC bias has the following time-dependence: ^ 1 ÞDIðt ^ 2 Þ þ DIðt ^ 2 ÞDIðt ^ 1 Þi Sðt1 ; t2 Þ ¼ hDIðt þ1 X ¼ S n ðt1  t2 Þ einOðt1 þt2 Þ=2 , n¼1

Fig. 1. The principal scheme to measure the low frequency photo-assisted noise in the band width f 1 of of 2 .

^ ¼ IðtÞ ^  hIðtÞi. ^ where DIðtÞ Its Fourier transform reads Sðo1 ; o2 Þ ¼

þ1 X

Sn ðo1 þ nO=2Þ

n¼1

2pdðo1  o2 þ nOÞ,

ð2Þ

where Sn ðoÞ is the Fourier harmonics of Sn ðtÞ. If the bandwidth of measurement is chosen such that 2pf 5 maxfO; E Th g, one then obtains the zero frequency limit of the spectral noise density S 0 ðo ! 0Þ. This part of the noise will be the object of our further study. An other possibility is to multiply the squared fluctuation DI 2 ðtÞ to cos Ot or sin Ot, before averaging it over time by means of low pass filter, then one would obtain the first harmonics of noise, Re S 1 and Im S 1 . The experiment of this type has been performed by Reulet and Prober [15,25]. The photo-assisted noise was predicted theoretically by Lesovik and Levitov [16]. For a short mesoscopic scatterer with a single channel of transmission T 0 they obtained S0 ðo ¼ 0Þ ¼ 2_OGQ T 0 ð1  T 0 Þ

þ1 X

nJ 2n ðAÞ,

(3)

n¼1

where G Q ¼ e2 =2p_ is the quantum of conductance A ¼ eV 0 =_O is an effective ‘‘flux’’, as defined in Ref. [16]. It is assumed that the temperature is small, kB T5_O. The function Pn ðAÞ ¼ J 2n ðAÞ can be associated with the probability to excite the electron–hole pair with absorption of n photons [J n ðxÞ are Bessel functions]. The above result is valid for a relatively short mesoscopic samples with Thouless energy E Th bO. The main idea of the paper is to extend this result for a longer or larger mesoscopic conductors with a Thouless energy E Th comparable or smaller than the driving frequency O. Two generic and important examples of such conductors are a chaotic quantum dot and a mesoscopic diffusive wire, which we consider in the following sections. 3. Noise through a chaotic quantum dot 3.1. Phenomenological description of charge transport

ð1Þ

We consider a chaotic cavity as shown schematically in Fig. 2 connected to two metallic leads through two

ARTICLE IN PRESS D. Bagrets, F. Pistolesi / Physica E 40 (2007) 123–132 o

V1 sin(Ω t)

o

[1]

[2]

Tn χ

1

Tn V ( t)

1 C1

V2 sin(Ω t) 2

χ

2

C2

Cg o

Vg sin(Ω t)

Fig. 2. Scheme of the system. A chaotic cavity connected to the electrodes through two coherent scatterers characterized by a set of transparencies fT n g.

scatterers characterized by a set of transparencies fT kn g where k takes P the value 1 or 2. We define the conductances G k ¼ GQ n T kn , the total conductance G ¼ G1 G 2 =ðG1 þ G 2 Þ, and a dwell time tD ¼ 2p_G Q =ðG 1 þ G 2 Þd, where d is the level spacing of the cavity. The cavity is coupled capacitively to the leads through the capacitances C k and to a gate through C g . For notational convenience we consider that three different time dependent voltage biases are applied to the gate and the two contacts (V g , V 1 , and V 2 ). Clearly, the current depends only on two voltage differences. The potential difference between the two leads is harmonically oscillating at frequency O, as shown in Fig. 2. Since the cavity resistance is negligible with respect to the contact resistances one can assume a uniform electric potential and electron distribution inside the cavity. In metals with diffusive electron dynamics there are different effects associated with Coulomb interaction. The very first and the simplest one is a classical charge relaxation, which leads to screening of an imbedded charge and provides the overall charge neutrality in the system. This is the classical effect which appears already at the mean field level, when making quantum mechanical calculations. In the following it will be the only interaction effect that we take into account. There are other effects related to interaction: (i) inelastic electron–electron collisions, defining the inelastic length or time scale; (ii) phase breaking effects, leading to the finite coherence time; (iii) quantum correction to the conductance due to the virtual elastic collisions, which is the precursor of Coulomb blockade in metallic systems. The consequences of these effects on the shot noise have been discussed for instance in Refs. [26,27]. The effects (i) and (ii) are negligible if the dwell time is much shorter that the inelastic scattering time due to the possible electron–electron or electron–phonon interaction. The effect (iii) is instead controlled by the conductance. We assume that GbGQ . Then the effect (iii) is negligible and the system can be treated by semiclassical theory. Bearing these ideas in mind we define the typical charge relaxation time as tRC ¼ C S =G S where C S ¼ C 1 þ C 2 þ C g . For the mesoscopic cavity its linear dimension L is much larger than the Fermi wavelength lF . Therefore, the time tRC is always much shorter than the dwell time, since the ratio tRC =tD  d=E C 51. Here E C ¼ e2 =C S is the

125

Coulomb energy and d ¼ 2 or 3 is the dimensionality of the cavity (d ¼ 2 for a quantum dot formed in a twodimensional electron gas in a semiconductor nanostructure and d ¼ 3 for a metallic grain). Indeed, the mean level spacing reads as d ¼ 1=ðnd Ld Þ, where nd is the density of states (n3 mkF and n2 m). The charging energy can be estimated as E C e2 =L. Therefore we find  d1 lF 51, (4) d=E C rs L where rs ¼ 1=ðkF aB Þ1 is the conventional electron gas parameter, aB ¼ _2 =ðme2 Þ being the Bohr radius. We start the discussion by considering an elementary model of charge transport, where diffusion and electric drift are described classically. The charge Q in the cavity is related to the electric potential V of the cavity itself by the relation X Q¼ C k ðV  V k Þ. (5) k¼1;2;g

On the other hand, the continuity equation for charge Q reads qt Q þ I ¼ 0,

(6)

where the electron current leaking from the cavity can be phenomenologically described as X G k ðV  V k Þ. (7) I ¼ Q=tD þ k¼1;2

The first term of Eq. (7) stems from the charge diffusion while the second is the standard Ohm’s law. For any given time-dependent voltages V k ðtÞ the system of Eqs. (5) and (6) can be conveniently solved in terms of V ðoÞ, the Fourier transform of the cavity potential. We find t=tD X V k ðoÞ V ðoÞ ¼ 1  iot k¼1;2;g   tD Ck  ak þ ð1  iotD Þ , ð8Þ tRC CS where we defined ak ¼ G k =G S for k ¼ 1 or 2, GS ¼ G1 þ G 2 , and ag ¼ 0. When tRC 5tD , the response time t ¼ tRC tD =ðtRC þ tD Þ is simply given by tRC . For tRC 5tD and Ot51 one thus finds the very simple result X ak V k ðoÞ (9) V ðoÞ ¼ k

and I 1 ðoÞ ¼ I 2 ðoÞ ¼ GS a1 a2 ðV 1  V 2 Þ.

(10)

Charge neutrality is perfectly enforced for low frequency drive. In the following we will mainly consider this experimentally relevant low driving-frequency limit OtRC 51, which will enable us to use the o51=tRC response for V ðoÞ. The method developed in the following is not limited by this condition, the extension to the case where o is not negligible with respect to tRC is straightforward.

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The admittance matrix at finite frequency can be evaluated employing the same model. One finds that its frequency dependence is similar to the cavity potential frequency dependence. This simple approach, however, fails to describe current fluctuations in the system, i.e. noise, which is the primary interest of our study. To achieve this goal, we resort to a general microscopic description of current fluctuations in mesoscopic conductors that allows access to the full counting statistics of charge transfer [28]. Within this method the low frequency current noise can be evaluated as the second moment of the number of transferred charges over the long time interval t0 b maxf_=eV ; 1=Og.

_  ðtÞ=e ¼ V  ðtÞ present the ‘‘classical’’ and derivatives j ‘‘quantum’’ fluctuating electrostatic potentials in the cavity. These fields appear in the action also in matrix form: j ¼ jþ ðtÞ þ j ðtÞs x , with s x the Pauli matrix in Keldysh space. The external time dependent voltage drive V k ðtÞ and the counting fields enter S through the Green function of the  k: two leads G

3.2. Microscopic description of current fluctuations

Here

Non-equilibrium transport through a chaotic cavity can be described by functional integral approach in Keldysh contour [26,29,30]. One can define a generating function U that depends on two counting fields w1 and w2 , such that the current and all higher moments averaged over a long time t0 can be obtained by differentiating U [28,31]:

F ðkÞ ðt1 ; t2 Þ ¼ eijk ðt1 Þ F eq ðt1  t2 Þeijk ðt2 Þ

I k ðw1 ; w2 Þ ¼ ie

qU ; qwk

Sk ðw1 ; w2 Þ ¼ e2

q2 U qw2k

(11)

where I k and Sk are the low frequency current and noise, respectively. The generating function can be expressed through the following functional integral: Z  Z   DG DjeiS½G;j , t0 Uðw1 ; w2 Þ ¼  ln (12) where the action S has a form  þ Sj .  j ¼ Scon  2pd1 Trðiqt GÞ S½G;

(13)

Here Scon describes the contacts [11,32]:   1X 1  k ; eij Ge  ij g  2Þ , Tr ln 1 þ T ½k ðf G Scon ¼ 2i n;k 4 n and Sj describes the capacitors Z þ1 X 2C k _ k Þj_  dt. ðj_ þ  j Sj ¼ 2 1 k¼1;2;g e The functional integral is performed over two fields  1 ; t2 Þ and f ðtÞ. The first is a 2  2 matrix and at mean Gðt field level coincides with the semiclassical Keldysh Green’s function for electrons in the cavity. It is constrained by the condition: X Z þ1  il ðt1 ; tÞG  lj ðt; t2 Þ ¼ dij dðt1  t2 Þ. dtG (14) l

1

The trace and product operation in Eq. (13) includes the summation over Keldysh indices and integration over time. In this compact notation the normalization constrain reads  2 ¼ 1. The second field is the pair of real functions G fjþ ðtÞ; j ðtÞg. They result from the Hubbard–Stratonovich decoupling of the Coulomb interaction term, and their time

 k ¼ eisx wk =2 G  ok eisx wk =2 G

(15)

and  ok ðt1 ; t2 Þ ¼ G

and F eq ðtÞ ¼

Z

dðt1  t2 Þ 2F ðkÞ ðt1 ; t2 Þ 0

dðt1  t2 Þ

! .

 e  de iet=_ T e , tanh ¼ 2p 2T i sinhðpTt=_Þ

(16)

(17)

(18)

where T is the temperature of the leads (kB ¼ 1Þ. The phase jk ðtÞ is related to voltage in contact k by the gauge relation _ k ðtÞ ¼ eV k ðtÞ and the function F ðkÞ ðt1 ; t2 Þ is connected to _j the non-equilibrium distribution function in the kth contact, as f ðkÞ ðt1 ; t2 Þ ¼ ½1  F ðkÞ ðt1 ; t2 Þ=2. Note that f ðkÞ ðt1 ; t2 Þ is a gauge dependent quantity. One should now calculate the current and noise starting from expression (12). We are considering the limit of large contacts G=G S 51 such that mesoscopic fluctuations are negligible. In this limit the electronic transport is then well described by the saddle point approximation. At this level of approximation we are also treating the interaction at mean field level. It means that we fully account for the classical charge relaxation (screening), but disregard other, more subtle effects, related to the Coulomb interaction, like (i) inelastic electron–electron collisions, defining the inelastic time scale and (ii) logarithmic renormalization of transmission coefficients T ½k n due to virtual elastic collisions. In zero-dimensional metallic system, which we consider, the impact of effects (i) and (ii) on the current and noise is small by the same parameter G Q =G51. If needed, these effects can be taken into account by considering Gaussian fluctuations of the phases j around the saddle point solution (see for instance Refs. [4,26,27,33]). The saddle point condition on Eq. (13) leads to two set of equations: dS ¼0  dG

and

dS ¼ 0, dj

(19)

where the first derivative has to be performed maintaining the normalization constraint (14). The first is a kinetic equation for the electron distribution function inside the cavity and the second is the Poisson equation for the charge distribution. It is instructive, first, to relate the least action principle to the elementary charge model discussed

ARTICLE IN PRESS D. Bagrets, F. Pistolesi / Physica E 40 (2007) 123–132

above, and to find the distribution function inside the  gives cavity. Minimization of action (13) with respect to G  the following equation that relates G to the current through the interfaces: X  ¼ 0,  k þ _tD ðqt þ qt ÞG (20) I 1 2

function F to the charge in the cavity: Qðtþ Þ ¼ ð2pe=dÞF ðtþ ; t ! 0Þ.

 k ¼ GQ I GS

lim eijþ ðt1 Þ F eq ðt1  t2 Þeijþ ðt2 Þ ¼

t !0

X n

ij  ij  T ½k ; G n ½e Gk e ½k   i j i j   2Þ  4 þ T n ðfe Gk e ; Gg

(21)

are the spectral currents found in Ref. [34]. One can readily verify by performing a derivative with respect to wk that at mean field level the wk dependent current is a trace of expression (21): qU eGS  k ðt; tÞg=2. I k ðw1 ; w2 Þ ¼ ie ¼ Trfs x I qwk GQ

(22)

The second set of equations is obtained by derivation with respect to j : d ðScon þ Sj Þ ¼ 0. dj ðtÞ

(23)

We begin by discussing the physical current that is simply I k for w1 ¼ w2 ¼ 0. In this case Eq. (21) has a solution of the form (16) and depends on a single function F ðt1 ; t2 Þ. The derivative with respect to jþ in Eq. (23) gives _  ¼ 0 that is solved by j ¼ 0, i.e. only the the condition j classical voltage fluctuations are left. It is convenient now to introduce new time variables: t ¼ t2  t1 and tþ ¼ ðt1 þ t2 Þ=2. Then Eq. (20) takes the form of a kinetic equation X qF tD þF ¼ ak eijþ ðt1 Þ F ðkÞ eijþ ðt2 Þ . (24) qtþ k¼1;2 The distribution function F ðtþ ; t Þ is a periodic function of tþ , which can be represented as the Fourier series X einOtþ F n ðt Þ, (25) F ðtþ ; t Þ ¼ n

1 j_ ðtþ Þ. p þ

(29)

The equation of motion for the charge is given by the derivative with respect to j in (23). It is composed of two terms. The first reads: dScon GS X  k ðt; tÞs x Þ ¼ TrðI dj ðtÞ GQ k¼1;2 _  tÞs x Þ ¼ 2 QðtÞ, ¼  2pd1 Trðqt Gðt; ð30Þ e where we have firstly used Eq. (20) and then the definition of charge (28). The second reads X Ck dSf ¼ 2 ðV_  V_ k Þ. (31) df ðtÞ e k¼1;2;g Combining Eqs. (30) and (31) we see that the saddle point Eq. (23) fixes the charge QðtÞ in accordance with relation (6). Note, that when the voltage time dependence (9) is enforced, QðtÞ vanishes identically, as it can be verified by calculating the limit t ! 0 in Eq. (26). On the otherR hand, the energy distribution function, F~ ðtþ ; eÞ ¼ dt F~ ðt ; tþ Þeiet =_ , varies periodically in time, and its dependence on the AC driving frequency, O, is on the scale of the inverse diffusion time t1 D . When an electron enters the cavity, after a very short time tRC , the charge rearranges to keep the cavity neutral, the distribution function, instead, will relax on a much longer time given by tD . As it will be shown in the following subsection, zero-frequency noise probes the electronic distribution function (see Eq. (42) in the following) and it does depend on the frequency O on the same scale. 3.3. Analytic derivation of the noise

and its explicit expression in terms of the Fourier component of F ðkÞ comes from the solution of the kinetic equation (24): P ak F ðkÞ n ðt Þ F n ðt Þ ¼ k . (26) 1 þ inOtD Here F ðkÞ n ðtÞ ¼ F eq ðtÞJ n ð2Ak sinðOt=2ÞÞ,

(28)

The kinetic equation (24) for t ! 0 reduces then to the continuity equation (6) for the total charge. This can be shown by exploiting the identity

k

where

127

(27)

where Ak ¼ eðV ðkÞ 0  V 0 Þ=ð_OÞ, the equilibrium distribution function F eq ðtÞ is given by Eq. (18), J n are the Bessel functions, V ðkÞ 0 are the amplitudes of the AC fields in the leads V k ðtÞ ¼ V ðkÞ 0 sinðOtÞ, and V 0 is the amplitude of the electric potential inside the cavity V ðtÞ ¼ f_ þ ðtÞ=e ¼ V 0 sinðOtÞ. In order to make a connection with the classical description given by Eq. (6) we relate the distribution

In order to obtain the noise we need to calculate the first derivative of the counting field dependent current S k ðtÞ ¼ ieqI k ðtÞ=qw for w1 ¼ w2 ¼ 0. To obtain the noise, it is thus  sufficient to calculate the linear correction in w to G: 1 þ ,  ¼G  0  2iwG G

(32)

and substitute this expression into Eq. (21) [11,12]. The  0 is given by Eq. (16) with the distribution function term G F given by Eqs. (25) and (26). In order to fulfill the  1 should antinormalization condition (14) the correction G  commute with G0 . This condition can be fulfilled automatically by using the parameterization proposed in Ref. [29]:     1 F 0 W 1 F  iwG1 ¼ . (33) 0 1 W0 0 0 1

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 on a cavity For the boundary conditions (15) the phase fðtÞ acquire the w dependent correction as well:  ¼ f ðtÞ þ wfyðtÞ þ y0 ðtÞs x g þ    . fðtÞ þ

(34)

By defining w ¼ w1  w2 we can calculate the noise as a derivative with respect to w and we are free to choose w1 ¼ a2 w and w2 ¼ a1 w. In this way W 0 ¼ 0 and y0 ¼ 0. Then from Eqs. (20) and (23) we obtain the system of coupled equations for W ið1 þ tD qtþ ÞW X ak eifþ ðt1 Þ F k eifþ ðt2 Þ ½yðt1 Þ  yðt2 Þ ¼Rw

ð35Þ

k¼1;2

and y: _ Q1 ðtÞ ¼ wC S yðtÞ=e.

(36)

Here the operator R reads X R¼ wk ak ½ð1  bk ÞðF F þ F k F k Þ ð37Þ

(38)

where bk ¼ G Q =Gk

X

ðkÞ T ðkÞ n ð1  T n Þ

(39)

n

are the Fano factors of the two junctions and with circle we indicate a time convolution. We now expand the w-dependent current (21) up to linear in w terms in order to evaluate the noise: ðbÞ ðcÞ I k ðwÞ ¼ I ðaÞ k þ I k þ I k , where pG k _ ½TrðW Þ  wy=p, e ipGk w Trf2  F F k  F k F g, I ðbÞ k ¼ 2e k ipG k w ð1  bk ÞTrfðF  F k Þ ðF  F k Þg. I ðcÞ k ¼ 2e k I ðaÞ k ¼

ð40Þ

As we can see from this expansion one needs to know TrW only, i.e. W ðt; tÞ averaged over one period in time in order to obtain the low frequency noise. It follows from the kinetic equation (35) and the relation (29), that _ ¼ i TrðRÞ, TrðW Þ  wy=p

þ 2pGð1  2a2 b1 ÞTr½f ð1  f Þ þ ð122Þ.

ð42Þ

Here the terms in the first and the second lines are the thermal and partition noise due to contact resistances, while the third line represents the noise of the chaotic cavity with open contacts, which stems from the fluctuation of the distribution function inside the dot. Evaluating the traces we finally obtain: X _Oðl þ rÞ coth ð_Oðl þ rÞ=2TÞ S¼G 1 þ O2 t2D n2 n;l;r ½FJðA1 ; A2 Þ þ F1 JðA1 ; A1 Þ þ F2 JðA2 ; A2 Þ þ 2GTð1  b1 a2  b2 a1 Þ,

ð43Þ

JðA1 ; A2 Þ ¼ J nþl ðA1 ÞJ l ðA1 ÞJ rn ðA2 ÞJ r ðA2 Þ, F1 ¼ ða1 þ b1 a2  FÞ=2, F2 ¼ ða2 þ b2 a1  FÞ=2,

and Q1 ðtÞ is the correction to the charge, given by Q1 ðtþ Þ ¼ ð2pe=dÞW ðtþ ; t ! 0Þ,

S ¼ GT ð1  2b1 a2 Þ þ 4pGa2 b1 Tr½f ð1  f 1 Þ þ f 1 ð1  f Þ

where

k

þ bk ðF F k þ F k F Þ,

f k ðt1 ; t2 Þ ¼ ½1  F~ k ðt1 ; t2 Þ=2

(41)

thus the actual time dependence of the phase yðtÞ is not important for the evaluation of the low frequency noise. _ have the evident We note here that the variables W and y=e physical meaning. They represent the most probable fluctuation of the distribution function and the electrostatic potential on the dot, which generate the given fluctuation of the current I k , parametrized by the counting field w. Using relation (41) between two fluctuations W and y and the current expansion (40) one can relate the noise S with distribution functions f ðt1 ; t2 Þ ¼ ½1  F~ ðt1 ; t2 Þ=2 and

F ¼ a1 a2 þ b1 a32 þ b2 a31 ,

ð44Þ

and the amplitudes are A1 ¼ a2 A, and A2 ¼ a1 A, with A ¼ eðV 01  V 02 Þ=_O. Expression (43) is the main result of this section. For OtD 51 it reduces to the result by Lesovik and Levitov [16] with the effective Fano factor F, appearing at the place of the quantum point contact Fano factor. The photonassisted shot noise in a chaotic cavity in the limit of small A has been considered recently by Polianski et al. [21]. They found that at A2 order there is no frequency dispersion. One can recover this result, expanding the general expression (43) up to the second order in A. For all other cases the frequency dispersion is present, as it can be seen from Fig. 3, where the dependence of dS=dO on both O and A ¼ eV 0 =_O is shown for low temperature T5_O. Two features are clear: the first is the frequency dispersion for fixed A, the presence of a crossover from a low frequency and high frequency results with a clear maximum for O  tD ; the second is the oscillatory behavior as a function of A, that is due to the Bessel functions, and that was already present in the short junction studied by Lesovik and Levitov. We also show in Fig. 4 the derivative of the noise with respect to the voltage, also known as differential noise. Also in this case a dependence on both the driving frequency and electromagnetic field intensity is clearly present. 4. Photon-assisted shot noise in diffusive wires 4.1. Maxwell relaxation of charge We pass now to discuss the photon-assisted noise in a diffusive wire. We characterize the wire by a length Lx in

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129

/Ω)/dΩ dS (Ω, eV )/dΩ dS (0, eV/Ω

described by the set of coupled equations of the classical electrodynamics in the diffusive media (see, e.g. Ref. [35]): Z ð45Þ fðr; tÞ ¼ Uðjr  r0 jÞrðr0 ; tÞ dr0 , 1.3 10

1.2 8 6 -1

4

eV

/h

1 -2

Ω

1.1

0

log

[Ω

10

2

1 D]

2 3

Fig. 3. Frequency and flux A (¼ eðV 01  V 02 Þ=ð_OÞ) dependence of the derivative of the photon-assisted noise with respect to the AC frequency (dS=dO) for the symmetric chaotic cavity (G1 ¼ G 2 ). The magnitude of dS=dO is normalized to its value at O ¼ 0. For symmetric cavity the curves appear to be independent of the transmission distribution of the contacts.

r_ þ r  j ¼ 0,

ð46Þ

j ¼ srf  Drr.

ð47Þ

Here rðr; tÞ and jðr; tÞ are the charge and current densities, fðr; tÞ is the electrostatic potential, s is the conductivity and UðjrjÞ is the (possibly screened by nearby gates) Coulomb potential. The conductivity s of the wire is frequency independent in the wide frequency range oo1=ttr , ttr being a transport scattering time. It is related to the diffusion coefficient by the Einstein relation s ¼ 2e2 nD (the factor 2 is the spin multiplicity). It is worth to mention, that Eqs. (45)–(47) are analogues of previously discussed Eqs. (5)–(7). One can solve Eqs. (45) and (46) in terms of the Fourier transform rq ðtÞ of the charge excitation as

1 10

0.8 0.6 0.4 -1

log

4

0

10

[Ω

1 D]

/h

-2

Ω

8 6

eV

1 dS eGF dV

ðqt þ Dq2 þ sq2 UðqÞÞrq ðtÞ ¼ 0,

2 2

3

Fig. 4. Frequency and flux dependence of the differential photon-assisted shot noise dS=dV in the symmetric chaotic cavity (G1 ¼ G2 ). The magnitude of dS=dV is normalized by eGF:

the direction of the current, a diffusion coefficient D, and a conductance GbG Q . These parameters define the Thouless energy of the wire, Th ¼ _D=L2x , which gives the typical diffusion time, tD _=Th , through the structure. The wire is connected to two leads which are kept at oscillating voltage difference eV ðtÞ ¼ eV 0 sinðOtÞ. In order to observe strong non-equilibrium effects, we consider not too long samples, where the diffusion time tD is much shorter than the energy relaxation time tE in the system. At low temperature tE stems primary from the inelastic electron–electron collisions. It can be estimated as tE ðD=EÞ1=2 n for the case of quasi-one dimensional geometry (n being a density of states per spin) and tE ðG=G Q Þ_=E for the case of quasi-two dimensional film, where E ¼ maxfeV 0 ; _Og. Following the same procedure of Section 3.1 it is instructive to start the discussion by considering the classical charge relaxation in the diffusive wire. It is

(48)

where UðqÞ is the Coulomb potential form factor. Typically the short diffusive wire can be made of the film with the size Lx 4Ly and thickness d5fLx ; Ly g, where the length scales Lx;y are of the same order of magnitude, so that one effectively has a two-dimensional metal [18]. In this situation for the non-screened Coulomb interaction we put UðqÞ ¼ 2p=jqj. It follows then from Eq. (48) that a charge fluctuation, spread over the system size with a wave vector q2p=Lx , will relax on the typical time scale t1 ¼ Th þ oM , where oM s& =Lx is the Maxwell relaxation rate. Taking into account that nmðkF dÞ, the definition for the sheet conductivity s& and Th , one estimates for a good metal, that Th =oM 

l2F 51, Lx d

(49)

with lF being the Fermi wavelength. Thus the charge relaxation time t is set by the Maxwell time, t1=oM , playing now the role of RC-time, while the diffusion time tD drops out from the problem. Then at low drivingfrequency limit, Ot51, the system is charge neutral, rðtÞ ¼ 0, and as it follows from Eq. (47) the admittance Y ðoÞ is frequency independent, Y ðoÞ ¼ G s& ðLy =Lx Þ. At the same time the electric field E ¼ rf is constant along the wire, so that fðx; tÞ ¼ V 0 sinðOtÞðLx  xÞ=Lx ,

(50)

where x is the coordinate in the current’s direction. This expression is the analogue of result (9) in case of zerodimensional chaotic quantum dot. 4.2. General formula for the photon-assisted noise The photon-assisted shot noise in the diffusive wire in the limit of large driving frequency OtD b1 has been

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130

theoretically considered by Shytov [23]. In this limiting case the noise can be expressed in terms of the electron distribution functions averaged over time, and can be analytically evaluated when eV =_Ob1. To obtain the noise for arbitrary value of O we employ the same procedure used for a chaotic cavity, taking into account the spatial  dependence of the Green’s function G. We describe the wire by the following action [36]:  Z 1 ^ 2 S½G; f ¼ 2ipn drTr DðrGÞ 4  ^  ðqt þ iefðx; tÞÞG þ Sf , ð51Þ  ¼ fþ þ f s x and f are the ‘‘classical’’ and where f ‘‘quantum’’ components of the electrostatic potential. The term S f is the electromagnetic part of the action   Z X 1 þ 2 þ 2ne f Sf ¼ 2 dt fq (52) q. UðqÞ q The first term in this expression takes into account the electromagnetic energy, and the second one is the static compressibility of the electron gas. The action (51) has to be supplemented by the boundary conditions for the Green’s function at the ends of the wire: ! dðt1  t2 Þ 2F eq ðt1  t2 Þ  1 ; t2 Þjx¼L ¼ Gðt (53) x 0 dðt1  t2 Þ and  1 ; t2 Þjx¼0 Gðt ¼e

is x w=2

dðt1  t2 Þ

2F L ðt1 ; t2 Þ

0

dðt1  t2 Þ

! eis x w=2 .

ð54Þ

Here the equilibrium distribution function F eq ðtÞ is given by Eq. (18): F L ðt1 ; t2 Þ ¼ eijðt1 Þ F eq ðt1  t2 Þeijðt2 Þ ,

(55)

and jðtÞ ¼ ðeV 0 =_OÞ cosðOtÞ. Minimizing the action (51) with respect to G one obtains the Usadel equations  tÞ; G  x GÞ  ¼ 0.  þ i½iqt  efðx; Drx ðGr

(56)

By considering the physical limit of this equation at w ¼ 0 and taking a limit t1 ! t2 one can rederive Eqs. (46) and (47) with charge and current densities given by rðr; tÞ ¼ 2enfpF ðtþ ; t ! 0; rÞ þ efþ ðr; tÞg, jðr; tÞ ¼ Trfs xjðr; t; tÞg=2,

ð57Þ ð58Þ

where we have defined the spectral matrix current jðr; t1 ; t2 Þ ¼ ps ðGr ^ GÞ. ^ (59) e By the same token, if one takes the saddle point of the action (51) with respect to f , the Poisson relation (45) is reproduced. Note, that appearance of the f-term in the definition of the charge density stems from the fact that the

gauge divA ¼ 0 is now used, while the preceding definition of charge (28) implied the gauge fðtÞ ¼ 0 inside the cavity. Using the Usadel equation, one can express the noise S in terms of the non-equilibrium distribution function f ðt1 ; t2 ; xÞ ¼ ½1  F ðt1 ; t2 ; xÞ=2 of the electrons inside the wire: Z 8pG Lx S¼ Tr½f ðxÞ ð1  f ðxÞ dx, (60) Lx 0 where f ðt1 ; t2 ; xÞ satisfies the diffusion equation with the potential fðx; tÞ fluctuating in time (cf. Eq. (50)) qtþ f  Dq2x f þ ieðfðx; t1 Þ  fðx; t2 ÞÞf ¼ 0.

(61)

Expression (60) is the generalization of the result by Nagaev [37] to the case of arbitrary time-dependent bias V ðtÞ, provided its typical time scale of variation is longer than 1=oM . Let us now consider the derivation of the formula (60) in more details. Following the logic of (3.3) one will seek a solution of Usadel equation for an infinitesimal value of the counting field w, that enters the boundary condition (54). The noise can then be found using the relation SðtÞ ¼ ieqIðtÞ=qw. To simplify the calculations it is convenient to perform the following gauge transformation:  1 ; t2 ; xÞ ! eis x lðxÞ Gðt  1 ; t2 ; xÞeis x lðxÞ Gðt

(62)

with lðxÞ ¼ wðLx  xÞ=2. This gauge is the continuous generalization of the gauge for wk used in the cavity, and it allows to discard the W 0 for the correction to the Green’s function. The Usadel equation (56) is modified only by the substitution of the operator rx with the ‘‘long’’ derivative  is defined according to the rule e x . Its action on G r  G,  rx G   i½A;  e xG r (63)  with A ¼ s x w=ð2Lx Þ playing the role of the vector potential. The matrix current operator (59) is modified in a similar way. We now seek the solution of the modified Usadel equation in power series in w ! dðt1  t2 Þ 2ðF þ W þ   Þ  ¼ G , (64) 0 dðt1  t2 Þ where it is assumed that W is linear in w and fðx; tÞ ¼ fðx; tÞ þ wyðx; tÞ þ    .

(65)

Substituting these expressions into the Usadel equation (56) we obtain Eq. (61) and the following equation for W: qtþ W  Dq2x W þ ieðfðx; t1 Þ  fðx; t2 ÞÞW iwD ¼ fF qx F þ ðqx F Þ F g Lx  iewðyðx; t1 Þ  yðx; t2 ÞÞF .

ð66Þ

This diffusion equation with the source term in the r.h.s. has to be solved assuming the boundary conditions W ð0Þ ¼ W ðLx Þ ¼ 0. Its full solution is not required for our purpose here, since we are interested in the zero

ARTICLE IN PRESS D. Bagrets, F. Pistolesi / Physica E 40 (2007) 123–132

frequency noise S. Expanding the current (59) in w, one finds that the zero order term averaged over one period vanishes (i.e. there is no average current) while the linear term can be written as follows: IðxÞ ¼ I W ðxÞ þ I F ðxÞ and pGLx I W ðxÞ ¼ Tr½qx W ðxÞ e iwpG Tr½1  F ðxÞ F ðxÞ. I F ðxÞ ¼ e

ð67Þ ð68Þ

Taking the trace of Eq. (66) one finds the following equation for I W ðxÞ: qx I W ðxÞ ¼

2iwpG TrðF qx F Þ ¼ qx I F ðxÞ, e

131

It is thus convenient to define the energy variable E, so that  ¼ E þ k_O, such that k is an integer, and _O=2oEo _O=2. One can then represents A in a matrix form: emn ðE þ nOÞ. A nm ðEÞ ¼ A

(74)

In this way the time convolution between two operators becomes a simple matrix product. In the matrix representation the distribution function entering Eq. (53) for the right lead (x ¼ Lx ) takes a diagonal form: F R ðEÞnm ¼ dnm signðE þ n_OÞ. For the left lead at zero temperature it reads instead:

(69)

where we have taken into account, that the zero frequency _ and of y_ vanish for any periodic forcing components of f field. Eq. (69) is the mathematical statement of the physical fact that the zero frequency noise can be calculated in any point of the wire since qx IðxÞ ¼ 0. Solving Eq. (69) with the proper boundary condition for W one readily finds that Z Lx 1 IðxÞ ¼ I ¼ I F ðxÞ dx (70) Lx 0 that leads to the expression (60) for the noise, as anticipated above. Due to the extra spatial dimension the diffusion equation (61) cannot be solved in a closed analytical form as easily as it has been done in case of chaotic quantum dot. We give thus the final expression (60) because it is a good starting point for analytical approximations, but in the following we solve numerically directly Eq. (56). 4.3. Numerical derivation of the photon-assisted noise in diffusive wires To obtain the photon-assisted noise we now solve the Usadel equation (56) numerically, using the charge neutrality condition (50) for the electrostatic field fðx; tÞ. Such procedure is legitimate since we have seen in the preceding section, that the actual value of w-dependent ^ tÞ does not affect the zero-frequency fluctuation in fðx; noise. To accomplish this program technically we switch to the energy representation of Eq. (56). Since the driving is periodic we can single out the tþ dependence for any  operator A: X  1 ; t2 Þ ¼ Aðt (71) A n ðt ÞeinO0 tþ .

F L ðEÞnm ¼ dnm þ 2inm

m X

J l ðAÞJ mnl ðAÞ,

l¼1

where A ¼ eV 0 =_O. In practical calculations it is enough to restrict the matrix size to jnjp3A, since A gives a typical number of multiphoton processes involved. Following the ideas of the circuit theory [11,32], we solve Eq. (56) by representing the wire as a chain of N  1b1  is uniform) with N identical chaotic cavities (where G tunnel barriers between them. This approach leads to a finite difference version of the Usadel equation:   1    V k   ðGk1 þ Gkþ1 Þ þ i ; Gk ¼ 0. (75) 2 NðN  1ÞTh The operators iqt  and V k have matrix representations ðÞn;n ¼ E þ nO, V knm ðEÞ ¼ ieV 0 ðN  kÞðdn;m1  dn;mþ1 Þ=ð2NÞ.

ð76Þ

Eq. (75) can be efficiently solved by means of subsequent iterations. Then photon-assisted shot noise can be obtained by differentiating the w-dependent current, S ¼ ieqI k =qw at w ¼ 0, where  k; G  kþ1 s x g=8 I k ðwÞ ¼ ðG=GQ ÞN Trf½G

(77)

can be taken at any one of the N barriers due to the current conservation. Results for the differential noise dS=dO versus AC frequency are shown in Fig. 5. We find that diffusion due to impurities induces on the photon-assisted noise a similar frequency dependence as the transmission through a chaotic cavity. Again a maximum is present with the main difference that the energy scale is set by the Thouless energy instead of the inverse dwell time. 5. Conclusions

n

In the energy domain this implies that X  1 ; 2 Þ ¼ en ð1 Þ2pdð1  2 þ n_OÞ, Að A

(72)

n

en ð  n_O=2Þ is the Fourier transform of A n ðt Þ: where A Z þ1 en ð  n_O=2Þ ¼ A dt eit A n ðtÞ. (73) 1

In conclusion we have shown that in mesoscopic conductors without broken gauge symmetry non-equilibrium noise can depend on the diffusion time instead of the electromagnetic time. Specifically we have found that the zero frequency noise for a AC biased chaotic cavity and diffusive wire for fixed product A ¼ eV =_O depends on O on the scale of the diffusion time. This is due to the fact that the distribution function re-adjust to charge neutrality

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132

via the Contract no. W-31-109-ENG-38. This work is also a part of research network of the Landesstiftung BadenWu¨rttemberg gGmbH.

4 3

V sin( t) 2

L

1

Fig. 5. Frequency dependence of the differential photon-noise for a diffusive wire for given values of A ¼ eV 0 =_O. Here Th ¼ _D=L2x and the numerical calculation has been performed with 10 nodes. The magnitude of dS=dO is normalized to its value at O ¼ 0. Curve (1) A ¼ 1:0, (2) A ¼ 2:0, (3) A ¼ 3:0, (4) A ¼ 4:0.

on the very short time t, but keeping neutrality it fluctuates still on a scale given by tD . Our quantitative predictions can be verified experimentally. The experiment of Ref. [19] can be in principle repeated with two quantum point contacts and a chaotic cavity in the middle. By carefully choosing the transparencies of the cavity one can easily match the Thouless energy (_=tD ) in the range of frequencies attainable with present technology. To give an idea of the time scales, a Thouless energy of 10 meV, that is typically realized in mesoscopic conductors, corresponds to O=2p ¼ 2:4 GHz which is a frequency readily accessible in experiments. Acknowledgments We acknowledge useful discussions with F. Hekking, M. Houzet, Yu. Nazarov, L. Levitov, and V. Vinokur. The work of D.B., conducted as a part of the EUROHORCS/ ESF EURYI project ‘‘Quantum Transport in Nanostructures’, was supported by funds from the Participating Organizations of EURYI and the EC Sixth Framework Programme. F.P. acknowledges support from the french Agence Nationale Recherche (project JCJC06 NEMESIS). We acknowledge hospitality of Argonne National Laboratory where part of this work has been performed with support of the US Department of Energy, Office of Science

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