Abstract
HgTe quantum wells possess remarkable physical properties as for instance the quantum spin Hall state and the “singlevalley” analog of graphene, depending on their layer thicknesses and barrier composition. However, double HgTe quantum wells yet contain more fascinating and still unrevealed features. Here we report on the study of the quantum phase transitions in tunnelcoupled HgTe layers separated by CdTe barrier. We demonstrate that this system has a 3/2 pseudo spin degree of freedom, which features a number of particular properties associated with the spindependent coupling between HgTe layers. We discover a specific metal phase arising in a wide range of HgTe and CdTe layer thicknesses, in which a gapless bulk and a pair of helical edge states coexist. This phase holds some properties of bilayer graphene such as an unconventional quantum Hall effect and an electricallytunable band gap. In this “bilayer graphene” phase, electric field opens the band gap and drives the system into the quantum spin Hall state. Furthermore, we discover a new type of quantum phase transition arising from a mutual inversion between second electron and holelike subbands. This work paves the way towards novel materials based on multilayered topological insulators.
Introduction
Low energy band structure in graphene is formed by two spindegenerate massless Dirac cones at two inequivalent valleys, giving rise to four massless Dirac cones in total^{1,2}. With one additional graphene layer added, bilayer graphene (BG) has an entirely different band structure. Most notably, symmetric BG is a zeroband gap semiconductor with quadratic energymomentum dispersion^{3}. However, its band gap is continuously tunable through an electrical field applied perpendicularly to the sample^{4,5}. The electronic states in BG are also fourfold degenerate, taking into account both spin and valley degeneracies. Recently, it has been shown that HgTe quantum wells (QWs) with vanishing band gap possess a single spindegenerate Dirac cone at the Brillouin zone center^{6,7,8,9} and thus can be considered as the “singlevalley” analog of graphene in twodimensional (2D) semiconductor heterostructures.
The central feature of the HgTe QWs is the possibility of band inversion. The barrier material (e.g., CdTe) has a normal band ordering, with the stype Γ_{6} band lying above the ptype Γ_{8} band, while in the well material (HgTe) the Γ_{6} band lies below the Γ_{8} band, corresponding to an inverted band ordering. If the thickness d of the HgTe/CdTe QWs does not exceed a critical value d_{c}, at the Γ point, the first conduction subband (E1) is electronlike, i.e. it is formed by a linear combination of the Γ_{6}, m_{J} = ±1/2〉 and Γ_{8}, m_{J} = ±1/2〉 states, while the first valence subband (H1) arises from the Γ_{8}, m_{J} = ±3/2〉 states, corresponding to the heavyhole band. In wide HgTe QWs (d > d_{c}), the E1 subband falls below the H1 subband. The inversion between E1 and H1 subbands leads to the formation of a 2D timereversal invariant topological insulator (TI)^{10} denoted the quantum spin Hall (QSH) state^{6,7} with dissipation less edge channel transport at zero external magnetic field^{11}. At critical QW thickness d_{c}, corresponding to topological phase transition between a TI and a trivial band insulator (BI), a lowenergy band structure mimics a massless Dirac cone at the Γ point^{6}.
Two tunnelcoupled HgTe QWs of critical thickness may, therefore, share some properties of natural BG. If each of the HgTe QW has d > d_{c}, this system offers a practical realization of tunnelcoupled layers of 2D TI. Moreover, the strained HgTe thick films have been proven to be threedimensional (3D) TIs^{12,13}. Therefore, a wide double HgTe QW can be considered as two thick layers of 3D TI separated by BI (CdTe material) with the surface states arising at the interfaces^{14}. A tensile strain in the HgTe films, which opens a topological gap, is induced by the difference in the lattice parameters of HgTe and CdTe. Thus, double HgTe QWs is a realistic system, which potentially paves the way towards physics of multilayered topological insulator materials.
So far, there are just a few works devoted to double HgTe QWs^{15,16,17}. All these works are based on the approach, in which each QW is described within BernevigHughesZhang (BHZ) model^{6} coupled by a spinconserved tunneling Hamiltonian. The latter will be shown to be irrelevant on double HgTe QWs, which feature a much richer physics than previously assumed.
In our theoretical investigation of double HgTe QWs (see Fig. 1), we first start from band structure calculations on the basis of an eightband Kane model^{18}. By using realistic material parameters, we obtain the phase diagram for symmetrical double QW as a function of the layer thicknesses. We discover a specific metal phase, in which the band properties in perpendicular magnetic and electric fields are very similar to the ones of natural BG. Then, we deduce an effective 2D Hamiltonian, involving two electronlike (E1, E2) and two holelike (H1, H2) subbands, to describe quantum phase transitions in the system. On the basis of such a simplified Hamiltonian, we calculate dispersion of edge states in different phases. We show that inversion between E2 and H2 subbands induces an additional pair of helical edge states, putting the system into BI phase even in the case of inverted band structure. The crossing between E2 and H2 levels yields a Dirac cone in the Γ point of the Brillouin zone.
Considering double HgTe QW, shown in Fig. 1, as a whole system and by using expansion in the planewave basis set (see Supplementary Information), we calculate energy dispersions of electronic subbands. Figure 2A,B show positions of electronlike and holelike subbands at k = 0 as a function of middle barrier thickness t. Intuitively, it is clear that at infinitely large barrier, calculated states correspond to the subband positions in two separated HgTe QWs, while at finite values of t each pair of the subbands is connected with evenodd state splitting in double QWs. If t tends to zero, the energy values transform into positions of electronic subbands in single HgTe QWs of 2d thickness. It is seen that the splitting between electronlike levels exceeds significantly the one of the holelike levels. This is due to significant difference in effective masses at k = 0 in electronlike and holelike subbands, which values determine the tunnelcoupling between the states and their evenodd splitting at given t.
There are two types of the band structure ordering, which arises depending on the HgTe layers thickness d. The first case shown in Fig. 2A takes place if d varies in the range from d_{c}/2 to d_{c}, where d_{c} corresponds to the thickness of single QW with Dirac cone at the Γ point. In this case E2 subband always lies above H2 subband and inversion between E1 and H1 subband can take place. At higher values of t, the E1 subband gains a higher energy than the H1 subband and the system has the normal band structure. When the thickness t is decreased, the energy of the E1 subband reduces, whereas the H1 subband energy practically does not change. The different dependence of E1 and H1 subbands on t implies that at some barrier thicknesses the band gap closes. In fact, the crossing point between E1 and H1 subbands yields a lowenergy band structure with Dirac cone in the vicinity of k = 0 (Supplementary Information). If the values of t are small enough to induce a gap between H1 and H2 subbands, it implies a quantum phase transition between BI and TI states, as in single HgTe QW^{6}. However, if H1 and H2 subbands coincide at k = 0, it gives rise to an additional “massive” branch of valence subband at the crossing point, similar to the pseudospin1 DiracWeyl 2D systems^{19}, as well as a quantum phase transition into a specific metal phase with the following subband ordering: E2H1H2E1. The reasons to assign this metal phase as a BG phase are discussed later.
The second type of band structure ordering, shown in Fig. 2B, is realized when d > d_{c}. In this case E1 subband always lies below H1 subband, i.e. they are inverted. At lower barrier thickness, E2 subband lies above H2 subband, while at high values of t, they swap their positions. At specific values of t, E2 and H2 subbands cross, which also leads to the appearance of a Dirac cone at the Γ point. As for the case shown in Fig. 2A, additional “massive” branch arises as well. However, such branch corresponds to conduction H1 subband. Up to date, it was never realized that crossing between E2 and H2 subbands also induces a quantum phase transition. In this work we explicitly demonstrate it for the first time. In particular, we show that this crossing point in double HgTe QWs correspond to the quantum phase transition between BG and BI phase.
Figure 2C shows the phase diagram, in which two bold lines correspond to the Dirac cones at the Γ point. The left one results from the crossing between E1 and H1 subbands, while the rightside curve is connected with the crossing of E2 and H2 levels. If the middle barrier is thin enough, a gap between H1 and H2 subbands opens and the inversion between E1 and H1 levels induces a quantum phase transition between BI and TI phases. The latter is shown by orange region. For relatively wide QWs, the socalled semimetal (SM) phase, corresponding to the whitestriped region, is implemented. It is characterized by a vanishing indirect band gap, when the side maximum of the valence subband exceeds in energy the conduction subband bottom. This phase arises in single HgTe QWs as well when the thickness goes beyond some critical value, denoted in Fig. 2C by d_{SM}^{20,21}. BG metal phase mentioned above corresponds to the blue region in Fig. 2C. Dispersion curves at various values of d and t in the vicinity of phase transitions between BI, BG and SM phases are provided in Supplementary Information.
Let us now explain the reasons to call this specific metal phase with the ordering of electronic subbands E2H1H2E1 a BG phase. First, we consider in details the case when both QWs have a critical thickness d_{c} ≈ 6.5 nm, which at infinitely large barrier corresponds to two Dirac cones. The presence of the transparent barrier, for example of t = 3 nm, turns from two Dirac cones in the vicinity of the Γ point into a band structure very similar to the one of natural BG^{3} (c.f. Fig. 3A). In particular, it consists of two gapless isotropic parabolas, formed by H1 and H2 subbands. Moreover, as in BG in which nonzero band gap can be induced by breaking the inversion symmetry of two monolayers, in double HgTe QWs it could be obtained by using QWs of different thickness or by adding of oneside chemically doping. Moreover, the potential of a continuously tunable band gap through an electric field applied perpendicularly to the sample plane is of particular importance.
The double HgTe QW in BG phase also holds this property. Figure 3B displays energy dispersion for BG phase in perpendicular electric field of 20 kV/cm. Even in this case the dispersion curves are very similar to the band structure of natural BG in external electric field^{3,4}. However, strong spinorbit interaction in HgTe layers removes the spin degeneracy away from the Γ point due to Rashba effect^{22}. Figure 3C shows the band gap values in the double QWs in BG phase as a function of applied electric field. Indeed, the band gap in BG phase is electricallytunable, as it is in natural BG. However, its dependence on electric field has nonmonotonic behaviour in double HgTe QWs. The reason for this is related with the additional side maximum of valence subband, which is also responsible for the formation of SM phase in zero electric field (see the diagram, shown in Fig. 2C). The top of the side maximum is increasing with the strength of electric field and the band gap reduces (see the inset, shown in Fig. 3C). As a result, the band gap is closed in high enough electric field, giving rise to the formation of the SM phase. A critical electric field, corresponding to arising of indirectband gap, coincides with the maximum value of Δ, shown in Fig. 3C.
Another characteristic of natural BG is the unconventional quantum Hall effect, related with the absence of zeroLandau level (LL) plateaus in Hall conductivity^{23}. For natural BG, plateaus in the Hall conductivity σ_{xy}, occur at integer multiples of 4e^{2}/h. This is similar to a conventional semiconductor with level degeneracy g = 4 arising from the spin and valley degrees of freedom. Deviation from the conventional case occurs at low density, where there is a step in σ_{xy} of height 8e^{2}/h across zero density, arising from the eightfold degeneracy of the zeroenergy LL. This specific LL is formed by atomic orbitals of different sublattice sites from both layers^{24}.
Figure 3D shows LL fan chart in double HgTe QW with BG phase, calculated within the eightband Kane model. In the BG phase, in perpendicular magnetic field a specific zeromode LL with degeneracy two times higher than other LLs also arises. This zeromode LL is shown in Fig. 3D, by bold orange line. For double HgTe QW, plateaus in the Hall conductivity are expected to occur at integer numbers of e^{2}/h. However, the doubled degeneracy of zeromode LL requires twice as many carriers to fill them, so the transition between the corresponding plateaus should be twice as wide in density and the step in σ_{xy} between the plateaus are expected to be twice as high, 2e^{2}/h instead of e^{2}/h. Since the zeromode LL in double HgTe QW is formed by the states of both H1 and H2 subbands, the presence of inversion asymmetry of two HgTe layers not only opens the band gap between H1 and H2 subbands (see Fig. 3B) but also splits the zeromode LLs, removing the double degeneracy order. Therefore, we also expect recovering of sequence of equidistant plateaus in the Hall conductivity as it is for gatebiased natural BG^{5}.
Besides the zeromode LL, there are two additional specific LLs, which in high magnetic fields are formed only by the states from E1 and E2 subbands. These LLs are shown in Fig. 3D by blue curves. In moderate magnetic fields, these E1 and E2 LLs are mixed with the states from H2 and H1 subbands respectively. It results in anticrossing between E1 (E2) LL and LLs from H2 (H1) subband, which is clearly seen in magnetic fields below 1 T. The crossing between the E1 and zeromode LL, arising at critical magnetic field B_{c} ~ 3.5 T, corresponds to the transition from inverted into normal band structure, similar to this observed in single HgTe QW^{7}.
We have considered in details the case of double HgTe QW, when both HgTe layers have a critical thickness d_{c}. However, all the mentioned properties of BG phase hold for any double QW with the values of d and t, corresponding to the blue region in Fig. 2C. The only difference is the ratio M_{1} over M_{2}, where M_{1} parameter describes the energy gap between E1 and H1 subbands, while M_{2} corresponds to the half of the gap between E2 and H2 levels. If d < d_{c}, 2M_{2} exceeds 2M_{1}, while in the opposite case of d > d_{c}, the gap between E2 and H2 subbands is lower than 2M_{1}. An amazing property of double HgTe QW is that it shares some characteristics of natural BG even in the BI phase at d > d_{c}. In particular, the possibility of tuning the band gap by electric field and of observing the unconventional double step in plateaus in the Hall conductivity still persists.
We now discuss quantum phase transitions in double HgTe QWs. For this purpose, we derive an effective 2D Hamiltonian, which describes the band structure in the vicinity of k = 0 in the phases shown in Fig. 2C. To infer this simplified model, we start from eight Bloch basic states, combined into an eightcomponent spinor:
For the QWs grown in [001] direction, projection m_{J} of total angular momentum at k = 0 on the growth direction is still a good quantum number. At the Γ point all QW subband states are formed by linear combination of the mentioned eight bulk bands. To describe BI, TI and BG phases, one should consider E1, E2, H1 and H2 subbands (see Fig. 2A,B). At k = 0, the E1, ±〉 and E2, ±〉 subband states are formed from the linear combination of the Γ_{6}, m_{J} = ±1/2〉, Γ_{7}, m_{J} = ±1/2〉 and Γ_{8}, m_{J} = ±1/2〉, while the H1, ±〉 and H2, ±〉 QW states are formed from the Γ_{8}, m_{J} = ±3/2〉 states. Away from the Γ point, the E1, E2, H1 and H2 states are mixed. To construct the effective Hamiltonian one should carefully take into account different parities of the envelope function components from Γ_{6}, ±1/2〉, Γ_{7}, ±1/2〉, Γ_{8}, ±1/2〉 and Γ_{8}, ±3/2〉 bulk bands into formation of given subband state^{25}. For instance, E1 state is formed by even function corresponding to Γ_{6}, ±1/2〉 band and by odd envelope functions corresponding to Γ_{7}, ±1/2〉 and Γ_{8}, ±1/2〉 bands, while parities of their contributions into E2 state are changed. In previous works^{15,16,17}, this point was missed; it results in the wrong description of double HgTe QWs. Detailed explanation is given in Supplementary Information.
After straightforward calculations, we obtain the following form of the effective 2D Hamiltonian for the E1, E2, H1, H2 states, expressed in the basis of two Kramer’s sets E1, +〉, H1, +〉, H2, −〉, E2, −〉 and E2, +〉, H2, +〉, H1, −〉, E1, −〉:
where Θ is a “time reversal” operator, given by
with K stands for complex conjugation and σ_{y} is one of the Pauli spin matrices. Each block in (1) is described by a fourcomponent spinor with pseudospin J = 3/2 degree of freedom (see Supplementary Information). H(k_{x}, k_{y}) is written as
Here, k_{x} and k_{y} are momentum components in the plane of double QW and C, M_{1}, M_{2}, A_{1}, A_{2}, B_{E1}, B_{H1}, B_{H2}, B_{E2}, Δ_{H1H2}, R_{1}, R_{2}, S_{0} are specific heterostructure constants, being defined by QW geometry and materials. The Hamiltonian H_{eff} (k_{x}, k_{y}) has blockdiagonal form because we keep the inversion symmetry and axial symmetry around the growth direction (see Supplementary Information). We note that H_{eff} (k_{x}, k_{y}) is valid for any values of d and t. In particular for t = 0, it describes energy dispersion in the vicinity of the Γ point in single HgTe QW beyond the BHZ model. Parameters Δ_{H1H2}, R_{1}, R_{2}, S_{0} significantly depend on t and all tend to zero at large middle barrier thickness, while B_{E1} = B_{E2} coincides. As it is easy to see, in this case the system is described by two noninteracting BHZ models, written for two pairs formed by E1, H1 and E2, H2 subbands. We note that B_{H1} = B_{H2} if Δ_{H1H2} = 0.
The most important quantities in H(k_{x}, k_{y}) are two mass parameters M_{1} and M_{2}. Almost all phases in Fig. 2C can be grouped into three types according to the sign of M_{1} and M_{2}. The BI phase at normal band structure, shown in the left part of the diagram, corresponds to positive values of M_{1} and M_{2}. The BG and TI phases arise if M_{1} < 0 and M_{2} > 0 and the difference between these phases is connected with Δ_{H1H2}, which equals to zero in the case of BG phase. The second BI phase, shown in the right from the BG phase, corresponds to negative values of M_{1} and M_{2} and Δ_{H1H2} = 0. The SM phase, arising at all values of M_{2} and M_{1} > 0, cannot be described within the simplified model because it is only valid near the Γ point, while the SM phase is formed by nonlocal overlapping of the valence band and conduction band bottom^{20,21}. Comparison between calculations of subband curves, performed within the simplified model and on the basic of an eightband Kane Hamiltonian for different phases is given in Supplementary Information.
Zero values of M_{1} and M_{2} both conform to quantum phase transitions at which Dirac cone at the Γ point occurs. The left and right bold black curves in the phase diagram, shown in Fig. 2C, are related with M_{1} = 0 and M_{2} = 0 respectively. Figure 4 presents energy dispersion for the bulk and edge states in various phases in double HgTe QWs obtained within the simplified model. Dispersion of the bulk states are shown in black, while the orange and blue curves are the edge states, described by different blocks of H_{eff }(k_{x}, k_{y}). The electrons in the edge states, marked by different colors, move in opposite directions. Figure 4 demonstrates that each change of sign of mass parameter M_{1} or M_{2} yields a pair of helical edge states, providing a quantum phase transition at M_{1} = 0 or M_{2} = 0. In particular, BG phase arising at M_{1} < 0 and M_{2} > 0 has a single pair of the edge states, coexisting with the gapless bulk states. If M_{1} < 0 and M_{2} = 0, the double HgTe QW mimics a 2D system with the presence of both bulk and edge massless fermions, which energy dispersions are shown in Fig. 4C.
Figure 4D shows special case with inversion between E2 and H2 subbands, which was not considered in 2D systems so far. It perfectly illustrates that each crossing between electronlike and holelike levels of higher indexes, also results in the appearance of a pair of helical edge states. Topological properties of corresponding insulator phase are determined by amount of inverted levels, which are connected with the number of pairs of the edge states. In the particular case of inversion of both E1 with H1 subband and E2 with H2 subband, the terms proportional to R_{1}, R_{2} and S_{0} induce the coupling between counterpropagating states with the same spin orientation. The latter could be interpreted as a spindependent tunneling between two layers of 2D TI (see Supplementary Information). It is clear from the inset that such tunnelcoupling opens the gap in the energy spectrum of the edge states even without external oneparticle scattering processes. Thus, we prove that double HgTe QW with M_{1} < 0 and M_{2} < 0 has a BI phase.
Let us now remark the main difference between BG phase in double HgTe QW and natural BG. The electrons in natural BG are chiral particles with S = 1/2 pseudospin of freedom^{23,24} and pseudospin winding number of 2^{25}. It results from the commutation of operator with the lowenergy Hamiltonian of natural BG. As it is shown above, the simplest Hamiltonian, required for the description of bulk and edge states in the BG phase, has J = 3/2 pseudospin of freedom. Moreover, it can be shown that operator does not commute with H(k_{x}, k_{y}), proving the nonchiral character of electron in double HgTe QWs. Thus, even though BG phase mimics some characteristics of natural BG, it is a novel and fascinating state of matter, exhibiting the coexistence of gapless bulk states and spinpolarized edge channels.
Since we have demonstrated the existence of gapless spinpolarized counterpropagating edge channels in double HgTe QWs, they should exhibit QSH effect in both TI (Δ_{H1H2} ≠ 0) and BG (Δ_{H1H2} = 0) phases. Intuitively it is clear that QSH effect in the TI phase could be observed on the sample with a sixterminal Hall bar, as it has been previously proposed for single QW^{6,7}. However, the measurements of twoterminal conductance in the BG phase contain contributions both from the bulk states and from the helical edge modes. To separate these contributions, we propose to introduce inversion asymmetry between the HgTe layers. If the band gap for the bulk states is open, for example by an external electric field, the system is driven to the TI regime and the detection of the QSH effect becomes possible^{6}.
Additional Information
How to cite this article: Krishtopenko, S. S. et al. Phase transitions in two tunnelcoupled HgTe quantum wells: Bilayer graphene analogy and beyond. Sci. Rep. 6, 30755; doi: 10.1038/srep30755 (2016).
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Acknowledgements
We thank all members of the group of D. Carpentier in ENS Lyon for insightful discussions and comments of this work. The authors gratefully acknowledge Sandra Ruffenach for her assistance in manuscript preparation. This work was partially supported by the CNRS through LIA TeraMIR project, LanguedocRoussillon region via the “Terapole Gepeto Platform”, the Russian Academy of Sciences, the nonprofit Dynasty foundation, the Russian Foundation for Basic Research (Grants 150208274, 155216012, 160200672), Russian Ministry of Education and Science (Grant МК6830.2015.2) and by Polish National Science Center (HARMONIA project, decision number DEC2013/10/M/ST3/00705). Theoretical studies of the edge states were performed in the framework of project № 161210317 provided by Russian Science Foundation.
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All authors have contributed equally to the underlying theoretical investigations. The figures were prepared by S.S.K. The paper and Supplementary Information was written by S.S.K. and F.T. All authors reviewed the manuscript.
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Krishtopenko, S., Knap, W. & Teppe, F. Phase transitions in two tunnelcoupled HgTe quantum wells: Bilayer graphene analogy and beyond. Sci Rep 6, 30755 (2016). https://doi.org/10.1038/srep30755
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DOI: https://doi.org/10.1038/srep30755
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