Transport through waveguides with surface disorder

arXiv:cond-mat/0610521v1 [cond-mat.dis-nn] 18 Oct 2006 Transport through waveguides with surface disorder M. Martı́nez-Mares,1 G. Akguc,2, ∗ and R. A. Méndez-Sánchez2 1 2 Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, A. P. 55-534, 09340 México D. F., México Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, A. P. 48-3, 62210 Cuernavaca Mor., México We show that the distribution of the conductance in quasi-one-dimensional systems with surface disorder is correctly described by the Dorokhov-Mello-Pereyra-Kumar equation if one includes direct processes in the scattering matrix S through Poisson’s kernel. Although our formulation is valid for any arbitrary number of channels, we present explicit calculations in the one channel case. Ours result is compared with solutions of the Schrödinger equation for waveguides with surface disorder calculated numerically using the R-matrix method. PACS numbers: 42.25.Dd, 73.23.-b, 84.40.Az The prediction of statistical properties of transport through disordered systems is one of the fundamental problems in mesoscopic physics [1, 2]. The transmission coefficient, the dimensionless conductance if we are concerned with electronic devices, or in general any scattering quantity varies from sample to sample due to fluctuations on the microscopic configuration of disorder. Then, its distribution over an ensemble of systems macroscopically equivalent, but microscopically different, is of most interest [1]. Transport through quasi-one dimensional waveguides (quantum wires in the electronic case) with bulk disorder has been tackled theoretically yielding to a Fokker-Planck equation, known as Dorokhov-MelloPereyra-Kumar (DMPK) equation [3, 4]. This equation gives the evolution of the probability density distribution of the transfer (scattering) matrix M (S) when the length L of the system increases. The parameter appearing as time scale in the DMPK equation is the diffusion time across the disordered region. This leads to a natural assumption, the isotropic model of uniformly distributed random phases for M (S) [1]. In the jargon of nuclear physics, it means that there are not prompt responses that could arise from direct processes in the system [5]. An analytical solution of the DMPK equation, for any degree of disorder, was found in the absence of timereversal symmetry (β = 2 in Dyson’s scheme [6]), by mapping the problem to a free-fermion model [7, 8]. In the presence of time-reversal invariance (β = 1), solutions are known only in the localized [9] and metallic regimes [10]. Those solutions were recently checked with extensive numerical Monte Carlo simulations [11, 12]. It was also found in those references that waveguides with surface disorder are correctly described by the solutions of the DMPK equation in the localized regime. Interestingly, the solutions of the DMPK equation does not fit with the numerical results for surface disorder in the crossover to– nor in– the metallic regime. Up to now, a complete theory that include wires or waveguides with rough surfaces is missing. In this letter we show that systems with surface disorder are correctly described by the DMPK equation as the direct processes, due to short-direct trajectories connecting both sides of γ γ’ W L FIG. 1: Sketch of a flat waveguide of width W supporting N channels with surface disorder in a region of length L. Two trajectories are also sketched. The trajectory γ passes directly from one to the other side while trajectory γ ′ passes randomly. the waveguides, are taken into account. We introduce prompt responses in a global approach as explained below. The direct processes are quantified by the ensemble average hSi of the S-matrix, known in the literature as the optical S-matrix [13, 14]. Our theoretical results are compared with numerical calculations based on the Rmatrix theory [15] to solve the Schrödinger equation for the one-channel case. The waveguide with surface disorder is sketched in Fig. 1. It consists in a flat waveguide of width W that support N open modes (or channels) with a region of surface disorder of length L. The scattering problem is studied in terms of a 2N × 2N scattering matrix S which has the structure   r t′ S= , (1) t r′ where r (r′ ) and t (t′ ) are the reflection and transmission matrices for incidence on the left (right) of the disordered region. The dimensionless conductance T = G/G0 , where G0 = 2e2 /h, is obtained from S as T = tr(tt† ), according with Landauer’s formula. From Fig. 1, it is clear that there could be direct transmission since there are trajectories, like γ, connecting both sides of the waveguide without any bounce in the rough surface although other trajectories like γ ′ connect randomly both sides. In order to verify this statement we perform a numerical computation for the particular case N = 1. We choose W = 1; length L = 100 is divided into 100 pieces. To implement the surface disorder the ending point of each piece is a random number between 0 and a constant δ that measures the strength 2 1 G/G 0 0.6 0.2 0.6 0.4 0 1 2 E/E 0 3 4 FIG. 2: Dimensionless conductance T = G/G0 as a function of E/E0 , where E0 = ~2 π 2 /2m, of a waveguide with (a) surface disorder and direct transmission and (b) surface disorder but without direct transmission. The insets show the respective waveguides. In (a) the conductance shows three regimes. In (b) the conductance shows a localized phase only. of the disorder. The ending points of each piece are connected with a spline interpolation to form a smooth disordered surface. The different realizations are done choosing different random displacements. We use a reaction matrix based method to solve the Schrödinger equation [15] in the scattering region with zero derivative at the leads. The basis states can be found using a metric for which scattering geometry transforms to a rectangular region. Finally, the scattering wavefunction is expanded in terms of those states. The result for the dimensionless conductance T , for one realization of disorder, is shown in Fig. 2(a) as a function of the incident energy E in units of the transverse energy E0 = ~2 π 2 /2m. One channel is open for E/E0 in the range 1 to 4. Within this range T shows three different regions and not one as expected in quasi-one-dimensional problems with bulk disorder. For low E/E0 (from 1 to 1.6 approximately) the conductance is small like in a “localized” regime. As shown in Refs. [11, 12] the distribution of the conductance in the localized regime of waveguides with surface disorder agrees with the solution of the DMPK equation. A “metallic” regime is obtained for energies between 2.5 and 4 for which the conductance is close to 1. Finally, there is a transition region between the “localized” and “metallic” regimes. The existence of direct transmission in the waveguides with surface disorder is clear when a similar waveguide is bent in such a way that the direct trajectories are forbidden. In Fig. 2(b) the conductance of a bent waveguide with surface disorder is plotted. It shows only a localized phase. The distribution over an ensemble of 10 elements agrees with the DMPK solution in the localized regime (see Fig. 3). The appearance of only the localized phase in the bent waveguide yields strong evidence that the “metallic” region in Fig. 2(a) is due to direct processes. In the phenomenological model we include the direct FIG. 3: Distribution of the conductance (histogram) for an ensemble of 10 bent waveguides with surface disorder. It agrees with the solution of the DMPK equation in the localized regime (solid line). processes in the solution of the DMPK equation. The 2N × 2N scattering matrix S of the system with hSi 6= 0 is related to a 2N ×2N scattering matrix S0 with hS0 i = 0 through [16, 17] S0 = 1 1 t†p . (S − hSiK ) † t′p (112N − hSiK S) (2) where 112N is the unit matrix of dimension 2N and tp can † be chosen as tp = t′p , where t′p is a 2N ×2N matrix which † satisfies t′p t′p = 112N − hSiK hSi†K [1]. Here, hSiK is the average of S taken with Poisson’s kernel, being the Jacobian of the transformation (2). The actual measured direct processes are quantified by hSi which is linearly related to hSiK (see below). Then S0 is a matrix that describes all the scattering process as S with exception of the direct processes. It has a microscopic configuration of disorder that gives rise to hS0 i = 0. Note that S is reduced to S0 when hSiK = 0 such that hSi = 0. Then the properties of the conductance in the “metallic” regime of the waveguide with surface disorder can be obtained from the localized regime by adding the direct processes. This means that the scattering matrices S and S0 of the waveguide in the “metallic” and localized regimes, respectively, are related through Eq. (2). Since the distribution of S0 is given by the solution of the DMPK equation in the localized regime, the distribution of S can be obtained from the distribution of S0 taking into account the Jacobian of the transformation (2). As in Eq. (1), S0 has the structure   r0 t′0 S0 = , (3) t0 r0′ and both, S and S0 , can be parameterized in a polar representation; for instance [4, 16, 18, 19] # # √ "  " (3) √ (1) 0 u0 − 1 − τ0 √ τ0 0 u0 √ , S0 = (4) (2) τ0 1 − τ0 0 u0 0 u0 (4) 3 where τ0 is a diagonal matrix whose elements are the (j) eigenvalues {τ0 n ∈ [0, 1]} of the matrix t0 t†0 and u0 , j = 1, 2, 3, 4, are N × N unitary matrices for β = 2, with (3) (1) T (4) (2) T and u0 = u0 the additional conditions u0 = u0 (j) for β = 1. In the isotropic model each u0 is distributed according  to the invariant measure of the unitary group,  (j) dµ u0 . The probability distribution of S0 can be written as (S0 ) = dP0 s(β) 0 ({τ0 n }) P0 s(β) 0 dµβ (S0 ), pβ ({τ0 n }) (5) where s0 = L/ℓ0 with ℓ0 the elastic mean free path without direct processes. P0 (β) s0 ({τ0 n }) is the solution of the DMPK equation [7, 8, 10] which depends on the elements of τ0 only, and dµβ (S0 ) = pβ ({τ0 n }) N Y dτ0 n n=1 4 Y j=1   (j) dµ u0 (6) where 0 ≤ φ0 , ψ0 < 2π and 0 ≤ τ0 ≤ 1. At the level of S0 , τ0 becomes the dimensionless conductance T0 , such that τ0 is replaced by T0 everywhere. Eq. (6) gives (see Ref. [1]) is the invariant measure of S0 where pβ ({τ0 n }) = Cβ Y a<b τ0 a − τ0 βb N Y . τ0 (β−2)/2 c (7) c=1 A similar parameterization as Eq. (4) holds for S and for its probability distribution we have (β) (S) = dPs(β) 0 Ps0 (S) dµβ (S), pβ ({τn }) a given value of hSiK , hSf i coincides with hSi − hSiK . Note that if hSiK = 0 (at the same time hSf i = 0), (β) ({τ0n }). Finally, S = S0 and Ps0 ({τn }) reduces to P0 s(β) 0 in Eq. (9), it remains to write the τ0 n ’s in terms of S using Eq. (2). As we mention before, we choose tp = t′p once we calculate t′p numerically from the measured value † of hSi; in general it can be done by diagonalizing t′p t′p . Then the distribution of the conductance T can be obtained by multiple integration of Eq. (9). An example is given below. The case β = 1 with N = 1. In what follows we are concerned with the β = 1 symmetry only and the index β becomes irrelevant such that we can suppress it everywhere. In this case, S0 is the 2 × 2 matrix [see Eq. (4)]  √ √ i(φ0 +ψ0 )  τ e − 1 − τ e2iφ0 , (10) S0 = √ i(φ00+ψ0 ) √ 0 τ0 e 1 − τ0 e2iψ0 (8) where dµβ (S) has a similar expressions as Eqs. (6) (β) and (7) suppressing the label “0”. Now, Ps0 (S) include the phases and it is obtained from the equation (β) dPs0 (S) = dP0 (β) s0 (S0 ), which gives our main result, P0 (β) s0 ({τ0 n (S)}) pβ ({τ0 n (S)})   (2N β+2−β)/2  †  det 112N − hSiK hSiK  ×  , (9) 2  det 112N − ShSi†K (S) = Vβ−1 pβ ({τn }) Ps(β) 0 where Vβ−1 is a normalization constant. The last term on the right hand side is the Jacobian of the transformation (2), known as Poisson’s kernel [14, 16, 17]. Here, we are postulating that this Jacobian is valid not only when S0 is uniformly distributed but also for S0 with uniformly distributed random phases. However, the average of S that appears in the Jacobian is not the actual value of hSi. In fact, it is satisfied the one-to-one correspondence hSi = hSiK + hSf i, where hSf i is the average of the fluctuating part of S, as can be easily seen inverting Eq. (2) to write S as S = hSiK + Sf . We have verified by numerically simulating DMPK S0 ’s, and hence S that, for dT0 dφ0 dψ0 . (11) dµ(S0 ) = √ 2 T0 2π 2π √ where we used that p(T0 ) = 1/2 T0 . As we mention above, the probability distribution of S0 is given by the solution of the DMPK equation. Although the solution in any phase can be used, we use the one in the localized regime which is well known. In this phase, the variable x0 , defined by T0 = 1/cosh2 x0 , is Gaussian distributed [2, 10],   1 s0 2 1  , (12) P0 s0 (x0 ) = √ x0 − exp − πs0 s0 2 with s0 = −4 lnhT0 i. Equivalent expressions to Eqs. (10) and (11) are valid for S without the label “0”. Those are used to write T0 in terms of S using Eq. (2) for a given hSiK . We will consider only the case hSiK = w σx , where σx is one of the Pauli matrices and w a complex number. p For t′p we choose t′p = 1 − |w|2 diag(eiθ1 , eiθ2 ), where θ1 and θ2 are arbitrary phases. From Eq. (2) we get 2 √ (1 + |w|2 ) T eiη − w∗ e2iη − w , T0 = 2 √ 1 − 2w∗ T eiη + w∗2 e2iη (13) where η = φ + ψ. We note that the result is independent of the arbitrary phases θ1 and θ2 . Finally, by direct substitution of hSiK , Eq. (12) and Eq. (13) into Eq. (9), we obtain the result for Ps0 (T, φ, ψ), from which the distribution of T is obtained by integration over the variables φ and ψ in the range 0 to 2π. Since φ and ψ always appear in the combination η = φ + ψ one integration can 4 10 (a) 0.1 0 Ps (T) 1 10 (b) 1 0.1 0 0.2 0.4 T 0.6 0.8 1 FIG. 4: (a) Distribution of T for the one channel case and β = 1 for s0 = 10, and intensities of the direct processes w = 0.2 (dashed), 0.5 (continuous), 0.8 (long-dashed). The crosses are random matrix simulations (106 realizations). For clarity we present only the case w = 0.5. The agreement is excellent. (b) Comparison between theory (random matrix simulations) and numerical solutions of Schrödinger equation as described in the text (histogram). Here, s0 = 33.6 and w = 0.93. The agreement is excellent but the experimental hSi does not fit the theoretical one. be done easily. The result is " 2 #  1 1 s0 1 −1 1 Ps0 (T, η) = cosh √ − exp − √ 2π πs0 s0 2 T0
3 1 − |w|2 1 1 × √ p . (14) 2 T T0 (1 − T0 ) 1 − 2w∗ √T eiη + w∗2 e2iη 3 The remaining integration over the variable η can be done numerically. The results for s0 = 10 and w = 0.2, 0.5 and 0.8 are shown in Fig. 4(a), as well as one random matrix simulation only (for clarity of the figure). The simulation of S is through Eq. (2) with S0 satisfying the DMPK equation in the localized phase. We observe an excellent agreement. As expected Ps0 (T ) moves from localized to metallic (from left to right) when |w| take values from zero to one. The peak in the distribution of T is probably reminiscent of the cut off observed in the numerical simulations with a higher number of channels of Ref. [12]. In Fig. 4(b), we compare the theory (simulations) with the numerical experiment of an ensemble of six realizations of flat waveguides with surface disorder, with the same parameters as those of Fig. 2. For s0 = 33.6, we have an excellent agreement for w = 0.93. The theory gives hSf 11 i = hSf 12 i = 0, hSf 21 i = (1 + i) × 10−4 , hSf 22 i = (−0.3 + 0.4 i) × 10−4 , and it is satisfied that hSi = 0.93σx + hSf i. However, the theoretical value of hSi does not coincide with the experimental one, which is a full matrix; for instance hS12 iexp = 0.51582 + 0.06476 i. This is probably due to the simplicity of the model we used for hSiK , a full matrix being more realistic. To conclude, we obtained the distribution of the conductance for waveguides with surface disorder. The direct transmission between both sides of the waveguide was taken into account in a global way. The results for the one channel case agree with numerical simulations, as well as with a numerical experiment made with Rmatrix method. When the direct processes are increased the distribution moves from localized to metallic. Then, the direct processes could be the cause of the disagreement between the Dorokhov-Mello-Pereyra-Kumar equation and the numerical results recently obtained. This project was supported by DGAPA-UNAM under project IN118805. MMM received financial support from PROMEP-SEP through contract No. 34392. We thank P. A. Mello for useful discussions. ∗ [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] Present address: Cyprus International University, Nikosia, Cyprus P. A. Mello and N. Kumar, Quantum Transport in Mesoscopic Systems: Complexity and Statistical Fluctuations, (Oxford University Press, New York, 2004). C. W. J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). O. N. Dorokhov, JETP Lett. 36, 318 (1982). P. A. Mello, P. Pereyra, and N. Kumar, Ann. Phys. (N.Y.) 181, 290 (1988). H. Feshbach Topics in the Theory of Nuclear Reactions, in Reaction Dynamics, (Gordon and Breach, New York, 1973). F. J. Dyson, J. Math. Phys. 3, 140 (1962). C. W. J. Beenakker and B. Rejaei, Phys. Rev. Lett. 71, 3689 (1993). C. W. J. Beenakker and B. Rejaei, Phys. Rev. B 49, 7499 (1994). J.-L. Pichard in Quantum Coherence in Mesoscopic Systems, edited by B. Kramer, NATO ASI Series B254 (Plenum, New York): p. 369 (1991). M. Caselle, Phys. Rev. Lett. 74 2776 (1995). García-Martín A. and J. J. Sáenz, Phys. Rev. Lett. 87, 116603 (2001). L. S. Froufe-Pérez, P. García-Mochales, P. A. Serena, P. A. Mello, and J. J. Saenz, Phys. Rev. Lett. 89 246403 (2002). G. López, P. A. Mello, and T. H. Seligman, Z. Phys. A 302, 351 (1981). P. A. Mello, P. Pereyra, and T. H. Seligman, Ann. Phys. (NY) 161, 254 (1985). G. B. Akguc and L. E. Reichl, Phys. Rev. E 67, 46202 (2003). L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains (translated from the russian by L. Ebner and A. Korányi) (Providence, RI: AMS, 1963). W. Friedman and P. A. Mello, Ann. Phys. (NY) 161, 276 (1985). H. U. Baranger and P. A. Mello, Phys. Rev. Lett. 73, 142 (1994). P. A. Mello and J.-L. Pichard, J. Phys. I 1, 493 (1991).