### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Symmetry analysis of the low-energy Hamiltonian
- 3 Symmetry analysis of quantum conductivity corrections
- 4 Microscopic calculation of the interference correction
- 5 Weak antilocalization in a quasi-one-dimensional strip
- 6 Summary and discussion
- Acknowledgments
- 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 – TIs –and thus opened research direction. Another realization of a 2D 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

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

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

- (5)

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.

### 3 Symmetry analysis of quantum conductivity corrections

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

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

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]):

- (15)

Thus, the behavior of the conductivity at the lowest *T* is governed by the single soft mode, which reflects the physical symplectic TR symmetry . 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]):

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

- (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

- (28)

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

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 . 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.

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

- (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

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

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

- (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 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

with

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 into a state

- (41)

where

- (42)

When , the system is in the orthogonal symmetry class (the scattering amplitude has no angular dependence due to Dirac factors), whereas the limit 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

- (43)

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

- (44)

where

- (45)

(here stands for disorder averaging),

- (46)

and

- (47)

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

#### 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 into a momentum 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 [which is the angle-averaged function ] 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 to is dressed by

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

- (50)

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

- (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 Indeed, as seen from Eq. (50)

- (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 has the form:

- (53)

Here, is the phase-breaking rate, and

- (54)

is the Fermi velocity at the Fermi energy

- (55)

The Fermi wave vector is given by

- (56)

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

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

- (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 ) in series over functions 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 ) ([29]). Since the term with 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 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 () and the terms in the Cooperon propagator, we find

- (71)

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

- (72)

- (73)

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

- (74)

where

- (75)

#### 4.2 Correction to the conductivity

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

- (77)

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

- (78)

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

- (79)

where are given by Eq. (75).

### 6 Summary and discussion

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

The common way to explore the quantum interference (WL or WAL) experimentally is to measure the magnetoresistivity 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.

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 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 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].