Quantum interference in HgTe structures
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 BernevigHughesZhang Hamiltonian and the relevant symmetrybreaking 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 lowfield magnetoresistance in a quasionedimensional geometry are analyzed.
I Introduction
Twodimensional (2D) and threedimensional (3D) materials and structures with strong spinorbit interaction in the absence of magnetic field (i.e. with preserved timereversal 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 spinorbit 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 timereversal 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 spinHall 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 spinorbit metal. The interference corrections to the conductivity and the lowfield magnetoconductivity of such a system reflect the Diracfermion character of carriers ostrovsky10 (); Tkachov11 (); OGM12 (); Richter12 (), similarly to interference phenomena in grapheneMcCann (); Nestoklon (); Ostrovsky06 (); aleinerefetov () 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 interferenceinduced quantum corrections to the conductivity of HgTebased 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 Diractype Hamiltonian and physically important symmetrybreaking mechanisms. In Sec. III we overview a general symmetrybased approach OGM12 () to the problem and employ it to evaluate the conductivity corrections within the diffusion approximation. Section IV complements the symmetrybased 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 onedimensional geometry is analyzed in Sec. V. Section VI summarizes our results and discusses a connection to experimental works.
Ii Symmetry analysis of the lowenergy Hamiltonian
The lowenergy Hamiltonian for a symmetric HgTe/HgCdTe structure was introduced in Ref. BernevigHughesZhang, in the framework of the method. The BernevigHughesZhang (BHZ) Hamiltonian possesses a matrix structure in the Kramerspartner space and E1 – H1 space of electron and holetype levels Qi11 (); Liu08 (); Rothe10 (),
(1)  
(2) 
Here the components of spinors are ordered as . It is convenient to introduce Pauli matrices for the E1—H1 space and for the Kramerspartner space (here and are unity matrices), yielding
(3) 
The effective mass and energy are given by
(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
(5) 
The eigenfunctions for each block are twocomponent spinors in E1H1 space:
(6) 
where spinors are different in different blocks
(7)  
(8) 
Here is the polar angle of the momentum and
(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 ()
(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 symmetrybreaking mechanisms Rothe10 (); OGM12 (). The Hamiltonian is characterized by the exact global timereversal (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 “blockwise” TR symmetry when . Around this point, the symmetry is approximate. An approximate orthogonal blockwise 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 symmetrybreaking length. In this regime, the diffusive logarithmic correction to the conductivity is insensitive to this symmetrybreaking mechanism and the system behaves as if this symmetry is exact. However, when the dephasing length becomes longer than the symmetrybreaking length, the relevant singular corrections are no longer determined by the dephasing but are cut off by the symmetrybreaking scale. This signifies a crossover to a different (approximate) symmetry class. Below we analyze the relevant symmetrybreaking 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 blockoffdiagonal elements in the full lowenergy 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 ()
(11) 
The BIA perturbation (11) contains the momentumindependent term with that connects the electronic and heavyhole bands Winkler () with opposite spin projections. The terms with and stem from the cubic Dresselhaus spinorbit interaction within and , respectively. Further, the symmetry is broken by the Rashba spinorbit interaction due to the structural inversion asymmetry (SIA):Liu08 (); Rothe10 ()
(12) 
Here only the linearinmomentum E1 SIA term is retained, as the SIA terms for heavy holes contain higher powers of . Finally, shortrange impurities and defects, as well as HgTe/HgCdTe interface roughness may also violate the symmetry of the QW, giving rise to a random local blockoffdiagonal perturbations.
Iii Symmetry analysis of quantum conductivity corrections
Here we overview the approach developed in Ref. OGM12, for the analysis of quantuminterference 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 antiunitary operators that act on a given operator according to
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:
(13) 
Each of these TR symmetries corresponds to a Cooperon mode contributing to the singular oneloop conductivity correction:
(14) 
Here is the phasebreaking 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 linearin 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
(16)  
(17) 
[where ] can be treated as weak perturbations to the massless (graphenelike) Dirac Hamiltonian:
(18) 
The latter possesses four TR symmetries:
(19)  
(20)  
(21)  
(22) 
These symmetries give rise to a positive weak antilocalization (WAL) conductivity correction
(23) 
corresponding to two independent copies of a symplecticclass 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
(24) 
At lowest temperatures, when , we find a nonsingularin result:
(25) 
For higher temperatures, when , these two copies of a unitaryclass system (2U) become two copies of the (approximately) symplectic class, with the correction given by Eq. (23).
In the presence of inversionasymmetry terms and , the only remaining TR symmetry is . The symmetry analysis yields the following expression for the conductivity correction in this (generic) case OGM12 ():
(26) 
Here is the symmetrybreaking rate due to the independent term in while describes the symmetry breaking governed by linearin 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 symmetrybreaking 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:
(27) 
The direction of the pseudospin within each block is almost frozen by the effective “Zeeman term” . The linearin terms of the BHZ Hamiltonian can then be treated as a weak spinorbitlike perturbation to the massive diagonal Hamiltonian
(28) 
Neglecting the block mixing, the conductivity is given by a sum of two weak localization (WL) corrections characteristic for an orthogonal symmetry class:
(29) 
Here is the symmetrybreaking 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:
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 symmetrybreaking times rather than by the sum of rates. Following this rule, the inclusion of , , and gives rise to the following interference correction: OGM12 ()
(30) 
The only true massless mode in Eq. (30) stems again from the physical symplectic TR symmetry . This means that the generic blockmixing terms drive the two copies of the (approximately) orthogonal class to a single copy of a symplecticclass system. The hierarchy of the symmetrybreaking rates , , and , generates the following three patterns of crossovers:OGM12 ()

: 2O 2U 1Sp.

and :
2O 1Sp. 
: 1Sp.
To summarize this section, we have analyzed the quantum conductivity correction in the diffusion approximation using the symmetrybased 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 “intrablock” 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:
(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 spinorbit 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 whitenoise 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 interferenceinduced 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:
(32)  
(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 E1H1 space which can be represented as a sum of the contributions of upper and lower branches:
(34) 
where the projectors are given by
(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,
(36) 
This allows us to retain in the matrix Green’s function only the contribution of the upper band:
(37) 
where is the disorderinduced selfenergy. From now on we will omit the branch index “+”. The spinors in the upper band of block II read:
(38)  
(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 whitenoise 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 E1H1 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 Rashbatype and Zeemantype terms in the Hamiltonian killing the quantum interference.
We see that the problem is equivalent to a singleband 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 selfenergy 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).
Analyzing the problem within the DrudeBoltzmann 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 shortrange scattering potential, the function turns out to be angledependent due to the “dressing” by the spinor factor Hence, for the case of a massive Dirac cone, the transport scattering rate
(48) 
differs from the total (quantum) rate
(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 angleaveraged function ] that enters the singleparticle 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:
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. Tkachov11 () Another important property is an asymmetry of Indeed, as seen from Eq. (LABEL:Wc),
(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 zerofrequency Cooperon has the form:
(53) 
Here, is the phasebreaking 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.
Introducing dimensionless variables
(57) 
where
(58) 
is the mean free path, we rewrite Eq. (53) as follows:
(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:
(60) 
where
(61)  
(62)  
(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
(64) 
Here
(65) 
(66) 
and
(67) 
From Eqs. (60), (64), and (65) we find
(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
(69) 
In particular, the conductivity can be expressed in terms of this probability taken at (return probability):
(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 interferenceinduced 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
Here we took into account that for Let us now find the return probability. To this end, we make expansions
(72)  
(73) 
in Eq. (IV.1), substitute the obtained equation into Eq. (69), take and average over . We arrive then to the following equation
(74) 
where
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:
(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 ()
(77) 
Here in the “Hikamibox” 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
(78) 
and, finally, with the use of Eq. (74), arrive at
(79)  
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