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Shot Noise Current-Current Correlations in Multi-Terminal Diffusive Conductors

We investigate the correlations in the current fluctuations at different terminals of metallic diffusive conductors. We start from scattering matrix expressions for the shot noise and use the Fisher-Lee relation in combination with diagram technique to evaluate the noise correlations. Of particular interest are exchange (interference) effects analogous to the Hanbury Brown--Twiss effect in optics. We find that the exchange effect exists in the ensemble averaged current correlations. Depending on the geometry, it might have the same magnitude as the mean square current fluctuations of the shot noise. The approach which we use is first applied to present a novel derivation of the 1/3-suppression of shot noise in a two-terminal geometry, valid for an arbitrary relation between the length and wire width. We find that in all geometries correlations are insensitive to dephasing.


I. INTRODUCTION
The shot noise in mesoscopic systems [1] continues to attract the attention of both theorists and experimentalists. For diffusive conductors, which are considered here, the two-terminal shot noise is quite well studied. The spectacular 1/3-suppression of the shot noise with respect to the Poisson value, S(ω = 0) = 1 3 eGV (here, as usual, S(ω) is the Fourier transform of the current-current correlator, S(t) = ∆I(t)∆I(0) , while G and V are the conductance of the wire and the applied voltage, respectively; ∆I = I(t) − I ), was derived in three different ways: from the distribution of transmission eigenvalues in a wire [2], semi-classically from the Langevin equation [3], and through a microscopic calculation of local current densities [4]. Later, Nazarov [5] claimed that this 1/3 suppression holds for an arbitrary two-terminal geometry (not necessarily quasi-onedimensional). Subsequent to experiments by Liefrink et al [6], which demonstrated shot-noise suppression close to 1/3 even for conductors much longer than the dephasing length, de Jong and Beenakker [7] provided a semi-classical discussion which showed that the 1/3suppression is insensitive to dephasing. More recent experiments by Steinbach et al [8] demonstrated the transition from the 1/3-suppression regime in wires short compared to an inelastic length through an interactiondominated regime [9,10] to a regime where shot noise is suppressed by inelastic scattering [2,11,12]. A macroscopic metal exhibits no shot noise [13].
Thus, S C = S A + S B : experiments A and B are not additive due to the interference terms Ξ 3 and Ξ 4 . It was shown in Ref. [14] that these terms have different signs for fermions and bosons; hence we will call them exchange terms. We define an exchange contribution as It follows from the unitarity of matrices s λν that the quantities Ξ 1 and Ξ 2 , which represent the classical result, are positively defined [17]. At the same time, traces Ξ 3 and Ξ 4 can have either sign. This means that exchangeinterference may either suppress or enhance the classical value. In a disordered system all these quantities should be averaged over impurity configuration. Naively, one might think that due to the phases contained in the quantities Ξ 3 and Ξ 4 these will average to zero, and thus the average of the exchange term ∆S vanishes (here angular brackets are used to indicate the disorder average). Below we explicitly calculate disorder-averaged correlation functions S j , and demonstrate that it is not the case. The average exchange correlator ∆S generally has a nonzero value. An analysis of the exchange correlator for chaotic cavities, reported elsewhere [18], leads to a similar conclusion. The paper is organized as follows. First, we investigate disorder averages of scattering matrices starting from Eqs. (2) and using the Fisher-Lee relation which connects scattering matrices and Green's functions. We then use diagram techniques developed for disordered systems to find the ensemble averages. As a simple check of the method developed, we give a novel derivation of the 1/3-suppression of the two-terminal shot noise for an arbitrary (not necessarily quasi-one-dimensional) geometry, thus confirming the result by Nazarov [5]. Then we turn to exchange-interference experiments and consider the two particular four-terminal geometries, shown in Fig. 1. We demonstrate that the geometry of Fig. 1a implies a negative exchange correlation, with the quantity ∆S being of the same order of magnitude as correlators S A and S B themselves. In contrast, the cross geometry of Fig. 1b shows a strong suppression of exchange effects, and gives a positive sign of the latter, provided the motion through the center of a cross is ballistic. Otherwise the exchange effect is governed by the scattering inside the cross center only.
In the calculations below we disregard electronelectron interaction. The latter is known not to produce an essential effect on two-terminal shot noise [9,10] provided the wire is short in comparison with the inelastic scattering length. We will show that the origin for this is that in the ensemble averaged quantities the effect is local and electron trajectories enclosing a large area are suppressed. This explains why the shot noise is not sensitive to dephasing. Hence, we believe that electron-electron interactions are not important for the exchange effects in shot noise. Note, however, that non-linear noise is affected by interactions, as was shown recently [19]. Interactions are also expected to affect the frequency dependence of the shot noise power.

II. GENERAL FORMALISM AND TWO-TERMINAL SHOT NOISE
We consider a disordered two-dimensional system, connected to reservoirs by ideal leads. Transverse motion of electrons in each lead is quantized, and we assume that all leads are wide, i.e. the number of transverse channels at the Fermi surface in the lead λ is large, Here p F is the Fermi momentum, while W λ is the width of the lead.
General relations [20][21][22] allow one to express scattering matrices for an arbitrary geometry through retarded and advanced Green's functions of the system. The standard procedure [22] is as follows. One chooses arbitrary cross-sections of the leads C λ , and introduces local coordinates related to these cross-sections (Fig. 2). Since nothing depends on the choice of these cross-sections, it is convenient to choose them as boundary between disordered region and leads. One obtains and Here v m = k m /M , M being the effective electron mass. The longitudinal wavevectors in the lead λ are , and those in lead ν are denoted by k n . Furthermore,n λ is the unit vector in the direction x λ , while χ m and χ n are wavefunctions of transverse motion in the leads λ and ν respectively; for simplicity we choose them to be real. Finally, D denotes a double-sided derivative, f Dg = f ∇g − g∇f.
In principle, Eqs. (3) and (4) allow one to average arbitrary combinations of scattering matrices over disorder, using the standard diagram technique [23]. It seems that the approach outlined here has not so far been used for the (analytical) calculation of any physical properties. However, it is rather close to the Hamiltonian approach, employed extensively for the calculation of conductance and conductance fluctuations [24][25][26][27][28]. Below we demonstrate that our formalism reproduces the 1/3-suppression of two-terminal shot noise; in particular, as a simplest check, we also reproduce the Drude formula for conductance.
The rest of the Section is devoted to the two-terminal geometry -a diffusive wire of the length L and width W , connected to two ideal leads α and β; L, W ≫ l, with l being the mean free path. For a moment we assume also L ≫ W , a restriction, which eventually will be lifted. We introduce an axisx directed along the wire, 0 ≤ x ≤ L, and an axisŷ directed across the wire. The general expressions (3), (4) can be rewritten as and Two-terminal shot noise power S ≡ S(ω = 0) can be conveniently expressed through scattering matrices evaluated at the Fermi level [29,30], eV Tr s +αβ s αβ − Tr s +αβ s αβ s +αβ s αβ . (7) Note that the first trace on the rhs is related to the conductance, Tr s +αβ s αβ .
It is convenient to calculate both traces separately.
A. Evaluation of Tr s +αβ s αβ .
Using Eqs. (5) and (6), we find for the conductance where the Green's functions are taken at the Fermi energy. Since the averaged Green's functions decay on scales of the mean free path, the average product of two Green's functions, each of them taken in remote points, is only due to the diffusion (see e.g. [31]): The diffusion propagator P (r, r ′ ) is a solution of the equation with appropriate boundary conditions (P = 0 at the contact to the ideal leads; n∇P = 0 at the walls). Here ν = M/2π, D = v F l/2 and τ are density of states, diffusion coefficient, and elastic lifetime, respectively. Under the assumption L ≫ W the diffusion can be considered to be one-dimensional, and the diffusion propagator does not depend on y, Now we insert Eq. (9) into Eq. (8). The diagram for g is shown in Fig. 3. One can approximate the short-ranged Green's functions as follows, Then, integrating over transverse coordinates, we obtain Taking into account that Multiplied by e 2 /2π, Eq. (14) gives the Drude formula, as it should be.
The trace of a product of four scattering matrices can be written as t ≡ Tr s +αβ s αβ s +αβ s αβ = 1 (4M ) 4 klmn 1 k k k l k m k n × Cα dy 2 dy 3 dy 6 dy 7 χ l (y 2 )χ l (y 3 )χ n (y 6 )χ n (y 7 ) Employing Eq. (9) again, we find the diagrams shown in Fig. 4. We omitted all diagrams containing a single electron line connecting two different leads, since these are exponentially small; the diagrams (a) and (e) contain also counterparts, similar to (c) and (d). The diagrams (b), (c) and (d) turn out to give the leading contribution, whereas others carry small factors. Thus, for diagram (a) points y 1 and y 4 should lie not further apart than a mean free path, which due to the orthogonality of transverse wavefunctions implies k = m. Therefore, the contribution of this diagram is suppressed by a factor (p F W ) −1 ≪ 1. The diagram (e), which is topologically equivalent to (b), is suppressed as (p F W ) −3 . Taking into account the explicit form (11) for the diffusion propagator, and integrating over coordinates y i and over one of two pair of coordinates in the diffusion propagators (those lying close to one of the ends of the wire), we arrive at the expression Here F is the Hikami box [32]. It is short-ranged (all points r a , r b , r c , and r d should be close to each other), and in the Fourier-space has the form Integration of the Hikami box over the cross-section of the wire yields Inserting Eq. (18) into Eq. (16) and performing the remaining integrations, we obtain which immediately gives the 1/3-shot noise suppression.

C. Universality
Now we lift the requirement L ≫ W , but still consider a diffusive system, W, L ≫ l. The result (14) for g = Tr[s +αβ s αβ ] is equivalent, in fact, to the Drude formula, and is therefore valid for an arbitrary relation between W and L. In the derivation of t = Tr[s +αβ s αβ s +αβ s αβ ] we should now take into account that the diffusion is not onedimensional any more, and write the diffusion propagator in the form instead of Eq. (11). Here φ q (r) and −q 2 are eigenfunctions and eigenvalues of the Laplace operator with appropriate boundary conditions. In our particular geometry one obtains with integers n x > 0 and n y ≥ 0. It is easy to see that the integration over y 1 and y 8 in the diagrams of Fig. 4 (b),(c),(d) places a constraint on the wavevector n 1y of the diffusion propagator connecting these two points, n 1y = 2k (unless n 1y = 0). In the same way, the other integrations over y i imply other constraints, which due to the δ-function in the expression for the Hikami box (17) yield a constraint on the channel indices k, l, m, n. Therefore all terms with non-zero transverse harmonics are small as (p F W ) −1 . Up to terms proportional to this small parameter the result (19) is exact. Thus, the 1/3shot noise suppression is, indeed, universal, and does not depend on the ratio W/L, provided the system is diffusive, in accordance with the conclusion of Ref. [5].
To conclude this section, we compare the method used above with other derivations of the 1/3-shot noise suppression [2][3][4]. As is well known, there exist two principally different methods of calculating conductance. One can first evaluate conductivity (which is a local quantity), starting from the Kubo formula, and then, after integration over a cross-section one obtains the conductance. Alternatively, one can calculate conductance directly, starting from the Landauer formula. (In fact, our derivation of the quantity g given above is of this kind). Both derivations are equivalent, although at intermediate stages they have not much in common.
A similar situation happens in the calculation of shot noise. On one hand, one can calculate the microscopic correlator of currents, and upon integration over a crosssection obtains the shot noise power. The derivation of Altshuler, Levitov and Yakovets [4] is exactly of this type [33]. It can be generalized to an arbitrary geometry, and, in principle, can be used for a broad class of problems. The local current correlator contains more information than is necessary for the calculation of the shot noise power. The method of Nagaev [3] and de Jong and Beenakker [7], who employ the quantum Langevin equation, is somewhat similar, although the equivalence between these two approaches is not evident. The generalization of the latter approach for a multi-terminal geometry does not seem to be quite obvious.
The derivation of Beenakker and one of the authors [2], as well as the present method, belong to another, scattering (or Landauer) type of approaches. Ref. [2] derives the shot-noise power with the use of the distribution of transmission eigenvalues of diffusive wire. This proof seems to be the most elegant. However, one should not forget that the distribution of transmission eigenvalues itself is derived by sophisticated methods such as the DMPK equation [34]. Although Nazarov [5] succeeded in extending this derivation to the case of an arbitrary two-terminal geometry, most probably it can not be generalized to multi-terminal case: for conductors with four (or more) probes the shot noise is not expressed through eigenvalues of the scattering matrix s + s. The derivation given in this paper is more general, and self-contained; it does not require the distribution of transmission eigenvalues, and hence allows a generalization to an arbitrary scattering geometry. We repeat that it differs from the microscopic calculation of ref. [4]. Certainly, although these two approaches can not be easily compared at intermediate stages, in the end they should lead to identical results for arbitrary scattering geometries.

III. MULTI-TERMINAL SHOT NOISE
Now we turn to the exchange-interference experiment described in the Introduction. We consider a fourterminal geometry (examples are shown in Fig. 1; for convenience, we still use the coordinates of Fig. 2), and calculate the current correlation for experiments A, B, and C.
The quantity Ξ 1 = Tr(s +αβ s αβ s +γβ s γβ ) is determined by diagrams of Fig. 4, where now points y 1 , y 4 , y 5 , y 8 belong to the contact with lead β; points y 2 , y 3 and y 6 , y 7 belong to the contacts with leads α and γ, respectively. Therefore the diagram (e) is exponentially small, while the diagram (a) is suppressed in the parameter (p F W β ) −1 . Hence, the quantity Ξ 1 is given by the same diagrams (b),(c), and (d), as the quantity t. Using Eq. (12) and integrating over the cross-section of the leads, we obtain (Fig. 5) ×P (r c , r d )P (r e , r f )P (r g , r h )F (r b , r e , r d , r g ).
Here the points r a , r c , r f , r h are given in the coordinates of the contacts β, β, α, γ, respectively. In the same way, for the quantity Ξ 3 one obtains ×P (r c , r d )P (r e , r f )P (r g , r h )F (r b , r e , r d , r g ), (22) and the points r a , r c , r f , r h are given in the coordinates of the contacts β, δ, α, γ, respectively. Expressions for the quantities Ξ 2 and Ξ 4 can be obtained from Eqs. (21) and (22), respectively, by interchanging β ↔ δ.
Expressions (21) and (22) are valid for an arbitrary four-terminal geometry and can be used for numerical calculations. It is important that not only traces Ξ 1 and Ξ 2 , as one could expect, but also quantities Ξ 3 and Ξ 4 are phase insensitive. Indeed, the electron motion which Eqs. (21) and (22) imply is just the diffusion between different leads. No closed paths are formed, except for ballistic motion due to the scattering described by the Hikami box somewhere in the middle of the sample. Since the size of this loop is very small, of order of the mean free path, dephasing is not expected to have an effect on the exchange noise. Certainly, some effects similar to weak localization exist, however, as for conductance [31], they are relatively weak (as (p F l) −1 ) in comparison with the main effect. We do not discuss these effects here.
To make further progress we have to solve the diffusion equation in a given geometry with appropriate boundary conditions. We turn now to the two different geometries, shown in Fig. 1.

A. Box geometry
First, we consider the geometry of Fig. 1a. We assume all leads to be wide, W λ ≫ l. Then points r a , r c , r f , and r h are typically far from the lead's boundaries. This means that, for example, in the integral over y f one can replace the diffusion propagator, P (r e , r f ) by another functionP (r e , r f ), which is also a solution to the diffusion equation, but with another boundary conditions, appropriate for an open surface, We do not need to specify boundary conditions for P (r e , r f ) on the other boundaries, since the point r e is typically in the middle of the sample. Consequently, we may substitute for all "true" diffusion propagators P the functionsP , the solution withP = 0 everywhere on the boundary, as is appropriate for an open system. The solutionP is Furthermore, the functionsP vary considerably on the scale of the size of a sample, L x and L y . If we assume W λ ≪ L x , L y , the functionP (r e , r f ) in the integral over y f may be taken independent of y f . Thus, we obtain and Here Y α , X β , Y γ , and X δ denote the positions of the corresponding leads. We see already from Eqs. (24) and (25) that the results are not-universal in the sense that they depend on the geometry of the sample. Indeed, within the approximation in which we replace P byP , the quantity Ξ 1 does not contain any information on the location and width of lead δ; at the same time, it depends essentially on the location and width of other leads. The quantity Ξ 2 contains information of all leads except β, whereas both Ξ 3 and Ξ 4 are governed by the geometry of all leads. Therefore all ratios Ξ i /Ξ j depend essentially on the geometry of the sample. This is in contrast with the case of a chaotic cavity [18], where one obtains Ξ 1 = Ξ 2 = −3Ξ 3 = −3Ξ 4 irrespectively of geometry, provided the leads are wide enough.
Performing the integration and taking into account that the remaining sums are converging rapidly for L x ∼ L y (the case we assume from now on), one obtains cumbersome expressions for the quantities Ξ i . In the symmetric case, L x = L y = L, W λ = W , Y α = Y γ = X β = X δ = L/2 they simplify. We obtain with positive constants It is seen that the exchange effect exists, and has a negative sign (i.e. exchange suppresses the result of experiment C in comparison with the sum of the results of experiments A and B). Although the relative value of the effect is Ξ 3 /Ξ 1 ∼ 0.1, the effect should be clearly observable.

B. Cross geometry
We consider now the cross geometry of Fig. 1b. We assume that all arms of the cross have equal [35] lengths L and widths W . For L ≫ W we can consider diffusion as one-dimensional. We also assume that the center of the cross is described by a reflection coefficient R and a transmission coefficient T = (1 − R)/3 between any two different arms.
The diffusion propagator is a solution of Eq. (10). We move to the coordinate system of Fig. 1b and fix the point r near the origin of the lead α, x ≃ L. We introduce which is proportional to the time-integrated probability of diffusion from point x in the arm α to point x ′ in the arm λ. The solution satisfying the boundary conditions and the condition of current conservation in the cross, The constant ǫ, defined as the ratio of diffusion probabilities, is calculated in the Appendix. The result is Now we substitute Eq. (27) into the general expressions (21) and (22). Since the area of the cross is negligible in comparison with the areas of the arms, we can neglect the possibility of finding the Hikami box inside the cross, and allow it to be situated only in one of the arms. Upon integration we obtain Thus, in the case T ≫ l/L, when the overall transmission through the sample is governed by the diffusive arms rather than by the center of the cross, one has ǫ ∼ 1. The quantities Ξ 1 and Ξ 2 are regular for ǫ = 1, and therefore assume the finite value, Ξ 1 = Ξ 2 = (5/192)(p F W l/L). At the same time, the exchange terms Ξ 3 and Ξ 4 are strongly suppressed in the parameter l/L, Ξ 3 = Ξ 4 = (1/64)(p F W l 2 /L 2 T )(1 − 2T ). In the less realistic case T ≪ l/L (the transmission is determined by the center of the cross) one obtains ǫ ≫ 1. All quantities Ξ 1 are small, since now all channels are nearly closed (cf. the situation for two-terminal shot noise [29,30]), however exchange terms are additionally suppressed in the parameter ǫ −1 .
Thus, in the cross geometry of Fig. 1b the exchange noise ∆S is suppressed in comparison with the regular terms S A + S B irrespectively of the transmission properties of the center of the cross. It is also quite remarkable that for the cross geometry the exchange contribution is positive, although small: the total effect is enhanced by the exchange.

IV. CONCLUSIONS
We have investigated shot noise in diffusive conductors on the basis of Eq. (2) and the Fisher-Lee relation, which expresses scattering matrices through advanced and retarded Green's functions. In this way, one can reduce disorder averages of various combinations of scattering matrices to standard diagram technique for Green's functions [31]. Although this approach resembles previously published calculations of conductance and conductance fluctuations [24][25][26][27][28], we believe it to be more transparent. We are not aware of any applications of this approach to noise problems.
As a check of the method, we first reproduced the 1/3-shot noise suppression in the two-terminal geometry and confirmed the statement of Ref. [5] that it is in fact super-universal and holds for an arbitrary relation between length and width of a wire, provided the system is diffusive. Our proof bears some similarity with other ones existing in the literature [2][3][4]; however, it is novel, and a direct equivalence to any of the existing proofs is not evident (see the discussion in the end of Section 2).
Then we turned to the multi-terminal geometry and investigated the interference experiment, similar to the Hanbury Brown and Twiss experiment known in optics [15]. We obtained general expressions for scattering matrix combinations (21), (22), determining noise intensities (1); then we investigated them for the two different geometries of Fig. 1. The important point we make is that the exchange effect, even when averaged over disorder, does not vanish. The reason is that typical electron trajectories, contributing to all averaged traces of scattering matrices, considered above (i.e. quantities g and t for the twoterminal geometry, and Ξ i in the four-terminal case) do not contain large closed loops. In particular, it is valid for the "exchange" traces Ξ 3 and Ξ 4 . A typical trajectory for the quantity Ξ 3 is shown in Fig. 6. It is a direct translation of diagrams contributing to this quan-tity. The electron motion is essentially diffusion between different leads with ballistic propagation (described by disorder-averaged single-particle Green's function) close to the leads and somewhere in the middle of the sample (the later motion described by the Hikami box in Eq. (22)). Thus, closed loops are related to ballistic motion over distances of an elastic scattering length only, and therefore neither the shot noise in two-terminal conductors nor the shot noise in multi-terminal structures should be sensitive to dephasing.
Another observation is that exchange corrections are not universal, in contrast to what is found in the chaotic case [18]: the ratio ∆S / S A + S B depends on the geometry of a sample in an essential way. Even the sign of the effect may change: for the box geometry of Fig. 1a it is negative, i.e. interference suppresses the total effect, while for the cross geometry (Fig. 1b) interference enhances the effect (although weakly).
The results obtained for the cross geometry allow us to make predictions for experiments in real systems. Indeed, we found that the exchange contribution is suppressed strongly with respect to the average noise intensities S A and S B . This result was obtained by assuming that the intermediate scattering, described by the Hikami box, does not happen in the center of the cross, i.e., strictly speaking, for ballistic propagation through the center. In more complicated situations the entire exchange effect will be determined by properties of the center of the cross. If the motion within the center is diffusive, one can apply the results obtained above for the box geometry. The total exchange effect is expected to be negative. However, since the arms of the cross (which correspond to disordered leads in the real experiments) contribute to the intensities S A and S B , but not to the exchange contribution, the latter will still be suppressed, if disorder extends far into leads. Finally, if the center of the cross is a chaotic cavity, one may use the results of Ref. [18]. The exchange contribution in the chaotic cavity separated from ideal leads by high barriers (disordered arms play the role of these barriers) is positive: the interference enhances the effect.

ACKNOWLEDGMENTS
We thank S. van Langen, who calculated the exchangeinterference correlator for chaotic cavities, for useful discussions. The work was supported by the Swiss National Science Foundation.

APPENDIX
To find the coefficient ǫ defined by Eq. (28) it is instructive to consider a discrete model of diffusion [36]. Each arm is modeled by a one-dimensional array of scatterers, placed at a distance l from each other; the total number of scatterers in each arm is N = L/l. Each scatterer is described by transmission t = 1/2 and reflection r = 1/2 probabilities. We denote the carrier flux densities in the arm α between sites n and n+ 1 away from the center of the cross by a n , and the flux towards the center of the cross by b n . Corresponding amplitudes in other arms are denoted by a ′ n and b ′ n (Fig. 7). The total flux at each site is given by ρ n = a n + b n , and ρ ′ n = a ′ n + b ′ n . The coefficient ǫ can be expressed as ǫ = ρ 0 /ρ ′ 0 . The diffusion equation implies that all densities should be linear functions of n; furthermore, matching conditions at each scatterer require b n−1 = a n ; b ′ n−1 = a ′ n . Thus, we write a n = A + B(n − 1), a ′ n = A ′ + B ′ (n − 1), The four constants A, B, A ′ , B ′ obey four equations: 1) Boundary condition for the arm β: b ′ N = 0. 2) Matching conditions at the center of the cross: and The fourth equation is Eq.(10), however, it is not required for the calculation of the constant ǫ. We obtain and the limiting cases given by Eq. (29) follow immediately.