Pseudo-Band-Structure in a Disordered Quantum Wire Array

Transport properties of quantum-wire arrays with disorder in the period are studied theoretically within a self-consistent Born approximation. Although no band gap opens to the lowest order because of the flat average potential, a pseudo-band-structure is recovered in higher order due to effects of strong diffuse Bragg scattering as long as the disorder remains small. A pseudo-band-gap manifests itself in the density of states, the spectral function, and the anisotropic conductivities, and modifies the conductivity in the quantum-wire direction qualitatively in the vicinity of the zone boundary.


Introduction
][3][4] This was achieved 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, zigzag corrugations have some fluctuations. 1,2) he purpose of this paper is to theoretically explore the transport of such strongly modulated two-dimensional systems with some disorder in the period.
In transport experiments a huge anisotropy in the electron mobility was observed. 4,5) he mobility in the quantum wire direction is 30 ∼ 70 times larger than that in the perpendicular direction.An opening-up of a band gap at the zone boundary due to a periodic potential 6) was suggested as the origin. 4)However, a self-consistent calculation of electronic states showed that the gap is not so appreciable and the Fermi wave number does not touch the zone boundary, giving almost circularly symmetric dispersion. 7)As a result, the band structure effects like anisotropic effective masses cannot explain the observed behavior.
Actual zigzag corrugations are disordered.Effects of fluctuations in the period and the potential height have been considered to the lowest order using a Boltzmann transport equation and the large anisotropy was successfully explained. 8,9) n the lowest order approximation, the effective potential becomes flat because of averaging and no band structure appears even in the vicinity of the zone boundary.This lowest-order calculation was extended also to the energy region near zone boundaries and showed that the conductivities in parallel and perpendicular directions exhibit a prominent dip structure.
Intuitively, however, an electron can feel the presence of a coherent potential for several periods when the disorder is not extremely large, leading to a formation of band structure locally.It is an interesting and important issue whether and how this pseudo-band-structure formation is restored in disordered quantum-wire arrays.
When the energy approaches the zone boundary, an electron suffers from strong diffuse Bragg scattering.In this paper we shall demonstrate the presence of a pseudoband-structure by taking into account higher order effects of the diffuse Bragg scattering.
We calculate the density of states, the spectral function, and the conductivities of this disordered onedimensional superlattice including effects of fluctuating corrugation in a self-consistent Born approximation and study effects of such pseudo-gap formation.The paper is organized as follows: In §2 the model and the method of calculations are described briefly.Explicit numerical results are presented in §3.A short summary is given in §4.

Disorder in periodic corrugation
The schematic illustration of the periodic quantum wire array with period a and minimum and maximum well width d 1 and d 2 , respectively, is given in Fig. 1.We choose the x and y direction in the direction perpendicular and parallel, respectively, to the quantum wires.Let Δ(r) be a periodic interface corrugation. 7)Then,  with where a 1 + a 2 = a and a 1 /a 2 = 1/2 for a GaAs (775)B substrate.The corrugation 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) is approximated well by with where E 0 ( d) is the energy of the lowest subband in a quantum well with width d = (d 1 + d 2 )/2. 10,11) [14] Next, we introduce small amount of disorder in the period, height, and direction of the quantum-wire array.The average corrugation vanishes identically, i.e., Δ(r) = 0, (7)   leading to the absence of a periodic modulation on average, where • • • means a sample average.The correlation function for the corrugation has been discussed previously. 8)The approximate but analytic expression is with (10)   where The correlation function is characterized by four parameters, α, ∇ξ, Δ 0 , and λ.They are summarized as follows: The tilt 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 corrugation is likely to vary almost at random between neighboring corrugations.For such a random fluctuation of the bottom and peak heights Δz = z 2 , 8) we have which give Δ 0 /Δ = α/2a.Note that this expression is more appropriate than eq.(2.33) of ref. 8 and makes the introduction of the constant factor β there unnecessary.
Figure 2 shows some examples of the correlation function.An important feature is the presence of a large diffuse Bragg peak at (±2π/a, 0) broadened in the q x direction by a Lorentzian function corresponding to the exponential decay in the real space.In the q y direction the correlation is essentially given by a Gaussian function, but its actual form varies as a function of q x .
There are some other scatterers in quantum wells.In the following we shall consider impurities with shortrange potential as an representative example.The impurity potential is given by Its correlation function is given by where n i is the concentration of impurities.The strength of the impurity scattering is characterized by mean free path l = v 0 τ 0 with the velocity at zone boundary v 0 = π /m * a and the relaxation time τ 0 given by /2τ 0 = n i u 2 m * /2 2 , where m * is the effective mass.

Self-consistent Born approximation
The Green's function averaged over configurations of disordered corrugations and impurities is written as with z being complex, ε(k) = 2 k 2 /2m * , and self-energy Σ(z, k).In the self-consistent Born approximation, the self-energy is given by The retarded and advanced Green's functions are defined by respectively.The density of states is given by Using the Kubo formula, 15) the conductivity σ μν with μ, ν = x, y is calculated as where f (ε) is the Fermi distribution function and with vertex functions Γ RA ν (ε, k), etc.The vertex function where F RA ν (ε, k) satisfies a Bethe-Salpeter type equation, 16) 22) The other vertex functions satisfy similar equations.
The huge Bragg peaks present in the correlation function of the corrugation makes the scattering probability strongly dependent on the electron wave vector.It is necessary, therefore, to solve the self-consistent equations ( 16) and (17) by discretizing the wave-vector space.The Bethe-Salpeter type equations should also be solved in a similar manner.Because 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 sufficient convergence is achieved so far is l/a = 16.In a high-quality two-dimensional system in a GaAs quantum well the mean free path can be larger than 10 μm, which gives l/a ∼ 10 3 for a = 12 nm.Therefore, the present system corresponds to a "bad quality" quantum well.The parameter F eff Δ giving the strength of the periodic potential is characterised by the gap ε G = 2|F eff Δ n | with n = 1 (see eq. ( 4)).We shall choose ε G /ε 0 = 0.2, i.e., |F eff Δ|/ε 0 = 0.5065 • • • , with ε 0 = ( 2 /2m * )(π/a) 2 being the energy at the first zone boundary, ∇ξ = 0.1 and λ/a = 5, and vary α and Δ 0 keeping the relation Δ 0 /Δ = α/2a.

Density of States
Figure 4 shows some examples of calculated density of states in the vicinity of ε 0 .The density of states is normalized by that in the ideal two-dimensional free electrons system g 0 = m * /π 2 .For a small disorder in the corrugation α/a = 0.06, the density of states exhibits a characteristic feature corresponding well to the formation of a gap, although broadened considerably due to the disorder and the presence of short-range scatterers.This feature is smoothed out with the increase of the disorder and disappears completely for α/a > 0.24.In Fig. 5 spectral functions, −ImG R (ε, k), are shown in the (k x , k y ) space.In the vicinity of the Bragg plane the spectral function decreases and has a tendency to follow the equi-energy line in the ideal quantum wire array.A careful analysis reveals further that this feature is largest for small disorder α/a = 0.06 and smaller for large disorder α/a = 0.24.Therefore, the spectral function shows also that electrons have pseudo-band-gap as long as the disorder of the corrugation remains small.

Conductivity
Figure 6(a) shows corresponding results for the conductivity in the x (σ xx , perpendicular to quantum wires) and y direction (σ yy , parallel to quantum wires) as a function of the Fermi energy.The conductivity is normalized by the Boltzmann result σ 0 = n 0 e 2 τ 0 /m * determined by short-range scatterers, where the electron concentration is given by n 0 = ε 0 g 0 .In Fig. 6(b), examples of the conductivity calculated using Boltzmann equation 8) are shown.
When the energy approaches the zone boundary, the conductivity σ xx has a prominent dip due to the strong diffuse Bragg scattering.The amount of the dip is about the same as that calculated using the Boltzmann equation.The most notable feature is that σ yy obtained in the self-consistent Born approximation has a broad and weak peak in the vicinity of the zone boundary quite in contrast to a slight dip in the Boltzmann result.This ap- pearance of the peak in σ yy arises due to the formation of a pseudo-gap due to disordered corrugations.In fact, the results shown in Fig. 6 have features same as those of the conductivities in the ideal quantum-wire array with impurities with short-range potential.
Figure 7 shows the conductivities of an ideal quantum wire array in the presence of short-range impurities, obtained by solving a Boltzmann equation.The band gap is chosen as ε G /ε 0 = 0.2 and the mean free path l/a = 16 in this example also.In the energy range corresponding to the band gap 0.9 < ε/ε 0 < 1.1 σ xx has a valley structure and σ yy has a peak.The parallel conductivity σ yy is dominated by states with large |k y | with high velocity in the y direction and the peak appears due to the reduction of the final-state density contributing to scattering.The perpendicular conductivity σ xx decreases considerably due to the reduction of the number of states with high velocity in the x direction in the band gap.A sharp  decrease at ε/ε 0 = 0.9 is due to the logarithmic divergence of the density of states.Figures 6(a) and (b) show that the Boltzmann conductivity is valid except for the qualitative (but quantitatively not so significant) difference in the behavior of σ yy in the vicinity of the zone boundary.They show also  that the anisotropy of the conductivity when the Fermi energy is about a half of ε 0 (roughly corresponding to the actual experimental conditions 4,5) ) is much smaller than that obtained previously. 8,9) n fact, σ yy /σ xx ≈ 1 for α/a = 0.06 and σ yy /σ xx < 2 even for α/a = 0.24.This reduction of the anisotropy is a direct consequence of the presence of strong short-range scatterers giving isotropic transport.In Fig. 2 the strength of impurity scattering is denoted by horizontal dotted lines for l/a = 16.It shows clearly that diffuse Bragg scattering is much smaller than impurity scattering around k F a/π ∼ √ 2 for α/a = 0.06 and becomes similar for α/a = 0.24.
Figure 8 shows the Boltzmann conductivity for varying strength of short-range scatterers.It shows that the conductivity parallel to quantum wires is very sensitive to the presence of short-range scatterers and the large anisotropy in the conductivity requires the mean free path determined by short-range scatterers be l/a > 10 3 for α/a = 0.06.

Summary
We have calculated the density of states, spectral functions, and conductivities of a disordered quantum wire array in the self-consistent Born approximation.When the disorder in the corrugation is small, both density of states and spectral function exhibit characteristic features corresponding to the formation of a pseudo-bandgap.The conductivities obtained by solving a Bethe-Salpeter equation show also that there is a pseudoband-structure.An important issue left for the future is whether the conductivity parallel to wires continues to have a peak in contrast to the Boltzmann result obtained previously 8,9) even when the strength of shortrange scatterers is sufficiently small and the transport is dominated by the diffuse Bragg scattering.

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

αFig. 2 .
Fig.2.Some examples of correlation function D(q) and n i u 2 /F 2 eff .(a) α/a = 0.06 and (b) α/a = 0.24.The solid lines represent D(q) and the dotted line represents n i u 2 /F eff for the mean free path l/a = 16.

Fig. 3 .
Fig. 3.The diagrams of (a) the self-energy and (b) the vertex functions.The solid lines denote Green's function and the dashed lines denote the correlation function of the disorder of interface corrugation and impurities.

Fermi 2 Fig. 4 .
Fig. 4. The density of states for different α/a as a function of Fermi energy, in units of m * /π .The solid lines are that in the disordered quantum wire array and the dotted line is that in the ideal case.The constant parameters a 1 /a = 1/3, a 2 /a = 2/3 come from the experiments.

Fig. 5 .
Fig.5.The spectral functions in the case of (a) α/a = 0.06 and (b) α/a = 0.24 for ε/ε 0 = 1.4,1.06, and 0.84.The dot-dashed lines represent the equi-energy curve in the system with an ideal corrugation and the dotted line the Bragg plane.The thin lines are the contour map of the spectral function for values (j/10) × (2τ 0 / ) with j = 1, 2, 3, 4. 2τ 0 / = 2 2 /m * n i u 2 is the peek value of the spectral function when the scatterer is impurity only.

Fermi 2 τFermi 2 τFig. 6 .
Fig. 6.Examples of calculated conductivities for different values of α/a.The solid lines represent the conductivity parallel to quantum wire direction (y direction) and the dotted lines represent that in the perpendicular direction (x direction).(a) Self-consistent Born approximation.(b) Lowest Born approximation using Boltzmann equation.

Fig. 7 .
Fig.7.Conductivities calculated using a Boltzmann equation in an ideal quantum-wire array in the presence of short-range impurities.The solid line is the conductivity in a direction to parallel to quantum wire, and dotted line is that in a direction to perpendicular to quantum wire.

Fig. 8 .
Fig.8.Effects of short-range scatterers on the Boltzmann conductivity.The solid lines represent the conductivity parallel to quantum wire direction (y direction) and the dotted lines represent that in the perpendicular direction (x direction).