Quantum interference in HgTe structures

# Quantum interference in HgTe structures

I. V. Gornyi Institut für Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia    V. Yu. Kachorovskii A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia    A. D. Mirlin Institut für Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Institut für Theorie der kondensierten Materie and DFG Center for Functional Nanostructures, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia.    P. M. Ostrovsky Max-Planck-Institut für Festkörperforschung, Heisenbergstr. 1, 70569, Stuttgart, Germany L. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
###### Abstract

We study quantum transport in HgTe/HgCdTe quantum wells under the condition that the chemical potential is located outside of the bandgap. 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.

## I Introduction

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 Hasan10RMP (); Qi11 (); kane (); BernevigHughesZhang (); Koenig07 (); Fu07 (); hasan (). In the 2D case, the TI behavior was experimentally discovered by the Würzburg group Koenig07 () 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. Koenig07, ; Roth09, . These experiments have shown that HgTe/HgCdTe structures realize a novel remarkable class of materials— topological insulators—and thus opened a new exciting research direction. Another realization of a 2D TI based on InAs/GaSb structures proposed in Ref. Liu08, was experimentally discovered in Ref. Du, .

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 ostrovsky10 (); Tkachov11 (); OGM12 (); Richter12 (), similarly to interference phenomena in grapheneMcCann (); Nestoklon (); Ostrovsky06 (); aleiner-efetov () and in surface layers of 3D TI ostrovsky10 (); Glazman (); Koenig13 (). Recently, the magnetoresistivity of HgTe/HgCdTe structures was experimentally studied away from the insulating regime in Refs. kvon, ; minkov, ; bruene, , 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 Sec. II symmetry properties of the underlying Dirac-type Hamiltonian and physically important symmetry-breaking mechanisms. In Sec. III we overview a general symmetry-based approach OGM12 () to the problem and employ it to evaluate the conductivity corrections within the diffusion approximation. Section IV 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 Sec. V. Section VI summarizes our results and discusses a connection to experimental works.

## Ii Symmetry analysis of the low-energy Hamiltonian

The low-energy Hamiltonian for a symmetric HgTe/HgCdTe structure was introduced in Ref. BernevigHughesZhang, in the framework of the method. The Bernevig-Hughes-Zhang (BHZ) Hamiltonian possesses a matrix structure in the Kramers-partner space and E1 – H1 space of electron- and hole-type levels Qi11 (); Liu08 (); Rothe10 (),

 HBHZ=(h(k)00h∗(−k)), (1) h(k)=(ϵ(k)+m(k)A(kx+iky)A(kx−iky)ϵ(k)−m(k)). (2)

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

 HBHZ=ϵ(k)σ0s0+m(k)σzs0+Akxσxsz−Akyσys0. (3)

The effective mass and energy are given by

 m(k)=M+Bk2,ϵ(k)=C+Dk2, (4)

The normal insulator phase corresponds to (which is realized in thin QWs, nm), whereas the TI phase is characterized by (realized in thick QW). Koenig07 ()

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

 E±k=ϵ(k)±√A2k2+m2(k). (5)

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

 ψ(±)k(r)=χ(±)keikr (6)

where spinors are different in different blocks

 χ(I,±)k =(1+μ2±)−1/2 (1μ±e−iϕk), (7) χ(II,±)k =(1+μ2±)−1/2 (1−μ±eiϕk.) (8)

Here is the polar angle of the momentum and

 μ±=±[A2k2+m2(k)]1/2−m(k)Ak (9)

corresponds to the upper and lower branches of the spectrum.

Disorder potential is conventionally introduced in the BHZ model by adding the scalar term Tkachov11 ()

 Hdis=V(r)σ0s0 (10)

to the Hamiltonian . 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 Rothe10 (); OGM12 (). The Hamiltonian is characterized by the exact global time-reversal (TR) symmetry . Further, this Hamiltonian commutes with , which we term the “spin symmetry”. Finally, an additional approximate symmetry operative within each Kramers block emerges for some regions of energy. Specifically, in the inverted regime acquires the exact symplectic “block-wise” TR symmetry when . Around this point, the symmetry is approximate. An approximate orthogonal block-wise TR symmetry emerges near the band bottom for and for high energies .

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 spin symmetry is violated by perturbations that do not preserve the reflection () symmetry of the QW. Such perturbations yield nonzero block-off-diagonal elements in the full 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 Liu08 ()

 HBIA=⎛⎜ ⎜ ⎜⎝00δek+−Δ000Δ0δhk−δek−Δ000−Δ0δhk+00⎞⎟ ⎟ ⎟⎠. (11)

The BIA perturbation (11) contains the momentum-independent term with that connects the electronic and heavy-hole bands Winkler () with opposite spin projections. The terms with and stem from the cubic Dresselhaus spin-orbit interaction within and , respectively. Further, the symmetry is broken by the Rashba spin-orbit interaction due to the structural inversion asymmetry (SIA):Liu08 (); Rothe10 ()

 HR=⎛⎜ ⎜ ⎜⎝00irek−00000−irek+0000000⎞⎟ ⎟ ⎟⎠, (12)

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

## Iii Symmetry analysis of quantum conductivity corrections

Here we overview the approach developed in Ref. OGM12, 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 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

 T:O↦U−1OTU.

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

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

 Tij:O↦σisjOTsjσi,i,j=0,x,y,z. (13)

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

 δσij=−cije22πhln(τϕτ),cij=−1, 0, 1. (14)

Here is the phase-breaking time due to inelastic scattering and is the transport time. The factors in Eq. (14) are zero when the TR symmetry is broken by the Hamiltonian; otherwise, for the orthogonal and symplectic type of the TR symmetry, respectively. The above perturbative loop expansion is justified by the large parameter , where 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. Ostrovsky06, . In Ref. OGM12, this approach was generalized to the case of massive Dirac fermions in a HgTe QW. By choosing the basis H1+,E1+,E1-,H1-, the linear-in- term in the BHZ Hamiltonian acquires the same structure as in Ref. Ostrovsky06, :

 (15)

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

 HBIA =Δ0σzsx+δ+(kxσx+kyσy)sx +δ−(kxσy−kyσx)sy, (16) HR =(re/2)[(kxσy+kyσx)sx−(kxσx−kyσy)sy], (17)

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

 HA=A(kxσx+kyσy)sz. (18)

The latter possesses four TR symmetries:

 Txx: O↦σxsxOTσxsx,T2xx=1, (19) Ty0: O↦σys0OTσys0,T2y0=−1, (20) Tyz: O↦σyszOTσysz,T2yz=−1, (21) Txy: O↦σxsyOTσxsy,T2xy=−1. (22)

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

 (23)

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

The mass term violates and symmetries footnote_mass () on the scale determined by the symmetry breaking rate Tkachov11 (); OGM12 () (see Sec. IV below for the microscopic derivation). The two out of four soft modes acquire the gap , yielding

 δσ=−2×e22πhln(ττϕ+ττm). (24)

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

 δσ≃2×e22πhln(τmτ). (25)

For higher temperatures, when , 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 and , the only remaining TR symmetry is . The symmetry analysis yields the following expression for the conductivity correction in this (generic) case OGM12 ():

 δσ=e22πh[ln(ττϕ+ττΔ+ττSO)−ln(ττϕ+ττm+ττΔ+ττSO)−ln(ττϕ+ττm+ττSO)−lnττϕ]. (26)

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

Thus the behavior of the conductivity at the lowest 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: OGM12 ()

• : 2Sp 2U 1Sp;

• : 2Sp 1Sp.

• or : 1Sp.

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:

 E+(k) ≃ |M|+Bk2+ϵ(k)+A2k2/|M|. (27)

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

 HM=−m(k)σzsz. (28)

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

 δσ=2×e22πhln(ττA+ττϕ). (29)

Here is the symmetry-breaking rate due to “relativistic” correction . The microscopic derivation of is performed in Sec. IV below.

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

 T00∼Tzz,T0z∼Tz0,Txx∼Tyy,Txy∼Tyx.

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.

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 , , and gives rise to the following interference correction: OGM12 ()

 δσ=e22πh[2ln(ττϕ+ττA+ττSO)+ln(ττϕ+ττSO+ττA+τΔ)−lnττϕ]. (30)

The only true massless mode in Eq. (30) stems again from the physical symplectic TR symmetry . 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 , , and , generates the following three patterns of crossovers:OGM12 ()

• : 2O 2U 1Sp.

• and :
aa 2O 1Sp.

• : 1Sp.

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 -dependence of the conductivity correction is given by , where , and , 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: and . 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 . Near the band bottom (and for very high energies, where the spectrum is no longer linear) the “intra-block” rates satisfy: , while . In the region of linear spectrum the relations are opposite: , while .

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

 δσ = e24π2ℏ×[2ln(ττϕ+ττA+ττSO) (31) − 2ln(ττϕ+ττm+ττSO) + ln(ττϕ+ττSO)−ln(ττϕ)].

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).

## Iv Microscopic calculation of the interference correction

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. Tkachov11, 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. Tkachov11, in the end of Sec. IV.2.

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

 HI = [MA(kx+iky)A(kx−iky)−M], (32) HII = [M−A(kx−iky)−A(kx+iky)−M]. (33)

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

The bare Green’s function of the system is a matrix in E1-H1 space which can be represented as a sum of the contributions of upper and lower branches:

 ^G(E,p)=^P+(k)E−E+k+^P−(k)E−E−k, (34)

where the projectors are given by

 ^P±(k)=|χ(±)k⟩⟨χ(±)k|. (35)

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

 E+k=√M2+A2k2. (36)

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

 ^G(E,k) ≃ ^P+(k)E−E+k−Σ+=^P+(k)G+(E,k), (37)

where 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:

 χk =1√1+μ2 (1−μeiϕk), (38) μ =AkM+√M2+A2k2. (39)

While the diffusive behavior of the quantum interference correction is universal, the precise from 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

 ⟨V(r)V(r′)⟩=W0δ(r−r′). (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

 …^G(E,k)V(k−k′)^G(E,k′)…

with

 …|χk⟩G(E,k)⟨χk|V(k−k′)|χ′k⟩G(E,k′)⟨χk′|…

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

 ~V(k,k′) = ⟨χk|V(k−k′)|χ′k⟩ (41) = V(k−k′)1+ηexp(ϕk′−ϕk)1+η,

where

 η=μ2. (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

 GR,A(E,k)=1E−Ek±iγ/2 (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:

 γ=∫2π0dϕ2π γD(ϕ)=γ01+η2(1+η)2, (44)

where

 γD(ϕk−ϕk′) = 2πℏ∫kdk′2π⟨|~V(k,k′)|2⟩δ(Ek−Ek′) (45) = γ01+2ηcos(ϕk−ϕk′)+η2(1+η)2

(here stands for disorder averaging),

 γ0=2πνFℏW0 (46)

and

 νF=M2πℏ2A21+η1−η (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 is the rate of scattering from the momentum to the momentum 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 transferred through disorder lines in these factors.) Though we consider the short-range scattering potential, the function turns out to be angle-dependent due to the “dressing” by the spinor factor Hence, for the case of a massive Dirac cone, the transport scattering rate

 γtr=∫2π0dϕ2π γD(ϕ)(1−cosϕ)=γ01+η2−η(1+η)2 (48)

differs from the total (quantum) rate

 γγtr=1+η21+η2−η. (49)

### iv.1 Kinetic equation for the Cooperon

It is well known that the Cooperon propagator obeys a kinetic equation. AA (); schmid (); AAG () 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 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:

 γC(ϕk−ϕk′) = γ01+2η e−i(ϕk−ϕk′)+η2e−2i(ϕk−ϕk′)(1+η)2.

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

 ∫dϕγC(ϕ)≠∫dϕγD(ϕ), (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. Tkachov11 () Another important property is an asymmetry of Indeed, as seen from Eq. (LABEL:Wc),

 γC(ϕ)=γ∗C(−ϕ)≠γC(−ϕ). (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 (LABEL:Wc) 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:

 [1/τϕ+iqvF)]Cq(ϕ,ϕ0)=γδ(ϕ−ϕ0) +∫dϕ′2π[γC(ϕ−ϕ′)Cq(ϕ′,ϕ0)−γD(ϕ−ϕ′)Cq(ϕ,ϕ0)]. (53)

Here, is the phase-breaking rate, and

 vF=2√η1+ηA (54)

is the Fermi velocity at the Fermi energy

 EF=M1+η1−η. (55)

The Fermi wave vector is given by

 kF=2MℏA√η1−η. (56)

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

Introducing dimensionless variables

 Γ=1/γτϕ,Q=ql, (57)

where

 l=vF/γ=2ℏ3A3MW0√η(1−η)1+η2 (58)

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

 (1+Γ+iQn)CQ(ϕ,ϕ0)=δ(ϕ−ϕ0) +∫dϕ′2π[1+ηe−i(ϕ−ϕ′)]21+η2CQ(ϕ′,ϕ0), (59)

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

 CQ(ϕ,ϕ0)=11+Γ+iQn ×[C0+ei(ϕQ−ϕ)C−1+e2i(ϕQ−ϕ)C−2+δ(ϕ−ϕ0)], (60)

where

 C0 = 11+η2∫dϕ2πCQ(ϕ,ϕ0), (61) C−1 = 2η1+η2∫dϕ2πCQ(ϕ,ϕ0)ei(ϕ−ϕQ), (62) C−2 = η21+η2∫dϕ2πCQ(ϕ,ϕ0)e2i(ϕ−ϕQ), (63)

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

 ^M⎡⎢⎣C0C−1C−2⎤⎥⎦=12π(1+Γ+iQn0)⎡⎢⎣1ei(ϕ0−ϕQ)e2i(ϕ0−ϕQ)⎤⎥⎦. (64)

Here

 ^M=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1+η2−P0−P1−P2−P11+η22η−P0−P1−P2−P11+η2η2−P0⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦, (65)
 Pn =∫dϕ2πe−inϕ1+Γ+iQcosϕ =(−i)|n|P0[1−P0(1+Γ)1+P0(1+Γ)]|n|/2, (66)

and

 P0=1√(1+Γ)2+Q2. (67)

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

 CQ(ϕ,ϕ0)=δ(ϕ−ϕ0)1+Γ+iQn +12π(1+Γ+iQn)(1+Γ+iQn0) (68)

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

 C(ϕ,ϕ0,r−r0)=1l2∫d2Q(2π)2eiQ(r−r0)/lCQ(ϕ,ϕ0). (69)

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

 W(ϕ−ϕ0)=C(ϕ,ϕ0,0). (70)

The first term in the r.h.s. 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 of collisions (the zeroth term in this expansion corresponds to ).nonback () 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.comment () 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

 CQ(ϕ,ϕ0)=12π(1+Γ+iQn)(1+Γ+iQn0) ×⎡⎢⎣1ei(ϕQ−ϕ)e2i(ϕQ−ϕ)⎤⎥⎦T(^M−1−^M−1Q=∞)⎡⎢⎣1ei(ϕ0−ϕQ)e2i(ϕ0−ϕQ)⎤⎥⎦.

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

 11+Γ+iQn =∞∑n=−∞Pnein(ϕ−ϕQ), (72) 11+Γ+iQn0 =∞∑m=−∞Pme−im(ϕ0−ϕQ) (73)

in Eq. (IV.1), substitute the obtained equation into Eq. (69), take and average over . We arrive then to the following equation

 W(ϕ)=12πl2∞∑n=−∞wnei(n−1)ϕ, (74)

where

 wn=∫d2Q(2π)2⎡⎢⎣Pn−1PnPn+1⎤⎥⎦T(^M−1−^M−1Q=∞)⎡⎢⎣Pn−1PnPn+1⎤⎥⎦.

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

 ^M−1=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1η20002η(1−η)2000η2⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (76)

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

### iv.2 Correction to the conductivity

The quantum correction to the conductivity is given by GKO14 ()

 δσ=−e2ℏl2tr ∫dϕ2π γC(π−ϕ)γW(ϕ)(1+cosϕ). (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 nonback ()). Using Eqs. (44), (49), and (LABEL:Wc), we obtain

 δσ =−e2ℏ(1+η2)l2(1+η2−η)2 ×∫dϕ2π(1−2ηeiϕ+η2e2iϕ)W(ϕ)(1+cosϕ), (78)

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

 δσ = −e22πℏ1+η2(1+η2−η)2[(1−η)w1+1+η2−4η2w0 (79) + w22+(η2−η)w−1+η22w−2],

where are given by Eq. (LABEL:wn).

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

 W(ϕ)≈12πl2∫d2Q(2π)2P301+η2−