Hyperfine interactions in two-dimensional HgTe topological insulators
We study the hyperfine interaction between the nuclear spins and the electrons in a HgTe quantum well, which is the prime experimentally realized example of a two-dimensional topological insulator. The hyperfine interaction is a naturally present, internal source of broken time-reversal symmetry from the point of view of the electrons. The HgTe quantum well is described by the so-called Bernevig-Hughes-Zhang (BHZ) model. The basis states of the BHZ model are combinations of both - and -like symmetry states, which means that three kinds of hyperfine interactions play a role: (i) The Fermi contact interaction, (ii) the dipole-dipole-like coupling and (iii) the electron-orbital to nuclear-spin coupling. We provide benchmark results for the forms and magnitudes of these hyperfine interactions within the BHZ model, which give a good starting point for evaluating hyperfine interactions in any HgTe nanostructure. We apply our results to the helical edge states of a HgTe two-dimensional topological insulator and show how their total hyperfine interaction becomes anisotropic and dependent on the orientation of the sample edge within the plane. Moreover, for the helical edge states the hyperfine interactions due to the -like states can dominate over the -like contribution in certain circumstances.
A topological insulator (TI) host gapless surface or edge states, while the bulk of the material has an insulating energy gap.Qi and Zhang (2011); Kane and Mele (2005a, b); Hasan and Kane (2010) In three-dimensional TIs the gapless surface states are spin-polarized two-dimensional (2D) Dirac fermions, whereas 2D TIs contain one-dimensional (1D) helical edge states. The helical edge states appear in counterpropagating pairs, and the states with equal energy and opposite wave numbers, and , form a Kramers pair. Thus, elastic scattering from one helical edge state (HES) to the other one within a pair cannot be induced by time-reversal invariant potentials e.g. stemming from impurities.Xu and Moore (2006) Therefore, the transport through a 2D TI is to a large extend ballistic with a quantized conductance of per pair of HESs. This highlights the central role of time-reversal symmetry in TIs.
Quantized conductance have recently been measured in micron-sized samples in HgTe quantum wells,König et al. (2007); Roth et al. (2009); König et al. (2008); Buhmann (2011); Brüne et al. (2012); König et al. (2013) which to date is the most important experimental demonstration of a 2D TI. Evidence of edge state transport was found in both two-terminalKönig et al. (2007) and multi-terminalRoth et al. (2009) devices. Moreover, clever experiments combining the metallic spin Hall effect and a 2D TI in a HgTe quantum well (QW) demonstrated the connection between the spin and the propagation direction.Brüne et al. (2012) However, also deviations from perfect conductance have been observed in longer HgTe devices,König et al. (2007); Roth et al. (2009); König et al. (2013); Gusev et al. (2011) which could stem from e.g. inelastic scattering mechanisms.Schmidt et al. (2012); Budich et al. (2012); Lezmy et al. (2012); Crépin et al. (2012); Maciejko et al. (2009); Ström et al. (2010) The effect of external magnetic fields have also been considered.König et al. (2007, 2008); Tkachov and Hankiewicz (2010); Maciejko et al. (2010); Scharf et al. (2012); Delplace et al. (2012); Kharitonov (2012); Gusev et al. (2013) The TI state in HgTe QWs was predicted by Bernevig, Hughes and Zhang (BHZ)Bernevig et al. (2006) by using a simplified model containing states with - and -like symmetry, respectively. They found that beyond a critical thickness of the HgTe QW, the TI state would appear as confirmed experimentally.König et al. (2007); Roth et al. (2009); König et al. (2008, 2008); Buhmann (2011) Furthermore, interesting experimental progress on 2D TI properties has also be achieved in InAs/GaSb QWsKnez et al. (2011); Suzuki et al. (2013); Du et al. (2013) as proposed theoretically.Liu et al. (2008)
Hyperfine (HF) interactions between the electron and nuclear spins can play an important role in nanostructures – even though it is often weak.Coish and Baugh (2009); Schliemann et al. (2003); Slichter (1996); Stoneham (1975) For instance in quantum dots, HF interactions can limit the coherence of single electronic spinsKhaetskii et al. (2002); Koppens et al. (2008); Petta et al. (2005); Cywiński (2011) and, moreover, it can even lead to current hysteresis due to bistability of the dynamical nuclear spin polarization.Ono and Tarucha (2004); Pfund et al. (2007); Rudner and Levitov (2007); Lunde et al. (2013) HF-induced nuclear spin ordering in interacting 1DBraunecker et al. (2009a, b); Scheller et al. (2013) and 2DSimon and Loss (2007); Simon et al. (2008) systems have also been discussed. Most studies consider the so-called contact HF interaction,Coish and Baugh (2009); Schliemann et al. (2003); Slichter (1996); Stoneham (1975) which is relevant for electrons in orbital states with -like symmetry e.g. the conduction band in GaAs. However, for -like orbital states – such as the valence band in GaAs – the contact HF interaction is absent. Nevertheless, other anisotropic HF interactions are present for -like states such as the dipole-dipole-like HF interaction,Fischer et al. (2008); Fischer and Loss (2010); Fischer et al. (2009a, b); Testelin et al. (2009) which can play a significant role e.g. for the decoherence of a hole confined in a quantum dot.Fischer et al. (2008); Fischer and Loss (2010); Testelin et al. (2009); Chekhovich et al. (2013)
HF interactions and dynamical nuclear spin polarization have also been investigated in the context of integer quantum Hall systems,Dobers et al. (1988); Wald et al. (1994); Kim et al. (1994); Dixon et al. (1997); Deviatov et al. (2004); Würtz et al. (2005); Nakajima et al. (2010); Nakajima and Komiyama (2012) which contain unidirectional edge states. Here HF-induced spin-flip transitions between the unidirectional edge states can create nuclear spin polarization locally at the boundary of the 2D sample.Wald et al. (1994); Kim et al. (1994); Dixon et al. (1997); Deviatov et al. (2004); Würtz et al. (2005); Nakajima et al. (2010); Nakajima and Komiyama (2012) Recently, we have predicted a similar phenomenon for a 2D TI, namely that embedded fixed spins such as the nuclear spins in a 2D TI can polarize locally at the boundary due to a current through the HESs.Lunde and Platero (2012) Interestingly, the 2D TI with localized spins remains ballistic,Tanaka et al. (2011); Lunde and Platero (2012) except if additional spin-flip mechanisms for the localized spins are present.Lunde and Platero (2012) However, combining localized spins and Rashba spin-orbit coupling in the 2D TI can produce a conductance change.Eriksson et al. (2012); Del Maestro et al. (2013); Eriksson (2013) In the previous works,Lunde and Platero (2012); Tanaka et al. (2011); Eriksson et al. (2012); Del Maestro et al. (2013); Eriksson (2013) the interaction between the fixed spins embedded into the 2D TI and the HESs were modelled phenomenologically. In contrast, here we pay special attention to the detailed forms of the HF interactions within a 2D TI.
In this paper, we find the different HF interactions within the BHZ model for a HgTe QW. To this end, we take into account both the - and -like states of the BHZ model, which couple differently to the nuclear spins. We show that all the HF Hamiltonians couple the time-reversed blocks of the BHZ model. However, only HF interactions relevant for -like states couple states within a time-reversed block as illustrated in Fig.1. Moreover, we estimate the different HF coupling constants. The derived Hamiltonians are general in the sense that they can be used to find the HF interactions for any kind of nanostructure in a HgTe QW, e.g. quantum dots,Chang and Lou (2011) ring structures,Michetti and Recher (2011) quantum point contactsZhang et al. (2011) or hole structures.Shan et al. (2011) As an illustrative example, we find the HF interactions for a pair of HESs. Remarkably, the intra HES transitions coupled to all the nuclear spin components perpendicular to the propagation direction of the HESs. This kind of coupling is unusual compared to e.g. an ordinary Heisenberg model. Interestingly, the details of the HF interactions depend on the spacial direction of the boundary at which the HESs propagate.
The paper is structured as follows: First the HF interactions and the BHZ model are outlined in secs. II and III. Then the HF interactions are found within the BHZ model for the simplest case of a 2D QW (sec.IV). From this, we derive the HF interactions for a given nanostructure in sec. V. Finally, the HF interactions for the HESs are found and discussed (sec. VI). Appendices A-E provide various details for completeness.
Ii The Hyperfine Interactions
The HF interaction between an electron at position with spin and the nuclear spin of the lattice atom at can be derived from the Dirac equationStoneham (1975); Fischer et al. (2009b) to be (in SI units)
where is the Fermi contact interactionFermi (1930), is the dipole-dipole like coupling between the electrons spin and the nuclear spin, and is the coupling of the electrons orbital momentum and the nuclear spin. Here is the electrons position relative to the th nucleus, , and is the vacuum permeability. The gyromagnetic ratios of the electron and the th nuclear spin of the isotope are, respectively, given by () , where is the Bohr magneton and the electron g factor; and , where is the nuclear magneton and the g factor of the th isotope. Here and are the bare electron and proton mass, respectively.foo (a) Moreover, is a length scale related to the finite size of the nucleus and therefore much smaller than all other length scales in the system. It can be found to beStoneham (1975) fm, where is the number of protons in the nucleus.foo (b) Thus, the total HF interaction between an electron and all the nuclear spins in the lattice is
where only those lattice points with a non-zero nuclear spin are included in the sum.
Not every atom in a HgTe crystal has a non-zero nuclear spin in contrast to e.g. GaAs. The amount of stable isotopes with a non-zero spin in Hg and Te are aboutSchliemann et al. (2003)
Hence, about 19 of all the atoms in HgTe have a non-zero nuclear spin. By isotope selection processes, this number can be varied somewhat experimentally.
The contact interaction is the only important HF interaction for -like states due to their spherical symmetry around the atomic core. On the other hand, -like states vanish at the atomic core and therefore the contact interaction does not affect electrons in those states. In contrast, the two other terms and can indeed play a role for -like states such as heavy-holes.Fischer et al. (2008) Moreover, Fischer et al.Fischer et al. (2008) found the atomic HF coupling constants to be about one order of magnitude lower for -like compared to -like states in GaAs.
Iii The Bernevig-Hughes-Zhang (BHZ) model
Bernevig, Hughes and ZhangBernevig et al. (2006) constructed a simple model describing the basic physics of a HgTe QW. The effective BHZ Hamiltonian is derived using methodsWinkler (2003); Bastard (1992); Fabian et al. (2007) and valid for close to the point, i.e. close to . The basis states of the model are the two Kramer pairs and . Details on the derivation of the BHZ model are found in Refs. Bernevig et al., 2006; Qi and Zhang, 2011; Rothe et al., 2010. For a 2D QW the BHZ Hamiltonian is
|where is a vector of creation operators and|
|with being a zero matrix and|
Here , , and have been introduced.foo (c) The parameters , , and depend on the QW geometry.Bernevig et al. (2006); Qi and Zhang (2011) Importantly, varying the QW width changes the sign of , which in turn makes the system go from a non-topological to a topological state with HESs.Bernevig et al. (2006)
The hamiltonian (4a) a priori has periodic boundary conditions and thereby does not contain any edges. By introducing boundaries into the model, it is possible to derive explicitly the HESs in the TI state of the QW.Zhou et al. (2008); Wada et al. (2011) This will be discussed further in sec. VI.1.
where are the transverse envelope functions in the -direction perpendicular to the 2D QW and are the lattice periodic functionsfoo (d) at for the band with projection of the total angular momentum, , on the -axis. Here is the electron spin and is the orbital angular momentum (see Appendix A). The time-reversal operator connects states within a Kramer pair ( and ), and the two blocks in (4b) are related by time-reversal. Here we choose phase conventions of the envelope functions such that time-reversed partners have equal envelope functions. Moreover, and are chosen real, whereas is chosen purely imaginary. (Appendix A gives more details on the envelope functions and the lattice periodic functions.)
The states are seen to be mixtures of the -like band and the -like band with , whereas consist only of the -like band with . Hence, the states have a definite total angular momentum projection,
but are not eigenstates of .
The HF interactions can only induce transitions between states with a difference of angular momentum projection of one unit: . Therefore, we can already at this point see that only particular combinations of the BHZ states can be connected by HF interactions as seen in Fig. 1. Furthermore, it is evident that HF interactions relevant for both - and -like states need to be included to have a full description of the HF interactions in a HgTe TI.
The real-space basis functions of (4a) for the 2D QW with periodic boundary conditions are
where , () is the QW length in the () direction and are the real-space lattice periodic functions at . Moreover, we have included the atomic volumefoo (e) explicitly here as it is often done for HF related calculations.Fischer and Loss (2010); Fischer et al. (2009a, b); Fischer et al. (2008) It depends on the choice of the individual normalization of the envelope functions and the lattice periodic functions, respectively, if should be included explicitly,Coish and Baugh (2009) as discussed in Appendix B.
Iv Hyperfine interactions within the BHZ model
Next, we find the HF interactions within the BHZ model by using the states (7) for a 2D QW with periodic boundary conditions. As we shall see, these results are useful, since they allow us to find the HF interactions for any nanostructure created in a HgTe QW (sec. V).
iv.1 Outline of the way to find the hyperfine interaction matrix elements
The HF interactions (1) are local in space on the atomic scale, so the important part of the wavefunction with respect to the HF interactions is the behavior around the nucleus. Hence, in the envelope function approximation, it is the rapidly varying lattice periodic functions that play the central role, whereas the slowly varying envelope functions only are multiplicative factors at the atomic nucleus, as we shall see below.
We set out to find the HF interactions
for in the basis (7), i.e. for with . We begin by describing the general way that we find the HF interaction matrix elements . To this end, the integration over the entire system volume is rewritten as a sum of integrals over each unit cell of volume , i.e.
This should be understood in the following way: Every space point can be reached by first a Bravais lattice vector and then a vector within the unit cell, i.e. . The superscript on the unit cell volume indicates that the integral is over the th unit cell. Thus, the matrix element is
Here one sum is over all unit cells , whereas the other sum is only over those atoms at position with a non-zero nuclear spin.foo (f) To proceed, we take as an illustrative example and obtain
where we have used the slow variation of the envelope functions on the atomic scale, , and the lattice periodicity of the lattice periodic functions, e.g. for all . Here , and the integral over is over the unit cell, whereas is for the nuclei. For a specific nuclear spin , we now include only the integral over that particular unit cell containing the nuclear spin, since the HF interactions are local in space. In other words, if the nuclei spin is not inside the integration volume of the unit cell , then the contribution is neglected,foo (g) i.e.
where the unit cell integral now is independent of the unit cell position . The sum is only over the lattice nuclei at with a non-zero nuclear spin. Therefore, the system does not have discrete translational symmetry, so the sum cannot simply be made into an integral. Hence, the matrix elements are not diagonal in due to the nuclear spins at random lattice points.
In order to proceed, we need to evaluate the integral of the lattice periodic function over the unit cell in Eq.(12). To this end, the symmetry of the lattice periodic functions are important: The contact interaction is zero for -like states, since they vanish on the atomic center, while matrix elements of for vanish for -like states due to their spherical symmetry. Here, we approximate the lattice periodic functions by a Linear Combination of Atomic Orbitals (LCAO) asGueron (1964); Fischer et al. (2008)
where and are atomic-like wave functions centered on the Te and Hg atom, respectively, and is only within a single two-atomic primitive unit cell of HgTe centered at . The atomic wave functions inherit the symmetry of the bandGueron (1964); Fischer et al. (2008) as indicated by the index . The atoms are connected by the vector , and the constants are determined by the lattice periodic function normalization , see e.g. Eq.(65). The electron sharing within the unit cell is described by , which fulfill .foo (h)
The LCAO approach (13) now facilitates evaluation of the unit cell integral in the matrix elements . Consider e.g. the unit cell integral in Eq.(12) for a non-zero spin on the Hg nucleus located on , i.e.
where only the important contribution of the atomic wave functions centered on the Hg atom is included. In other words, integrals involving atomic wave functions centered on different atoms are neglected. Fischer et al.Fischer et al. (2008) estimated that these non-local contributions are two to three orders of magnitude smaller for GaAs – even for the long-ranged potentials in in Eqs.(1b,1c).
Thus, we have now outlined how to find the matrix elements for all three kinds of HF interactions (1). The matrix elements of the types and follow the same lines as above. The essential ingredients are the locality of the HF interactions, the periodicity of and the slowly varying envelope functions. Next, we find the three HF interactions (1) within the BHZ model.
iv.2 The contact HF interaction for -like states
Now we find the contact HF interaction Eq.(1a) within the BHZ basis (7). We begin by noting that and , since the contact interaction is only non-zero at the atomic center (), where the -like atomic orbitals vanish. Hence, only the -like part of the states leads to non-zero matrix elements of . Using the approach in Sec. IV.1 to find the matrix elements, we get
where are the raising and lowering nuclear spin operators. In analogue to the case of a quantum dot,Fischer et al. (2008); Coish and Baugh (2009) we here introduce the position dependent contact HF coupling asfoo (i)
which includes the atomic contact HF coupling
for the nuclear spin at site of isotope . Here depends on the real space position of the nuclear spin. In contrast, does not depend on the nuclear position, since it can be given in terms of the atomic orbital by using Eq.(13) as , i.e. only depends on the nuclear isotope type at site . Moreover, at the present level of approximation, we can freely replace the Bravais lattice vector by the actual position of a nuclear spin within the unit cell in the envelope functions in Eq.(16) due to their slow variation. Finally, we arrive at the HF contact interaction in the BHZ basis as
where andfoo (j)
The sum is only over non-zero nuclear spins. Therefore, it is now clear that the contact HF interaction contains elements , which connect the time-reversed blocks in the BHZ hamiltonian (4b). Moreover, as illustrated in Fig.1, only the states are connected by , since only these states contain a -like symmetry part. In in table 1, estimates of the atomic contact HF couplings are given for the stable isotopes of HgTe with non-zero nuclear spin (see Appendix D for details).
iv.3 The HF interactions for -like states
To begin with, we argue that the -like states – part of the states – do not contribute to the matrix elements and for . (In contrast, the -like part of do contribute to these elements as will be shown below.) To understand this, the HF matrix elements are written in terms of the unit cell integrals over the atomic-like wave functions as outlined in Sec. IV.1. Firstly, for the dipole-dipole like HF interaction (1b), we have
due to the rotational symmetry of the -like orbitals around the atomic core.foo (b) Secondly, we have
due to opposite parities of the - and -like orbitals.foo (k) The same matrix elements containing instead of are also zero, because the -like states have zero orbital momentum, i.e. .
Therefore, only -like states contribute, so we are now left with (see Sec. IV.1)
and , where and . Using the LCAO approach Eq.(13), the unit cell integrals over the lattice periodic functions now become integrals over the atomic-like wave functions as in Eq.(14). We write the atomic wave functions as a product of a radial part and an angular part , i.e. , using spherical coordinates with the nucleus in the center. Since the integrals are over the two-atomic unit cell volume, they do not a priori factorize into a product of radial and angular integrals. However, due to the dependence of (), the important part of the unit cell integrals are numerically within one or two Bohr radii from the atomic core, which is certainly within the unit cell volume. Therefore, it is a good approximation to write the unit cell integrals (e.g. Eq.(14)) as
where the specific choice of is not important for the numerical value of the integral.foo (l) Therefore, we are now left with an essentially atomic physics problem, where the integral separates into a product of a radial and an angular part. The radial part is
which is the same for all the matrix elements of and and only depends on the type of atom Hg, Te. Due to the smallness of the nuclear length scale , it is not significant for the magnitude of .foo (b) Using the radial integral (25), we introduce the atomic -like HF coupling for isotope (at site ) as
which are estimated to be about one order of magnitude smaller than the atomic contact HF couplings (18), see table 1. Here it makes sense to have a common atomic HF coupling for the dipole-dipole like coupling and the orbital to nuclear-spin coupling , since the normalization constants for the LCAO lattice functions (13) are numerically approximately equal, , as discussed in Appendix D.foo (m) Moreover, we also use that are independent of the sign of , see Eq.(76). Calculating the angular integrals as discussed in Appendix C, the matrix elements (23) for the dipole-dipole like HF interaction become
where we introduce the position dependent -like HF couplings as
and . In comparison, for the contact HF interaction only a single position dependent HF coupling was introduced in Eq.(17). Here the explicit dependence on the position of the nuclear spin has been suppressed in the notation for simplicity, i.e. . Similarly, the matrix elements for the HF interaction between the electronic orbital momentum and the nuclear spins become