Two-dimensional electron systems beyond the diffusive regime

Two-dimensional electron systems beyond the diffusive regime

P. Markoš Department of Physics FEI, Slovak University of Technology, 812 19 Bratislava, Slovakia

Transport properties of disordered electron system can be characterized by the conductance, Lyapunov exponent, or level spacing. Two additional parameters, and were introduced recently which measure the non-homogeneity of the spatial distribution of the electron inside the sample. For the orthogonal, unitary and symplectic two dimensional disordered models, we investigate numerically the system size dependence of these parameters in the diffusive and localized regime. Obtained size and disorder dependence of and is in agreement with with single parameter transport theory. In the localized regime, independently on the physical symmetry of the model. In the diffusive regime, equals to the symmetry parameter . For the symplectic model we analyze the size dependence of in the critical region of the metal-insulator transition and found the non-universal critical value .

73.23.-b, 71.30.+h, 72.10.-d

I Introduction

Transport of electrons through disordered structures offers a broad variety of interesting universal phenomena. lee (); McKK-93 () With increase of the strength of the disorder the character of the transport changes from the ballistic to diffusive up to the insulating, where all electrons are localized. Anderson-58 ().

In the limit of weak disorder (diffusive regime) the transport can be studied analytically using, for instance, the Dorokhov Mello Pereyra Kumar (DMPK) equation dmpk () the Green’s function analysisucf () or random matrix theory. pichard-nato (); been () The existence of the metal-insulator transition in two and three dimensional models AALR (); evers () is a strong motivation to construct an analytical theory of the transport beyond the diffusive regime somoza-prl (); garcia (). Also, numerical data for the localized regime markos2 (); acta (); prior (); somoza (); Z () show that, contrary to theoretical expectation, the distribution of the logarithm of the conductance is never Gaussian for disordered systems in higher dimension. Therefore, a general transport theory must explain how the dimension of the system and physical symmetry of the model evers () influence the ability of electron to move through the sample.

The most elaborated analytical description of the transport in strongly disordered structures is based on the generalized DMPK equation (GDMPKE). mk () The theory takes into account that the spatial distribution of electrons in the regime of localization is not homogeneous. The last was confirmed by numerical simulations in Ref. etopim (); prior () In GDMPK, the non-homogeneity of electron distribution is measured by a large number of parameters (defined later); however, only two of them, and are decisive for the transport.mmwk ()

The GDMPKE is not exactly solvable, but approximate analytical solution for 3D disordered systems mmwk (); MMW (); douglas () agrees very well with numerical data. Numerical solution of GDMPKE Brnd-2007 () confirmed that it correctly describes disordered orthogonal systems and that parameters depend on the dimension of the system.

Detailed numerical analysis of parameters and in three dimensional model was performed in MMW (). The aim of this paper is to investigate how these parameters depend on the physical symmetry in two dimensional (2D) models. We present numerical data for the parameters and for the orthogonal model (O), unitary (U) and two symplectic (S)EZ (); Ando-89 () models in diffusive and insulating regime. For the S models, we also study the behavior of both parameters in the critical regime of the metal-insulator transition.

Ii Generalized DMPK equation

Consider a disordered system of the length connected to two semi-infinite ideal leads with open channels. Transmission parameters are given by the transfer matrix, which can be written in general form as dmpk ()


In Eq. (1), are matrices, and is a diagonal matrix, with positive elements . In systems with time reversal symmetry, matrices and can be represented in terms of and . MelloPichard () For the orthogonal system, and . For the symplectic symmetry, the scattering depends on the spin of the electron; the elements of matrices and are matrices which fulfill the symmetry relations pichard-nato (); MelloPichard ()


Statistical variables , and contain entire information about the transport. In the weak disorder limit,dmpk () the conductance (in units of ) is completely determined by eigenvalues . SE (); pichard-nato ()


In the last equation, we used the parametrization .

The probability distribution of s can be found as a solution of the DMPK equation.dmpk () The generalization of the DMPK for the orthogonal symmetry class, was done by Muttalib and Klauder mk () who introduced new parameters, which characterize the spatial distribution of the electron in the disordered sample. The generalized DMPK equation reads mk ()


where is the mean free path, and


This equation can be simplified when all are approximated by and for all , (). This approximation was confirmed by numerical work MMW (); Brnd-2007 ().

Although the conductance is still given by Eq. (3), it becomes implicitly a function of the spatial distribution of the electron.

Iii Models

In numerical work, disordered sample is represented by two dimensional (2D) square disordered lattice of the size . The orthogonal 2D model with on-site disorder is defined by the Hamiltonian


Here, is the lattice spacing, are random energies from the box distribution, , measures the strength of the disorder and defines the energy scale. To avoid closed channels in leads, we use . comment () In what follows we consider , the energy of the electron . With , we identify the number of channels


It is generally accepted AALR (); McKK-93 () that only localized regime exists in the model when the size of the system (the critical disorder ). Nevertheless, diffusive transport is observable for sufficiently weak disorder and small sample size.acta ()

The second model of interest is the symplectic model with spin dependent hopping. Here, the hopping of electron from one site to the neighboring one can be accompanied by the change of the sign of the spin and , become matrices. In numerical simulations, we study the Ando model with hopping hopping terms


The spin-orbit coupling is characterized by the parameter and . In this paper, . We also study the Evangelou-Ziman (EZ) model EZ () which uses the random hopping matrices : with help of three independent random variables, , distributed uniformly in interval




and consider .

Both Ando and EZ model exhibit the metal-insulator transition when the disorder reaches the critical value . EZ (); Ando-89 () Owing to the anisotropy of our models, the critical disorder differs from that obtained in previous works.EZ (); jpa () We found for the Ando model and for the EZ model.

The 2D model with external magnetic field can be obtained by including the Peierls hopping term , into the Hamiltonian (6).

Iv The matrix

The matrix is defined in terms of higher moments of the matrices :


Here, represents an ensemble average.

For the orthogonal system, the matrix is defined as mk ()


In the diffusive regime,dmpk ()


For the unitary models,




For the systems with symplectic symmetry the matrix is given MC (); Mello ()


In this equation, the matrices , and are defined in terms of the matrix :




In the diffusive regime, is degenerated diagonal matrixMC () with diagonal elements


Our numerical results discussed in Sect. V confirm that the same holds for any disorder strength.

From Eqs. (13,15,19) it follows that


We use these relations to test the numerical accuracy of our results.

V Results

We consider square samples of the size attached to two semi-infinite ideal leads. The size increases from to (S model) up to (O model). For each value of and , we analyze the statistical ensemble of typically samples ( for the largest system size). In numerical calculation, the sample and leads are represented by the transfer matrix and , respectively.xx () Following PMcKR (), the conductance is given as a trace of matrices and , where () are () matrices composed of right and left eigenvalues of , respectively. The upper index () indicate the direction of the propagation through the sample. Comparing with Eq. (1) we find




Thus, eigenvalues can be obtained numerically by diagonalizing of the matrices and . Matrices and consist of corresponding eigenvectors. Details of numerical method are given in Ref. MMW () Mean values, and were calculated as an average over the statistical ensemble


Obtained data for were also used for the calculation of probability distributions.

As noted in Section IV, are matrices. Numerical data confirm that, with the relative accuracy of , these matrices remain diagonal degenerate for each value of the disorder and all size of the system.

Figure 1: (Color online) and as a function of the system size for square lattice for the symplectic (S) and orthogonal (O) s and unitary (U) systems. The disorder . Solid lines are linear fits with slopes given in the legend.

v.1 Diffusive regime

We first verify the prediction of the DMPK equation for the diffusive regime. In Fig. 1 we show the dependence of parameters and for the orthogonal and symplectic system with disorder . The system is in the diffusive regime (the conductance varies between 4.9 and 5.03 for the orthogonal model, and increases from 7 to 11 for the S model). Linear fits shown by solid lines confirm that both and and equals to the symmetry parameter in the diffusive regime. The spatial distribution of electrons is homogeneous and no additional parameter must be introduced into the model. The transport is universal, the only model parameter in the DMPK is the ratio of the system length to the mean free path. Although the DMPK was derived only for the quasi-one dimensional systems, our data show that relations (13,15,19) are valid also for the square samples.

Figure 2: (Color online) The size dependence of (top) and (bottom) for the 2D orthogonal model for various strength of the disorder (given in the legend of bottom panel). converges to the non-zero limit when for each disorder strength. This limiting value, however, is too small to be observable numerically for weak disorder within the considered size of the system. decreases to zero for any value of the disorder .
Figure 3: (Color online) The size dependence and at the critical point for the symplectic (Ando and EZ model) models. Solid and dashed lines are power fits () with the exponent .
Figure 4: (Color online) The size dependence of the parameter for the Ando model. In the metallic regime, (), slowly increases when increases and converges to its isotropic value . In the localized regime (), decreases to zero when the size of the system increases. At the critical point, does not depend on . The -independent critical value of (dotted line) is plotted by dot line.
Figure 5: (Color online) The size dependence of the parameter for the 2D orthogonal model. We expect that decreases to zero for any value of the disorder . However, the size of the system is not sufficient to see this decrease when the disorder is small. Instead, we observe the diffusive limit . Only for sufficiently strong disorder, decreases to zero. In contrast to the symplectic model (Fig. 4) there is no indication for the existence of the critical point where converge to the -independent limit. Inset shows the disorder dependence of for the unitary ensembles (, ).

v.2 Insulating regime

In the limit of strong disorder, we expect that depend on the index and . Contrary, off-diagonal elements , , should decrease to zero, () so that decreases to zero when the size of the system ()

Figure 2 shows the dependence of and for orthogonal systems with various strength of the disorder. Similarly to the 3D orthogonal model discussed in MMW (), both and are linear functions of , Since no metallic regime exists for the non-zero disorder, we expect that converges to the nonzero value in the limit of for all values of ,


The limiting value can be easily calculated numerically for strong disorder. This is more difficult for weak disorder (), since becomes smaller than the inverse of the accessible sample size.

Similar data (not shown) were obtained also for the symplectic models.

v.3 Critical regime (symplectic models)

Critical regime exists only for the S systems. In the critical regime, we found that both and decreases at the critical point to zero


(Fig. 3), so that reaches a critical value, which does not depend on the size of the system


As shown in Fig. 3, the critical value is not universal but depend on the model. We obtain for the Ando model, and 1.795 for the Evangelou-Ziman model.

Figure 4 shows that the length and disorder dependence of parameter can be, at least in principle, used for the estimation of critical parameters in the same way as mean conductance of the smallest Lyapunov exponent. For very weak disorder, we find that only weakly depends on the size of the system and increases to the metallic limit when increases to infinity, indicating that the system is in the metallic regime. For stronger disorder, decreases to zero when the size of the system increases, in agreement with the prediction of the () We found the critical regime between these to limits, where converges to the size-independent constant (obtained already in Fig. 3) when .

For comparison, we show in Fig. 5 data for calculated for the 2D orthogonal model. We found no critical regime. Although for weak disorder, we expect that this is the finite size effect, and will decrease to zero for each disorder strength when increases.AALR ()

Figure 6: (Color online) as a function of the localization length (estimated as ) for the 2D orthogonal model (top) and the 2D symplectic model (bottom). The strength of the disorder varies between and in both Figures. For strong disorder () data converge to the universal curve (solid line is a function for the orthogonal model). Insets in both panels show that is an unambiguous function of .

v.4 The universality

With two new parameters and , we must verify if the transport properties of the system are still maintained by only a single parameter.AALR () In the metallic regime, the answer is trivial since the entire matrix reduces to model-independent numbers given by Eqs. (13,15,19). The universality of the critical regime was shown in the previous section. Here, we concentrate on the localized regime, where we expect that becomes an unambiguous function of the localization length. MMW () The last can be estimated from the smallest parameter ,


In Fig. 6 we plot as a function of for the orthogonal and symplectic Ando models. Data confirm that the parameter becomes an linear function of with increasing system size and converge to the system-size independent limit when .

Two inset of Fig. 6, show that the parameter is an unambiguous function of in all three regimes. In the localized regime, when , data confirm that , consistent with prediction of the Muttalib’s theory.

Figure 7: (Color online) The probability distribution of for the 2D orthogonal model with various disorder strength (given in the figure). The size of the system . Inset shows the distributions . Data confirm that is a good statistical variable with a well defined mean value and variance.
Figure 8: (Color online) The probability distribution of in the strongly localized regime for the Ando model. The mean value is given in the legend.
Figure 9: (Color online) The probability distribution of for the 2D Ando model at the critical point and in the metallic regime (, inset). Both the square samples and Q1D samples are considered.

v.5 Statistical properties of

In the previous analysis we dealt only with mean values of and . Since both and are statistical variables, we must also to study their statistical properties. Figure 7 shows the probability distribution of parameter and for the 2D orthogonal model. For each disorder, the mean value can be identified with the most probable value. In the localized limit, both and var  are of order of unity, and the distribution becomes size independent (Fig. 8).

In Fig. 9 we plot the probability distribution of for the symplectic Ando model in the critical and metallic regime. We demonstrate that the distributions for the square sample with quasi-one dimensional systems are almost identical.

Figure 10: 2D Ando model: as a function of the conductance statistical ensemble of samples with disorder (top) and (bottom). The size of the system .

v.6 Correlation vs

We have shown that in the metallic regime but in the insulator. Small values of indicate that the mean conductance of the system is large. Contrary, large values of correspond to systems with small mean conductance. This is in agreement with our expectation: small conductance means that the electron has problems to go through the sample. When it finally reaches the opposite side, its spatial distribution is not homogeneous any more.

However, the correspondence large - small holds only for mean values of these parameters. As shown in Fig. 10, the values of and for a given sample are not correlated within a given statistical ensemble: small values can be accompanied with any value of - either small or large . The absence of the correlation observed in both the metallic and in strongly localized regime, confirms that the statistical fluctuations of do not affect the mean value of the conductance.

Vi Conclusion

The electron transport through disordered system is determined by spatial distribution of the electron inside the disordered sample, which can be measured by parameters and . Our aim in this paper was to investigate how these two parameters depend on the size of the system, strength of the disorder and physical symmetry of the model. We concentrated on 2D disordered systems. In order to better understand the role of the disorder, we compare numerical data for the orthogonal and symplectic physical symmetry. For completeness, we add also a few data for the unitary ensemble.

In the diffusive regime, the size dependence of both parameters follow the analytical relations given by the theory of DMPK equation. In particular, equals to the symmetry parameter . In the localized regime, converges to the size independent limit and .

For the symplectic models, which exhibit the metal-insulator transition, we analyze the size dependence of both parameters and we found that possesses a critical value when disorder . Also, we found no significant difference between the values of for the two dimensional and quasi-one dimensional systems. No critical value was found for the orthogonal model.

We also found that is an unambiguous function of the localization length and is uniquely given by the parameter . Therefore, the use of these parameters does not contradict the single parameter scaling theory.

Since the elements of matrices are given by elements of statistical matrices , they are also statistical variables. Fortunately, analysis of their probability distributions confirm that their mean values are good representatives of the statistical ensembles. We found no statistical correlations between the conductance and . Therefore, we conclude that mean values, and , and, consequently, , are physical parameters for the description of disordered systems.

Acknowledgments: This work was supported by project VEGA 0633/09.


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