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Keywords:

  • electronic transport;
  • HgTe;
  • localization;
  • quantum interferences;
  • quantum wells;
  • spin-orbit interaction

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Symmetry analysis of the low-energy Hamiltonian
  5. 3 Symmetry analysis of quantum conductivity corrections
  6. 4 Microscopic calculation of the interference correction
  7. 5 Weak antilocalization in a quasi-one-dimensional strip
  8. 6 Summary and discussion
  9. Acknowledgments
  10. References

We study quantum transport in HgTe/HgCdTe quantum wells under the condition that the chemical potential is located outside of the band gap. We first analyze symmetry properties of the effective Bernevig–Hughes–Zhang Hamiltonian and the relevant symmetry-breaking perturbations. Based on this analysis, we overview possible patterns of symmetry breaking that govern the quantum interference (weak localization or weak antilocalization) correction to the conductivity in two-dimensional HgTe/HgCdTe samples. Further, we perform a microscopic calculation of the quantum correction beyond the diffusion approximation. Finally, the interference correction and the low-field magnetoresistance in a quasi-one-dimensional geometry are analyzed.

1 Introduction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Symmetry analysis of the low-energy Hamiltonian
  5. 3 Symmetry analysis of quantum conductivity corrections
  6. 4 Microscopic calculation of the interference correction
  7. 5 Weak antilocalization in a quasi-one-dimensional strip
  8. 6 Summary and discussion
  9. Acknowledgments
  10. References

Two-dimensional (2D) and three-dimensional (3D) materials and structures with strong spin–orbit interaction in the absence of magnetic field (i.e., with preserved time-reversal invariance) may exhibit a topological insulator (TI) phase [1-7]. In the 2D case, the TI behavior was experimentally discovered by the Würzburg group ([4]) in HgTe/HgCdTe quantum wells (QWs) with band gap inversion due to strong spin–orbit interaction. The band inversion results in emergence of helical modes at the edge of the sample. These modes are topologically protected as long as the time-reversal symmetry is preserved.

Application of a bias voltage leads to the appearance of a quantized transverse spin current, which is the essence of the quantum spin-Hall effect (QSHE). An interplay between the charge and spin degrees of freedom characteristic to QSHE is promising for the spintronic applications. The existence of delocalized mode at the edge of an inverted 2D HgTe/HgCdTe QW was demonstrated in Refs. [4, 8]. These experiments have shown that HgTe/HgCdTe structures realize a novel remarkable class of materials –inline image TIs –and thus opened research direction. Another realization of a 2D inline image TI based on InAs/GaSb structures proposed in Ref. ([9]) was experimentally discovered in Ref. ([10]).

When the chemical potential is shifted by applying a gate voltage away from the band gap, a HgTe/HgCdTe QW realizes a 2D metallic state, which can be termed a 2D spin–orbit metal. The interference corrections to the conductivity and the low-field magnetoconductivity of such a system reflect the Dirac fermion character of carriers [11-14], similarly to interference phenomena in graphene [15-18] and in surface layers of 3D TI [11, 19, 20]. Recently, the magnetoresistivity of HgTe/HgCdTe structures was experimentally studied away from the insulating regime in Refs. [21-23], both for inverted and normal band ordering.

In this article, we present a systematic theory of the interference-induced quantum corrections to the conductivity of HgTe-based structures in the metallic regime. We investigate the quantum interference in the whole spectrum, from the range of almost linear dispersion to the vicinity of the band bottom and address the crossover between the regimes. We begin by analyzing in Section 'Symmetry analysis of the low-energy Hamiltonian' symmetry properties of the underlying Dirac-type Hamiltonian and physically important symmetry-breaking mechanisms. In Section 'Symmetry analysis of quantum conductivity corrections', we overview a general symmetry-based approach ([13]) to the problem and employ it to evaluate the conductivity corrections within the diffusion approximation. Section 'Microscopic calculation of the interference correction' complements the symmetry-based analysis by microscopic calculations. Specifically, we calculate the interference correction beyond the diffusion approximation, by using the kinetic equation for Cooperon modes, which includes all the ballistic effects. A quasi-one-dimensional geometry is analyzed in Section 'Weak antilocalization in a quasi-one-dimensional strip'. Section 'Summary and discussion' summarizes our results and discusses a connection to experimental works.

2 Symmetry analysis of the low-energy Hamiltonian

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Symmetry analysis of the low-energy Hamiltonian
  5. 3 Symmetry analysis of quantum conductivity corrections
  6. 4 Microscopic calculation of the interference correction
  7. 5 Weak antilocalization in a quasi-one-dimensional strip
  8. 6 Summary and discussion
  9. Acknowledgments
  10. References

The low-energy Hamiltonian for a symmetric HgTe/HgCdTe structure was introduced in Ref. ([3]) in the framework of the inline image method. The Bernevig–Hughes–Zhang (BHZ) Hamiltonian possesses a inline image matrix structure in the Kramers-partner space and E1–H1 space of electron and hole-type levels [2, 9, 24],

  • display math(1)
  • display math(2)

Here, the components of spinors are ordered as E1+, H1+, E1 −, and H1 −. It is convenient to introduce Pauli matrices inline image for the E1–H1 space and inline image for the Kramers-partner space (here inline image and inline image are unity matrices), yielding

  • display math(3)

The effective mass inline image and energy inline image are given by

  • display math(4)

The normal insulator phase corresponds to inline image (which is realized in thin QWs, inline image nm), whereas the TI phase is characterized by inline image (realized in thick QW) ([4]).

The Hamiltonian inline image breaks up into two blocks that have the same spectrum

  • display math(5)

The eigenfunctions for each block are two-component spinors in the E1–H1 space

  • display math(6)

where spinors inline image are different in different blocks

  • display math(7)
  • display math(8)

Here, inline image is the polar angle of the momentum inline image and

  • display math(9)

corresponds to the upper and lower branches inline image of the spectrum.

Disorder potential inline image is conventionally introduced in the BHZ model by adding the scalar term ([12])

  • display math(10)

to the inline image Hamiltonian inline image. This model describes smooth disorder that does not break the spatial reflection symmetry of the structure and thus does not mix the two Kramers blocks of the BHZ Hamiltonian.

We are now going to discuss symmetry properties of the Hamiltonian of a 2D HgTe/CdTe QW and symmetry-breaking mechanisms [13, 24]. The Hamiltonian inline image is characterized by the exact global time-reversal (TR) symmetry inline image. Further, this Hamiltonian commutes with inline image, which we term the “spin symmetry.” Finally, an additional approximate symmetry operative within each Kramers block emerges for some regions of energy. Specifically, inline image in the inverted regime acquires the exact symplectic “block-wise” TR symmetry when inline image. Around this point, the symmetry is approximate. An approximate orthogonal block-wise TR symmetry emerges near the band bottom for inline image and for high energies inline image.

When employing the symmetry analysis to a realistic system, the symmetries of Hamiltonian (3) should be regarded as approximate. The term “approximate symmetry” here means that the corresponding symmetry-breaking perturbations in the Hamiltonian are weak, such that they violate this “approximate symmetry” on scales that are much larger than the mean free path. On the technical level, the gaps of the corresponding soft modes (Cooperons) are small in this case. This, in turn, implies that there exists an intermediate regime, when the dephasing length (or the system size) is shorter than the corresponding symmetry-breaking length. In this regime, the diffusive logarithmic correction to the conductivity is insensitive to this symmetry-breaking mechanism and the system behaves as if this symmetry is exact. However, when the dephasing length becomes longer than the symmetry-breaking length, the relevant singular corrections are no longer determined by the dephasing but are cut off by the symmetry-breaking scale. This signifies a crossover to a different (approximate) symmetry class. Below we analyze the relevant symmetry-breaking perturbations in HgTe structures.

The inline image spin symmetry is violated by perturbations that do not preserve the reflection (inline image) symmetry of the QW. Such perturbations yield nonzero block-off-diagonal elements in the full inline image low-energy Hamiltonian. One of possible sources for the block mixing is the bulk inversion asymmetry (BIA) of the HgTe lattice. The corresponding term in the effective Hamiltonian reads ([9])

  • display math(11)

The BIA perturbation (11) contains the momentum-independent term with inline image that connects the electronic and heavy-hole bands ([25]) with opposite spin projections. The terms with inline image and inline image stem from the cubic Dresselhaus spin–orbit interaction within inline image and inline image, respectively. Further, the inline image symmetry is broken by the Rashba spin–orbit interaction due to the structural inversion asymmetry (SIA) [9, 24]:

  • display math(12)

Here, only the linear-in-momentum E1 SIA term is retained, as the SIA terms for heavy holes contain higher powers of k. Finally, short-range impurities and defects, as well as HgTe/HgCdTe interface roughness may also violate the inline image symmetry of the QW, giving rise to a random local block-off-diagonal perturbations.

3 Symmetry analysis of quantum conductivity corrections

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Symmetry analysis of the low-energy Hamiltonian
  5. 3 Symmetry analysis of quantum conductivity corrections
  6. 4 Microscopic calculation of the interference correction
  7. 5 Weak antilocalization in a quasi-one-dimensional strip
  8. 6 Summary and discussion
  9. Acknowledgments
  10. References

Here we overview the approach developed in Ref. ([13]) for the analysis of quantum-interference corrections to the conductivity of an infinite 2D HgTe QW. Within the diffusion approximation, conductivity corrections that are logarithmic in temperature T are associated with certain TR symmetries. The TR symmetry transformations can be represented as anti-unitary operators that act on a given operator inline image according to

  • display math

Here, U is some unitary operator (note that the momentum operator changes sign under transposition).

When the Hamiltonian of the system is given by a inline image matrix, possible TR symmetry transformations can be cast in the form involving the tensor products of Pauli matrices:

  • display math(13)

Each of these TR symmetries corresponds to a Cooperon mode contributing to the singular one-loop conductivity correction:

  • display math(14)

Here, inline image is the phase-breaking time due to inelastic scattering and inline image is the transport time. The factors inline image in Eq. (14) are zero when the TR symmetry inline image is broken by the Hamiltonian; otherwise, inline image for the orthogonal and symplectic type of the TR symmetry, respectively. The above perturbative loop expansion is justified by the large parameter inline image, where inline image is the Fermi energy counted from the bottom of the band.

An analogous symmetry analysis of the interference effects was performed for a related problem of massless Dirac fermions in graphene in Ref. ([17]). In Ref. ([13]), this approach was generalized to the case of massive Dirac fermions in a HgTe QW. By choosing the basis H1+, E1+, E1 −, and H1 −, the linear-in-k term in the BHZ Hamiltonian acquires the same structure as in Ref. ([17]):

  • display math(15)

When the chemical potential is located in the range of approximately linear spectrum, inline image, the Dirac mass inline image and the inline image-symmetry-breaking terms

  • display math(16)
  • display math(17)

[where inline image] can be treated as weak perturbations to the massless (graphene-like) Dirac Hamiltonian:

  • display math(18)

The latter possesses four TR symmetries:

  • display math(19)
  • display math(20)
  • display math(21)
  • display math(22)

These symmetries give rise to a positive weak antilocalization (WAL) conductivity correction

  • display math(23)

corresponding to two independent copies of a symplectic-class system (2Sp).

The mass term inline image violates inline image and inline image symmetries1 on the scale determined by the symmetry-breaking rate [12, 13] inline image (see Section 'Microscopic calculation of the interference correction' below for the microscopic derivation). The two out of four soft modes acquire the gap inline image, yielding

  • display math(24)

At lowest temperatures, when inline image, we find a nonsingular-in-T result:

  • display math(25)

For higher temperatures, when inline image, these two copies of a unitary-class system (2U) become two copies of the (approximately) symplectic class, with the correction given by Eq. (23).

In the presence of inversion-asymmetry terms inline image and inline image, the only remaining TR symmetry is inline image. The symmetry analysis yields the following expression for the conductivity correction in this (generic) case ([13]):

  • display math(26)

Here, inline image is the symmetry-breaking rate due to the k-independent term inline image in inline image while inline image describes the inline image-symmetry breaking governed by linear-in-k terms in inline image and inline image.

Thus, the behavior of the conductivity at the lowest T is governed by the single soft mode, which reflects the physical symplectic TR symmetry inline image. This mode yields a WAL correction characteristic for a single copy of the symplectic class system (1Sp). At higher temperatures, depending on the hierarchy of symmetry-breaking rates, the folllowing patterns of symmetry breaking can be realized ([13]):

  •   inline image: 2Sp inline image 2U inline image 1Sp.

  •   inline image: 2Sp inline image 1Sp.

  •   inline image or inline image: 1Sp.

We now turn to the case when the chemical potential is located in the bottom of the spectrum, inline image. In this limit, the spectrum is approximately parabolic:

  • display math(27)

The direction of the pseudospin within each block is almost frozen by the effective “Zeeman term” M. The linear-in-k terms of the BHZ Hamiltonian can then be treated as a weak spin–orbit-like perturbation to the massive diagonal Hamiltonian

  • display math(28)

Neglecting the block mixing, the conductivity is given by a sum of two weak localization (WL) corrections characteristic for an orthogonal symmetry class:

  • display math(29)

Here, inline image is the symmetry-breaking rate due to “relativistic” correction inline image. The microscopic derivation of inline image is performed in Section 'Microscopic calculation of the interference correction' below.

The TR symmetries of the Hamiltonian inline image can be combined into four pairs:

  • display math

Symmetry-breaking perturbations can affect the symmetries from each pair in different ways. When both TR symmetries from the pair are respected by the perturbation, the full Hamiltonian decouples into two blocks corresponding to the eigenvalues of inline image. Such a pair contributes to the conductivity as if there is a single TR symmetry. If only one of the two TR symmetries is broken within the pair, the remaining symmetry yields a conventional singular contribution. Finally, when both symmetries within the pair are broken, such pair does not contribute.

Thus, when both symmetries are not simultaneously violated, each pair contributes as a single soft mode. Note that in this case the corresponding Cooperon mass is determined by the sum of symmetry-breaking times rather than by the sum of rates. Following this rule, the inclusion of inline image, inline image, and inline image gives rise to the following interference correction ([13]):

  • display math(30)

The only true massless mode in Eq. (30) stems again from the physical symplectic TR symmetry inline image. This means that the generic block-mixing terms drive the two copies of the (approximately) orthogonal class to a single copy of a symplectic-class system. The hierarchy of the symmetry-breaking rates inline image, inline image, and inline image generates the following three patterns of crossovers ([13]):

  •   inline image: 2O inline image 2U inline image 1Sp.

  •   inline image and inline image: 2O inline image 1Sp.

  •   inline image: 1Sp.

image

Figure 1. “Phase diagrams” showing the symmetry patterns for the quantum correction to the conductivity in a 2D HgTe QW when the chemical potential is located away from the band gap, inline image. The length L is the smallest of the system size or the dephasing length inline image. The transport scattering length inline image (shown by a vertical dotted line) and the BIA-splitting length inline image (D is the diffusion constant) are assumed to be independent of energy. The “phase boundaries” (solid lines) of the 2U-regions are defined by inline image for 2Sp inline image 2U crossover, and by inline image for 2O inline image 2U crossover. Left panel: Inverted band structure (thick quantum well) with no block mixing. Dashed line shows energy for which inline image Middle panel: Normal band structure (thin quantum well) with no block mixing. Right panel: Inverted band structure with block mixing characterized by inline image and inline image (dash-dotted), with inline image. The energies inline image (from bottom to top) mark different horizontal cross-sections of the “phase-diagram” corresponding to the patterns 2O inline image 1Sp, 2O inline image 2U inline image 1Sp, 2U inline image 1Sp, 2Sp inline image 2U inline image 1Sp, and 2Sp inline image 1Sp, respectively, that appear with increasing L. The perturbative one-loop results discussed in Section 'Symmetry analysis of quantum conductivity corrections' require the condition inline image, which introduces an additional horizontal line near the band bottom in all the panels. Further, we assume that the localization length in the 2U and 2O regions is smaller than the scale L. Adapted from Ref. ([13]); Copyright (2012) by the American Physical Society.

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To summarize this section, we have analyzed the quantum conductivity correction in the diffusion approximation using the symmetry-based approach. We have identified various possible types of behavior that include 2O, 2U, 2Sp, and 1Sp regimes. The T-dependence of the conductivity correction is given by inline image, where inline image, and inline image, respectively. The “phase diagram” describing these regimes is shown in Fig. 1.

In general, crossovers between the regimes are governed by four symmetry-breaking rates: inline image inline image inline image and inline image. The first two describe a weak block mixing in the BHZ Hamiltonian. They are present for arbitrary position of the Fermi energy and are assumed to be smaller than inline image. Near the band bottom (and for very high energies, where the spectrum is no longer linear) the “intrablock” rates satisfy: inline image, while inline image. In the region of linear spectrum, the relations are opposite: inline image, while inline image.

Assuming for simplicity the absence of the BIA splitting of the spectrum, inline image the general expression for the conductivity correction can be written as:

  • display math(31)

The first term here describes two copies (decoupled blocks) of WL near the band bottom, the second term describes two copies (decoupled blocks) of WAL in the range of linear dispersion, and the last two terms reflect a block mixing due to the spin–orbit interaction/scattering (they are present at any energy).

4 Microscopic calculation of the interference correction

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Symmetry analysis of the low-energy Hamiltonian
  5. 3 Symmetry analysis of quantum conductivity corrections
  6. 4 Microscopic calculation of the interference correction
  7. 5 Weak antilocalization in a quasi-one-dimensional strip
  8. 6 Summary and discussion
  9. Acknowledgments
  10. References

In this section, we present a microscopic calculation of the interference correction to the conductivity for white-noise disorder beyond the diffusive approximation. We first consider the model with decoupled blocks and later analyze the effect of block mixing.

The WAL correction for decoupled blocks was studied in Ref. ([12]) within the diffusive approximation for the case when the chemical potential is located in the almost linear range of the spectrum. It was shown there that the finite bandgap (leading to a weak nonlinearity of dispersion) suppresses the quantum interference on large scales. Here, we calculate the interference-induced conductivity correction in the whole range of concentrations and without relying on the diffusion approximation. This allows us to describe analytically the crossover from the WL behavior near the band bottom to the WAL in the range of almost linear spectrum. We compare our results to those of Ref. ([12]) in the end of Section 'Correction to the conductivity'.

For simplicity, we will consider the case inline image. Then the two blocks of the BHZ Hamiltonian read:

  • display math(32)
  • display math(33)

A generalization onto the case of k-dependent mass inline image is straightforward. For definiteness, we will consider the block inline image.

The bare Green's function of the system is a inline image matrix in the E1–H1 space, which can be represented as a sum of the contributions of upper and lower branches:

  • display math(34)

where the projectors inline image are given by

  • display math(35)

Making use of the condition inline image, we can neglect the contribution of the lower branch when considering the interference corrections for inline image residing in the upper band,

  • display math(36)

This allows us to retain in the matrix Green's function only the contribution of the upper band:

  • display math(37)

where inline image is the disorder-induced self-energy. From now on we will omit the branch index “+”. The spinors in the upper band of block II read:

  • display math(38)
  • display math(39)

While the diffusive behavior of the quantum interference correction is universal, the precise form of the correction in the ballistic regime depends on the particular form of the disorder correlation function. In what follows, we will assume a white-noise correlated disorder with

  • display math(40)

Within this model, the crossover between the diffusive and ballistic regimes can be described analytically.

Next, we notice that in the standard diagrammatic technique, each impurity vertex inline image is sandwiched between two “projected” Green's functions. Therefore, we can dress the impurity vertices by adjacent parts of the projectors, thus replacing in all diagrams

  • display math

with

  • display math

As a result, all the information about the E1–H1 structure as well as the chiral nature of particles is now encoded in the angular dependence of the effective amplitude of scattering from a state inline image into a state inline image

  • display math(41)

where

  • display math(42)

When inline image, the system is in the orthogonal symmetry class (the scattering amplitude has no angular dependence due to Dirac factors), whereas the limit inline image corresponds to the symplectic symmetry class with the disorder scattering dressed by the “Berry phase.” The intermediate case corresponds to the unitary symmetry class, with a competition between the Rashba-type and Zeeman-type terms in the Hamiltonian killing the quantum interference.

We see that the problem is equivalent to a single-band problem with the Green's functions

  • display math(43)

and effective disorder potential dressed by “Dirac factors,” Eq. (41). The quantum (total) scattering rate inline image entering the Green's function (43) as the imaginary part of the self-energy is related to the disorder correlation function (40) as follows:

  • display math(44)

where

  • display math(45)

(here inline image stands for disorder averaging),

  • display math(46)

and

  • display math(47)

is the density of states at the Fermi level (in a single cone per spin projection).

Analyzing the problem within the Drude–Boltzmann approximation, it is easy to see that the rate inline image is the rate of scattering from the momentum inline image to the momentum inline image This function enters the collision integral of the kinetic equation and, as a consequence, describes the vertex correlation function in the diffuson ladder. (In the quasiclassical approximation, we can disregard the momentum q transferred through disorder lines in these factors.) Though we consider the short-range scattering potential, the function inline image turns out to be angle-dependent due to the “dressing” by the spinor factor inline image Hence, for the case of a massive Dirac cone, the transport scattering rate

  • display math(48)

differs from the total (quantum) rate inline image

  • display math(49)

4.1 Kinetic equation for the Cooperon

It is well known that the Cooperon propagator obeys a kinetic equation [26-28]. The collision integral of this equation contains both incoming and outgoing terms describing the scattering from a momentum inline image into a momentum inline image Importantly, the rates entering these two terms are different for the case of a single massive cone. The outgoing rate is determined by the rate inline image [which is the angle-averaged function inline image] that enters the single-particle Green function (43). To find the incoming rate, we notice that the disorder vertex lines in the Cooperon propagator are also dressed by the Dirac spinor factors. Disregarding the momentum transferred through disorder lines in these factors, we find that the vertex line corresponding to the scattering from inline image to inline image is dressed by

  • display math

The corresponding rate inline image is given by Eq. (45) with inline image replaced by inline image yielding:

  • display math(50)

Let us make two comments, which are of crucial importance for further consideration. First, we note that

  • display math(51)

which means that the collision integral in the Cooperon channel does not conserve the particle number. This implies in turn that the Cooperon propagator has a finite decay rate even in the absence of the inelastic scattering ([12]). Another important property is an asymmetry of inline image Indeed, as seen from Eq. (50)

  • display math(52)

Once the projection on the upper band and the associated dressing of the disorder correlators in the Cooperon ladders have been implemented, the evaluation of the correction to the conductivity reduces to the solution of a kinetic equation for the Cooperon propagator in an effective disorder. The latter is characterized by the correlation functions (50) in the incoming part of the collision integral and by (45) in the outgoing term. The kinetic equation for the zero-frequency Cooperon inline image has the form:

  • display math(53)

Here, inline image is the phase-breaking rate, inline image and

  • display math(54)

is the Fermi velocity at the Fermi energy

  • display math(55)

The Fermi wave vector is given by

  • display math(56)

Diagrammatically, Eq. (53) corresponds to a Cooperon impurity ladder with four Green's functions at the ends.

Introducing dimensionless variables

  • display math(57)

where

  • display math(58)

is the mean free path, we rewrite Eq. (53) as follows:

  • display math(59)

where inline image As seen from Eq. (59), the incoming term of the collision integral contains only three angular harmonics: inline image, and inline image This allows us to present the solution of Eq. (59) in the following form:

  • display math(60)

where

  • display math(61)
  • display math(62)
  • display math(63)

and inline image is the polar angle of vector inline image Substituting Eq. (60) into Eqs. (61), (62), and (63), we find a system of coupled equations for inline image and inline image which can be written in the matrix form

  • display math(64)

Here

  • display math(65)
  • display math(66)

and

  • display math(67)

From Eqs. (60), (64), and (65) we find

  • display math(68)

where inline image The Fourier transform of the Cooperon propagator gives the quasiprobability ([27]) (per unit area) for an electron starting with a momentum direction inline image from an initial point inline image to arrive at a point inline image with a momentum direction inline image:

  • display math(69)

In particular, the conductivity can be expressed in terms of this probability taken at inline image (return probability):

  • display math(70)

The first term in the right-hand side of Eq. (68) describes the ballistic motion (no collisions). The second term can be expanded (by expanding the matrix inline image) in series over functions inline image Such an expansion is, in fact, an expansion of the Cooperon propagator over the number N of collisions (the zeroth term in this expansion corresponds to inline image) ([29]). Since the term with inline image does not contribute to the interference-induced magnetoresistance, we can exclude it from the summation in the interference correction and regard this contribution as a part of the Drude conductivity.2 Indeed, after a substitution into inline image we see that this term describes a return to the initial point after a single scattering, so that the corresponding trajectory does not cover any area and, consequently, is not affected by the magnetic field. Neglecting both the ballistic (inline image) and the inline image terms in the Cooperon propagator, we find

  • display math(71)

Here, we took into account that inline image for inline image Let us now find the return probability. To this end, we make expansions

  • display math(72)
  • display math(73)

in Eq. (71), substitute the obtained equation into Eq. (69), take inline image and average over inline image. We arrive then to the following equation

  • display math(74)

where

  • display math(75)

On a technical level, the logarithmic divergency specific for WL and WAL conductivity corrections comes from a singular behavior of the matrix inline image at inline image Before analyzing the solution in the full generality, let us consider the limiting case inline image, inline image. In this case inline image, inline image, and we find:

  • display math(76)

Then in the limit inline image (orthogonal class) the singular mode is inline image [see Eq. (60)] and inline image, while in the limit inline image (symplectic class) the singular mode is inline image and inline image.

4.2 Correction to the conductivity

The quantum correction to the conductivity is given by ([30])

  • display math(77)

Here in the “Hikami-box” factor inline image the unity comes from the conventional Cooperon diagram describing the backscattering contribution, while inline image arises from a Cooperon covered by an impurity line (nonbackscattering term ([29])). Using Eqs. (44), (49), and (50), we obtain

  • display math(78)

and, finally, with the use of Eq. (74), arrive at

  • display math(79)

where inline image are given by Eq. (75).

image

Figure 2. Left panel: Conductivity correction in the absence of block mixing, Eq. (79), as a function of inline image for infinite dephasing time (solid line); sum of the logarithmic WL and WAL asymptotics, Eqs. (81) and (83) (dashed line). Right panel: Conductivity correction for different values of dimensionless dephasing rate inline image.

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As discussed above, for inline image and inline image one of the modes becomes singular (corresponding to inline image and inline image respectively). Keeping the singular modes only, one can easily obtain the return probability and the conductivity in vicinities of the points inline image and inline image For inline image we find

  • display math(80)

and

  • display math(81)

For inline image we find

  • display math(82)

and

  • display math(83)

According to Eqs. (81) and (83), the symmetry breaking rates inline image and inline image introduced in Section 'Symmetry analysis of quantum conductivity corrections' are equal to

  • display math(84)

and

  • display math(85)

respectively.

Using Eqs. (44) and (49), the result (85) for the symmetry-breaking rate inline image in the range of almost linear spectrum (inline image) agrees with the result of Ref. ([12]), where this scale was first identified. In the opposite limit inline image (which was not analyzed in Ref. ([12])), the result (84) of the microscopic calculation confirms the estimate of Ref. ([13]).

image

Figure 3. Left panel: Conductivity correction in the absence of block mixing as a function of the dephasing rate for different values of inline image. Right panel: Conductivity correction as a function of dephasing rate for inline image.

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The leading logarithmic contributions (81) and (83) are exactly the WL and WAL corrections found in Section 'Symmetry analysis of quantum conductivity corrections' from the symmetry analysis for the regimes 2O and 2Sp, respectively. We are now in a position to evaluate corrections to these results. Setting for simplicity inline image (no dephasing), we get from Eqs. (79) and (75)

  • display math(86)

The regime inline image was considered in Ref. ([12]). However, our results for subleading terms in this regime differ from those obtained there. This difference is due to the fact that Ref. ([12]) only considered contributions of small momenta inline image. While this is sufficient to get the universal WL and WAL terms (81) and (83), evaluation of the corrections requires taking into account the “ballistic” momenta inline image.

We note that the last term, inline image in the asymptotics for inline image in Eq. (86) is, in fact, determined by momenta inline image and was identified within the diffusive approximation of Ref.  ([12]). However, the numerical coefficient in front of this term in Ref. ([12]), which in our notation would take the form inline image, differs from the prefactor inline image in the corresponding term in our Eq. (86). The difference stems from setting inline image everywhere (except for the term inline image in the denominators of the Cooperon propagators) in the calculation of Ref. ([12]). It is worth emphasizing that the ballistic terms inline image and inline image neglected in Ref. ([12]) give a much larger contribution in the limit inline image than this “diffusive-like” term.

The results for arbitrary values of inline image are shown in Figs. 2 and 3. In the left panel of Fig. 2, we plotted the conductivity correction calculated with the use of Eqs. (75) and (79) (solid line) in the absence of dephasing (inline image). The sign of the correction changes with increasing inline image so that the system undergoes a crossover from the orthogonal to symplectic ensemble as expected. In the same panel, we plotted by dashed line the sum of two asymptotes, given by Eqs. (81) and (83), respectively. A deviation of the exact result from the interpolating formula is mostly due to the nonsingular mode related to the eigenvalue inline image in matrix inline image given by Eq. (76). As a result, the difference between the solid and dashed lines vanishes in the limit inline image, while yielding a shift inline image1 near inline image. The right panel of Fig. 2 illustrates the crossover at different dephasing rates. In the left panel of Fig. 3, we plotted the conductivity correction as a function of dephasing rate for different inline image The most interesting feature of these curves is the nonmonotonous dependence of conductivity on inline image for inline image close to the point separating localization and delocalization behavior. This feature is emphasized in the right panel of Fig. 3, where dependence of conductivity on dephasing rate is plotted for inline image We expect that the conductivity in the 2U symmetry regime (energy inline image in Fig. 1) shows an analogous nonmonotonous dependence on the magnetic field due to ballistic effects.

4.3 Block mixing

Let us now briefly analyze the effect of a weak block mixing. As discussed above, on largest scales the mixing leads to a single copy of the symplectic class (1Sp). Below we assume the simplest form of the mixing potential:

  • display math(87)

where inline image is a short-range potential with the correlation function given by Eq. (40) and inline image is a parameter responsible for the block mixing. We also assume that inline image is small and real: inline image The potential Eq. (87) obeys the symmetry: inline image where

  • display math(88)

The result of the microscopic calculation reads ([30]):

  • display math(89)

We stress that this equation was obtained in the diffusion approximation under the assumption that dimensionless gaps of diffusive modes are small. Hence, though expressions for the last two logarithms in this equation are exact, the explicit form of the first two logarithms is exact only in a vicinity of the point inline image while the third logarithm is exact near the point inline image In this sense, Eq. (89) interpolates between the two limits (inline image and inline image) similarly to Eq. (31) (cf. Fig. 2). Away from the points inline image and inline image, the block mixing in the first three terms of Eq. (89) can be neglected, so that one can use Eq. (79) with Eq. (75) to describe these terms in the crossover range of inline image beyond the diffusive approximation. It is also worth noting that the gap inline image in the fourth logarithm is nonzero only when both inline image and inline image are nonzero (see discussion above Eq. (30)).

Introducing the symmetry breaking times inline image, inline image, and inline image according to

  • display math(90)
  • display math(91)
  • display math(92)

we can rewrite Eq. (89) in a form combining (for inline image) Eqs. (26) and (30):

  • display math(93)

This expression correctly reproduces all singular interference corrections in the whole range inline image provided inline image. Note also that for inline image, Eq. (93) reduces to the conventional expression for the symmetry pattern 2O inline image 1Sp with anisotropic spin relaxation (different for in-plane and out-of-plane components) described by inline image and inline image. In a similar way, one can also include ([30]) the block mixing described by inline image.

5 Weak antilocalization in a quasi-one-dimensional strip

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Symmetry analysis of the low-energy Hamiltonian
  5. 3 Symmetry analysis of quantum conductivity corrections
  6. 4 Microscopic calculation of the interference correction
  7. 5 Weak antilocalization in a quasi-one-dimensional strip
  8. 6 Summary and discussion
  9. Acknowledgments
  10. References

Motivated by recent experiment of Ref. ([23]), we will discuss in this section the magnetoresistance of a quasi-one-dimensional HgTe structure. The magnetotransport measurements in Ref. ([23]) were performed on a 2D QWs patterned in narrow 1D strips. While the width of the HgTe quantum well is of the order of few nanometers, its lateral width W can be less than hundred nanometers. In such a restricted quasi-1D geometry, boundary conditions play a crucial role for electron transport. Below we analyze possible symmetries of the boundary conditions and ballistic transport effects arising in quasi-1D samples.

5.1 Symmetry breaking mechanism due to boundary scattering

We start by assuming that the boundary does preserve the inline image-symmetry and the Hamiltonian splits into two independent blocks inline image and inline image related by the physical TR symmetry. It was demonstrated in Ref. ([13]) that in this case the block-wise TR-symmetry of inline image is necessarily broken by boundary conditions.

It is well known that a massless Dirac electron cannot be confined by any potential profile due to the Klein tunneling effect. The only way to introduce boundary conditions for a single flavor of Dirac fermions (without block mixing) is to open a gap at the boundary by introducing a large mass in the Hamiltonian. This boundary mass term violates the block-wise symplectic TR symmetry. On a technical level, each boundary scattering event in a Dirac system introduces an additional phase shift in the electron wave function. These scattering phases combine in such a way that the amplitudes of two mutually time-reversed trajectories pick up a relative phase difference inline image for each boundary scattering. Thus the contribution of a trajectory with N boundary scattering events to the conductivity correction contains a factor inline image. After the summation over all paths with arbitrary N, this alternating sign factor kills the singular correction to the conductivity.

More general boundary conditions are possible if the massive BHZ Hamiltonian includes higher-order (quadratic) terms in the k expansion. In particular, the hard-wall boundary conditions, inline image, are widely used in the literature in this case. The TR symmetry breaking mechanism discussed above is however still effective in this limit provided inline image. Quadratic terms become important only very close to the boundary, while at a longer distance the dynamics is well described by the linearized Hamiltonian. The hard-wall model effectively reduces to the same infinite mass boundary condition for the problem at these longer scales. An explicit computation of the corresponding scattering states can be found in Ref. ([13]).

We arrive at the conclusion that the block-wise symplectic TR-symmetry is inevitably broken by the boundary conditions. In a diffusive quasi-1D strip, the edge-induced symmetry-breaking rate reads

  • display math(94)

where W is the strip width. The time inline image is given by the average time for a particle to propagate diffusively between the edges and “to become aware” of the violation of the block-wise TR symmetry by the boundaries. When the mean free path is longer than the width of the quasi-1D sample, the corresponding symmetry breaking time is given by the typical flight time for ballistic propagation between the two boundaries. The total rate of breaking the block-wise symplectic TR symmetry at inline image close to unity is thus given by inline image.

In the opposite limit inline image, when the block-wise TR symmetry is of the orthogonal type, the vanishing of the wave function at the boundary does not lead to the TR symmetry breaking. Indeed, conventional “Schrödinger” particles can be confined by a scalar potential.

Above we have discussed boundary conditions that preserve the inline image-symmetry. The spin symmetry, however, can be broken at boundaries of a 2D sample if the edges violate the reflection symmetry in the z-direction. Such edges can be modeled by short-range impurities located near the boundaries. This situation is quite likely for realistic samples. In this case, the only remaining symmetry respected by the edges is the physical symplectic TR symmetry.

5.2 Interference corrections and magnetoresistance in a quasi-1D system

Let us now analyze the interference effects in a quasi-1D geometry, i.e., in a strip of width W. We assume a single copy of symplectic class since the block-wise TR symmetry is broken by the boundaries, so that only the patterns 1Sp or 2U inline image 1Sp are realized due to the block mixing with no room for 2Sp regime. The magnetoresistance depends on the relations between the width W, 2D mean free path l, dephasing length inline image, and magnetic length inline image. We start our analysis with the case of sufficiently weak dephasing inline image. In the opposite limit, when the dephasing length is shorter than the width of the strip, the WAL correction is given by 2D formulas. For inline image the diffusive results apply, whereas for inline image the WAL correction is described by nonuniversal ballistic 2D formulas; the magnitude of the correction is small in this case.

For sufficiently weak magnetic fields, the characteristic length scale at which the interference is suppressed is determined by the condition

  • display math(95)

For such lengths, the area inline image is pierced by one magnetic flux. We assume inline image, thus the motion along the strip is diffusive within inline image. This implies sufficiently weak magnetic field, inline image.

When the width of the quasi-1D strip is much larger than the 2D mean free path, inline image, the correction to the conductance of the strip of length L due to WAL reads [26, 28, 32, 33]:

  • display math(96)

Note that this expression assumes a phenomenological (B-independent) dephasing length; a more accurate formula including Airy function can be found in Refs. [26, 28, 34]. The difference between the two expressions is of the order of few percent, so that we use the formula (96). The measured correction to the longitudinal conductivity of the 2D array of strips is given by

  • display math(97)

neglecting the separation between the strips.

For narrow strips, inline image, the transverse motion is ballistic and the resulting correction is modified,

  • display math(98)

The function inline image incorporates information about boundary conditions. For diffuse boundaries in the quasi-2D geometry various limiting cases of this function were obtained in Ref. ([31]). In a later paper Ref. ([33]), this function was computed numerically for both quasi-1D and quasi-2D samples with both mirror and diffuse boundaries. In the limiting case inline image, analytical expression was also obtained for diffuse boundary conditions.

We represent the quasi-1D results in a closed integral form applicable for all values of inline image:

  • display math(99)

These functions are shown in Fig. 4. For inline image, the functions inline image approach the universal value corresponding to the diffusive result Eq. (96). Ballistic effects partly suppress WAL due to cancellation of magnetic flux piercing purely ballistic trajectories.

image

Figure 4. Functions inline image and inline image defined in Eq. (99).

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Remarkably, the magnitude of the magnetoconductivity peak in quasi-1D geometry is determined only by the ratio of the dephasing length inline image and the strip width W,

  • display math(100)

Details of transverse ballistic motion, encoded in the function inline image, are relevant only for the width of the peak.

Let us now consider the case of a relatively weak block mixing described by inline image. Then the conductivity correction takes the form

  • display math(101)

For a strong block mixing, this equation reproduces the 1Sp result (98), while in the opposite limit of weak mixing the conductivity correction gets strongly suppressed as illustrated in Fig. 5.

image

Figure 5. Conductivity correction in a quasi-1D strip of HgTe, Eq. (101), for inline image, inline image and different values of the block-mixing length: inline image from top to bottom.

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6 Summary and discussion

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Symmetry analysis of the low-energy Hamiltonian
  5. 3 Symmetry analysis of quantum conductivity corrections
  6. 4 Microscopic calculation of the interference correction
  7. 5 Weak antilocalization in a quasi-one-dimensional strip
  8. 6 Summary and discussion
  9. Acknowledgments
  10. References

To summarize, we have reviewed manifestations of quantum interference in conductivity of 2D HgTe structures. A symmetry analysis yields a rich “phase diagram” describing regimes with different types of one-loop quantum interference correction (WL, WAL, no correction). We have supplemented the symmetry analysis by a microscopic calculation of the quantum interference contribution to the conductivity. This approach allows us also to calculate the behavior of the conductivity in the crossover regimes and beyond the diffusive approximation. We have also discussed symmetry breaking mechanisms and the quantum interference correction in a quasi-1D (strip) geometry.

image

Figure 6. Schematic illustration of the magnetoresistivity inline image on a linear-logarithmic scale for energies inline image from the right panel of Fig. 1. The numbers denote the prefactors inline image (here inline image) of the logarithmic magnetoresistivity. Adapted from Ref. ([13]); Copyright (2012) by the American Physical Society.

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The common way to explore the quantum interference (WL or WAL) experimentally is to measure the magnetoresistivity inline image in a transverse magnetic field B. Each of the symmetry-breaking patterns discussed in Section 'Symmetry analysis of the low-energy Hamiltonian' then translates into a succession of regions of magnetic field with corresponding signs and prefactors of the low-field magnetoresistivity, see Fig. 6.

Let us now discuss available experimental data in context of the above theoretical findings. In Refs. [21, 22], the low-field magnetoresistivity was investigated in 2D HgTe samples both in the case of normal (thin QWs) and inverted (thick QWs) band ordering. For both types of samples, a weak positive magnetoresistivity was observed in the lowest magnetic fields B. This behavior can clearly be attributed to WAL. The coefficient in front of inline image (where inline image) was found to be consistent with 1/2, as expected for the 1Sp regime. In higher magnetic fields, a crossover to negative magnetoresistance was observed that could be presumably attributed to WL. Such a behavior corresponds, in our terminology, to the inline image symmetry pattern (characteristic for systems with relatively small carrier concentration, see regimes inline image and inline image in Fig. 6).3 For inverted structures with high carrier concentrations, Ref. ([21]) observed only positive magnetoresistance consistent with regimes inline image and inline image of Fig. 6.

In a recent work ([23]), the magnetoresistance of a HgTe structure patterned in an array of quasi-1D strips was experimentally studied. The WAL behavior was observed both for normal-gap and inverted-gap structures in the whole range of magnetic fields and gate voltages. This is consistent with the 1Sp behavior as expected from the above theory in the quasi-1D regime. Indeed, boundaries of the strips break the block-wise TR symmetry, as has been shown in Section 'Symmetry breaking mechanism due to boundary scattering'. Furthermore, diffuse boundary scattering due to short-range edge irregularities is favorable for the violation of the inline image symmetry at the boundary, which breaks down the spin symmetry, yielding the 1Sp regime. In addition, the block mixing arises also due to the BIA, see Ref. ([2]), and due to short-range impurities located within the quantum well. The difference in the magnitude of the effect for normal-gap and inverted-gap setups observed in Ref. ([23]) can be possibly related to a stronger block mixing in the inverted case (cf. Fig. 5) and/or stronger dephasing in the normal case. A stronger block-mixing in the inverted case might possibly be attributed to the higher probability of having irregularities (background short-range impurities or edge roughness) breaking the inline image symmetry of the sample within a thicker QW (inverted band ordering) as compared to a thin QW (normal band ordering).

Finally, it is worth emphasizing that, while we have focussed on HgTe/HgCdTe QWs, the above analysis is expected to be applicable to a broader class of structures with Dirac-type spectrum, including, in particular, InAs/GaSb structures [9, 10].

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Symmetry analysis of the low-energy Hamiltonian
  5. 3 Symmetry analysis of quantum conductivity corrections
  6. 4 Microscopic calculation of the interference correction
  7. 5 Weak antilocalization in a quasi-one-dimensional strip
  8. 6 Summary and discussion
  9. Acknowledgments
  10. References

We are grateful to C. Brüne, A. Germanenko, E. Hankiewicz, E. Khalaf, G. Minkov, L. Molenkamp, M. Titov, and G. Tkachov for useful discussions. The work was supported by DFG within SPP 1285 ”Semiconductor spintronics” and SPP 1666 ”Topological insulators”, by grant FP7-PEOPLE-2013-IRSES of the EU network InterNoM, by GIF, and by BMBF.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Symmetry analysis of the low-energy Hamiltonian
  5. 3 Symmetry analysis of quantum conductivity corrections
  6. 4 Microscopic calculation of the interference correction
  7. 5 Weak antilocalization in a quasi-one-dimensional strip
  8. 6 Summary and discussion
  9. Acknowledgments
  10. References
  1. 1

    A similar role is played by disorder that acts differently on E and H subbands. In the rotated basis used in Section 'Symmetry analysis of quantum conductivity corrections', the corresponding term in the Hamiltonian is proportional to inline image, similarly to inline image, and can be regarded as a random mass term. In the regime of an almost linear spectrum, inline image, such disorder would lead to an additional contribution to inline image. On the other hand, when the chemical potential is located near the band bottom, inline image, such type of disorder does not yield additional symmetry breaking and only contributes to the elastic scattering time inline image.

  2. 2

    In fact, this question is more subtle and the term inline image requires special attention. It results in an ultraviolet logarithmic divergency (coming from large momenta inline image) of the return probability inline image This divergency leads to ultraviolet renormalization ([16]) of the key problem parameters (such as inline image) and will not be discussed here. By default, we assume that all parameters of the problem are already renormalized and that inline image

  3. 3

    A weak negative magnetoresistance in relatively strong fields may also arise in the inline image regime of Fig. 6 due to ballistic effects, cf. Fig. 3 (right panel).