Dynamical Conductivity in Disordered Quantum Wire Array

The dynamical conductivity of quantum wire arrays with disorder in a corrugation period, direction, and height are calculated in a self-consistent Born approximation. Anisotropic scattering associated with diﬀuse Bragg peaks causes the formation of pseudo-band-structure and gives rise to inter-band optical absorption in the dynamical conductivity in the direction perpendicular to quantum wires.


Introduction
][3][4] This was made by the growth of a corrugated GaAs quantum well with a zigzag shape on top of a flat AlAs surface and covered by an AlAs barrier layer.Actually, the zigzag corrugations have some fluctuations. 1,2) he purpose of this paper is to theoretically explore the influence of such strongly modulated two-dimensional systems on the dynamical conductivity.
In transport experiments a huge anisotropic mobility was observed. 4,5) he mobility in the quantum wire direction is 30 ∼ 70 times as large as that in perpendicular direction.Band-gap opening at the zone boundary due to periodic potential 6) was suggested as the origin. 4)owever, a self-consistent calculation of electronic states showed that the gap is not so appreciable and the Fermi wave number dose not touch the zone boundary, giving almost circularly symmetric dispersion. 7)As a result, the band structure cannot be the origin of the observed huge anisotropy.
Actual zigzag corrugations are disordered.This disorder causes a diffuse Bragg peak in the correlation function of the corrugations and huge anisotropic scattering.Calculations to the lowest order in the anisotropic scattering within a Boltzmann equation gave a large anisotropic mobility in agreement with the experiments. 8,9) ater calculations with effects of higher order scattering included in a self-consistent Born approximation have shown that this system has a pseudo-bandstructure. 10) In fact, the density of states has a dip and the equi-energy lines in the wave-vector space have a characteristic feature near the energy corresponding to the Brillouin-zone boundary in a perfectly periodic system.
When a band gap is formed, inter-band optical transitions contribute to the dynamical conductivity.Therefore, the dynamical conductivity has important information on features of the pseudo-band-structure.In this paper, we calculate the dynamical conductivity in systems with anisotropic scattering and demonstrate the presence * E-mail: thiroshi@stat.phys.titech.ac.jp of inter-band optical transitions across a pseudo-bandgap.This paper is organized as follows: In §2 the model and the method of calculations are described briefly, in §3 numerical results are presented, and in §4 a short summary is given.Calculations using a Boltzmann equation are discussed in Appendix.

Disorder in periodic corrugation
A schematic illustration of the periodic quantum wire array is shown in Fig. 1.The period of the corrugation is a and the minimum and maximum well-width are d 1 and d 2 , respectively.We choose the x and y direction perpendicular and parallel, respectively, to the quantum wire direction.Let Δ(r) be the interface corrugation. 8)e have For a GaAs (775)B substrate we have a 1 /a 2 = 1/2.The corrugation Δ(r) is expanded as where Δ n is Fourier coefficients given by for n = 0 and Δ n = 0 for n = 0.It is known that effective potential V eff (r) corresponding to Δ(r) for electrons in the lowest subband is approximated well by with  where E 0 ( d) is the subband energy in a quantum well with d = (d 1 + d 2 )/2. 11,12) [15] Next, we introduce small amount of disorder in period, height, and direction of the quantum wire array.This has been discussed previously. 8)The average corrugation vanishes, i.e., Δ(r) = 0. ( The correlation function of the corrugation is represented as the approximate but analytic expression with   where This correlation function is characterized by four parameters, α, ∇ξ, Δ 0 , and λ.They are summarized as follows: α : fluctuations of the corrugation period ∇ξ : fluctuations of the local quantum-wire direction Δ 0 : fluctuations of the height of the corrugation averaged locally λ : correlation length of fluctuations of the height of the corrugation averaged locally in the quantum-wire direction In the case of the present corrugation, the tile angle θ 1 and θ 2 shown in Fig. 1 are determined by the microfacets appearing during the growth process.However, the height of the peak and the valley of the corrugations are likely to vary at random between neighboring corrugations.For such a random fluctuation of the bottom and peak height Δz = z 2 − z 2 (z is the height of one corrugation), 8,10) we have When a 1 /a 2 = 1/2, eqs.( 13) and ( 14) give Δ 0 /Δ = α/2a.Frequency (units of π 2 h/2m * a 2 ) α/a=0.06 α/a=0.12α/a=0.18α/a=0.24Frequency (units of π 2 h/2m * a 2 ) ) α/a=0.06 α/a=0.12α/a=0.18α/a=0.24There are some other scatterers in actual systems.In the following we shall consider impurities with shortrange potential as an representative example.The impurity potential is given by where u is strength of a short-range impurity.The correlation function is given by where n i is the concentration of impurities.The strength of the scattering is characterized by mean free path l = v 0 τ 0 with the velocity at the zone boundary v 0 = π /m * a and relaxation time τ 0 given by /2τ 0 = n i u 2 m * /2 2 , where m * is the effective mass.Thus, the total scattering potential V (r) is written by

Self-Consistent Born Approximation
The Kubo formula for the dynamical conductivity is written as 16) where J ν is the current operator and ν = x or y, Ω is the system volume, and δ +0.The real part of (18) is rewritten as with the vertex function defined by where Γ RA ν , for example, is written as ) Similar expressions are obtained for Γ AR ν , Γ RR ν , and Γ AA ν .Here, G R (E, k) is the retarded Green's function, written as is the advanced Green's function, and F RA ν (E, ω, k), etc. are the current vertex-functions to be defined below.
We shall use a self-consistent Born approximation in the following.This approximation is represented diagrammatically in Figs.2(a) and (b).It is known as the lowest-order approximation free from divergence and giving qualitatively reasonable results.In fact, it was successful in demonstrating the opening of a pseudo band gap in the present quantum wire arrays with disorder in their period. 10)In this approximation the self-energy is given by In the case of usual scatterers not causing strong anisotropic scattering, the frequency dependence is described by a Drude form where σ 0 is the static conductivity and τ is a relaxation time.In the following, we shall compare the results with this Drude conductivity in which τ is obtained by σ(0) and σ(ω) with ω/ε 0 = 0.02, i.e., the low-frequency limit, where ε 0 is the energy at the first-zone boundary, given by The dynamical conductivity can also be calculated by solving a Boltzmann transport equation.Details on the calculation of Boltzmann conductivity σ B (ω) are discussed in Appendix.The result shows that Reσ B xx (ω) decreases more rapidly than σ D (ω) with the increase of ω in the low-frequency region and at the same time has a much larger tail in the high-frequency region.
In the following, eqs.( 23) and ( 24) are solved numerically by simple iteration in the discrete wave vector space.Because the huge diffuse Bragg peaks present in the correlation function, the scattering probability has strongly dependent on the electron wave vector. 8,10) ecause the number of k points is limited due to computational time, impurities have to be introduced for the purpose of making actual calculations possible.In the following explicit calculations, the largest mean free path for which the sufficient convergence is achieved so far is l/a = 16.In a high quality two-dimensional system in GaAs quantum well, the mean free path can be larger than 10 μm, which given l/a ∼ 10 3 for a = 12 nm.Therefore, the present system corresponds to a "bad quality" quantum well.
The parameter F eff Δ which gives the strength of the periodic potential is characterized by the gap ε G = 2|F eff Δ n | with n = 1.We shall choose ε G /ε 0 = 0.2, i.e., |F eff Δ|/ε 0 = 0.5065 • • • , with ∇ξ = 0.1 and λ/a = 5, and vary α and Δ 0 under the relation Δ 0 /Δ = α/2a.These parameters are the same as those in the previous study. 10)igure 3 summarizes the density of states and the conductivities as a function of the Fermi energy ε F , calculated previously. 10)The density of states exhibits a clear feature of the pseudo-band-gap formation and the anisotropy in the conductivity is largest in the gap.In the following we shall calculate the dynamical conductivities when the Fermi level is at the zone boundary, i.e., ε F = ε 0 .Frequency (units of π 2 h/2m * a 2 ) ) 0.06 0.12 0.18 0.24

Ideal Case
Fig. 6.Some examples of σ inter−band (ω) together with the interband conductivity in an ideal quantum-wire array.

Numerical Results
Figure 4 shows some examples of calculated conductivities in the direction (a) perpendicular and (b) parallel to the quantum-wire direction.First, we should note that the static conductivity in the quantum-wire direction is larger than that in the perpendicular direction, although the anisotropy is small because of the presence of considerable amount of isotropic short-range scatterers.Further, the conductivity in the quantum-wire direction is featureless and exhibits a simple Drude behavior, i.e., the results obtained the self-consistent Born approximation and those of the Boltzmann equation are almost the same and in agreement with σ D (ω).
On the other hand, the conductivity in the direction perpendicular to the quantum-wire direction deviates considerably from the Drude conductivity in the highfrequency region ω/ε 0 > 0.1, in particular in the case of the small disorder α/a = 0.06.Further, the deviation from the Drude conductivity decreases with the increase of the disorder α.This deviation represents the contribution of inter-band optical transitions due to the formation of pseudo-band-structure by the corrugation.
This fact is demonstrated in Fig. 5 showing the vertex function Γ x (k) defined by eq.(20).In an ideal quantumwire array with completely periodic corrugation, interband transitions appear across the band gap as illustrated in the inset.The wave vector k x corresponding to this transition is denoted by the vertical dashed lines in Fig. 5(a) corresponding to the smallest disorder α/a = 0.06.The figure shows that the region in the k space corresponding to this k x gives a large contribution to the conductivity.In the case of large disorder α/a = 0.24, on the other hand, this feature disappears completely, showing that the pseudo-band-structure is destroyed by the disorder.
It is interesting to note that most of the high-frequency tail of σ xx (ω) can be reproduced quite well by the Boltzmann conductivity, in particular, in the case of the large disorder, although the deviation becomes larger with the decrease of α/a and is appreciable for the smallest disorder α/a = 0.06.This is likely to be expected because the pseudo-band-gap and the level broadening are about the same due to the disorder in the quantum-wire potential.It is, therefore, difficult to extract the pure contribution of inter-band optical transitions from σ xx (ω).
One way is to subtract the Drude conductivity, i.e., This σ inter−band (ω) is compared with the inter-band conductivity in an ideal system in Fig. 6.In the case of the smallest disorder α/a = 0.06, σ inter−band (ω) has a broad peak slightly below ω/ε 0 = 0.2, i.e., the band gap in the ideal system, and the integrated intensity is similar.For a slightly large disorder α/a = 0.12, the peak is shifted to the low-frequency side and at the same time the integrated intensity becomes smaller.With the further increase in the disorder the inter-band conductivity decreases and disappear.We can also subtract the Boltzmann conductivity to extract inter-band transitions.The results are qualitatively similar except that the integrated intensity becomes much smaller.

Summary
We have calculated the dynamical conductivity in quantum-wire arrays with disorder in their corrugation period, direction, and height within the self-consistent Born approximation.The conductivity in the wire direction is given by a featureless Drude form, but the perpendicular conductivity has a large and long high-frequency tail.When the disorder is sufficiently small, this highfrequency conductivity is caused by optical transitions across the pseudo-band-gap formed by the diffuse Bragg scattering.This has been demonstrated by the double peak structure of the vertex function in the wave-vector space.

Acknowledgment
This work was supported in part by a 21st Century COE Program at Tokyo Tech "Nanometer-Scale Quantum Physics" from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

Appendix: Boltzmann Equation
A Boltzmann transport equation is given by  where V k k is the matrix element of the scattering potential and ε k = k 2 /2m * .To the linear order in external electric field E(t) = E exp(−iωt), the distribution function can be written as where f 0 (ε k ) is the equilibrium distribution function.Then, we have Define g x k (t) and g y k (t) as g k (t) for the electric field in the x and y direction, respectively.The corresponding conductivity will be denoted by σ xx and σ yy .First, we consider the case of the field E in the x direction.Define h x (k, ω) by with relaxation time τ k defined by which depends strongly on k for the present anisotropic scattering.Then, eq.(A•5) is rewritten as The above equation shows that the contribution of states with large τ k , giving a dominant contribution to σ(0), rapidly diminishes and that with small τ k , giving only a negligible contribution to σ(0), becomes more and more dominant with the increase of frequency ω.As a result, Reσ(ω) decreases rapidly with the increase of ω in the low-frequency region and at the same time has a large tail in the high-frequency region.
When there are only short range scatterers causing isotropic scattering, h x (k, ω) becomes Thus, the real part of the conductivity σ xx (ω) is given by a simple Drude form: where σ xx (0) is the static conductivity.Because of the δ function giving the energy conservation, h x (k, ω) becomes a function for ω and θ which is the angle of k measured from the k x direction.Therefore, the integral equation (A•8) can be solved by discretizing θ along the Fermi line at zero temperature.When we consider σ yy (ω), we should change x into y in the above equations.
Figure 7 shows some examples of calculated h x (k, ω)τ k and h y (k, ω)τ k , proportional to the nonequilibrium distribution function.For isotropic scatterers, h x τ k ∝ cos θ and h y τ k ∝ sin θ.In the case of small disorder α/a = 0.06, they are considerably different from cosine or sine curve that are nonequilibrium distribution functions for the simple Drude conductivities because of the strong anisotropic scattering.In the case of large disorder α/a = 0.24, on the other hand, the distribution function is close to the Drude form.

Fig. 1 .Fig. 2 .
Fig. 1.A schematic illustration of a periodic quantum wire array consisting of a GaAs/AlAs heterostructure.The quantum wire are along the y direction.

Fig. 3 .
Fig. 3.The density of states (upper panel) and the static conductivities (lower panel) as a function of the Fermi energy, calculated in the self-consistent Born approximation.

Fig. 5 .
Fig. 5.Some examples of the contour maps of Γx(k) for (a) α/a = 0.06 and (b) 0.24.The numbers show values of Γx in units of 4m * 2 a 3 / 4 π 3 .The vertical dashed lines show kx for allowed optical transitions for a clean periodic quantum-wire-array.The inset shows the band structure and optical transitions near the gap.In this figure ω is measured in units of ε 0 / .

Fig. 7 .
Fig.7.Examples of calculated nonequilibrium distribution functions g x (θ, ω) and g y (θ, ω) in (a) and (b) respectively as a function of θ.The solid lines and the dotted line represent that in case of α/a = 0.06 and α/a = 0.24 respectively.In this figure ω is measured in units of ε 0 / .