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Robust level coincidences in the subband structure of quasi-2D systems

R. Winkler

a,b,c,n

, L.Y. Wang

c

, Y.H. Lin

c

, C.S. Chu

c,d

a

Department of Physics, Northern Illinois University, DeKalb, IL 60115, USA

b

Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

cDepartment of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan d

National Center for Theoretical Sciences, Physics Division, Hsinchu 30043, Taiwan

a r t i c l e

i n f o

Article history:

Received 14 August 2012 Accepted 5 September 2012 by F. Peeters

Available online 13 September 2012 Keywords:

A. Semiconductors A. Topological insulators

a b s t r a c t

Recently, level crossings in the energy bands of crystals have been identified as a key signature for topological phase transitions. Using realistic models we show that the parameter space controlling the occurrence of level coincidences in energy bands has a much richer structure than anticipated previously. In particular, we identify robust level coincidences that cannot be removed by a small perturbation of the Hamiltonian compatible with the crystal symmetry. Different topological phases that are insulating in the bulk are then separated by a gapless (metallic) phase. We consider HgTe/CdTe quantum wells as a specific example.

&2012 Elsevier Ltd. All rights reserved.

Recently level crossings in the energy bands of crystals have become a subject of significant interest as they represent a key signature for topological phase transitions induced, e.g., by tuning the composition of an alloy or the thickness of a quasi-two-dimensional (2D) system[1–4]. For example, it was proposed[5]

and soon after confirmed experimentally [6,7] that HgTe/CdTe quantum wells (QWs) show a phase transition from spin Hall insulator to a quantum spin Hall regime when the lowest electron-like and the highest hole-like subbands cross at a critical QW width of  65 ˚A; see also[2,8–11]. Here we present a systematic study of level crossings and anticrossings in the subband structure of quasi-2D systems. We show that the parameter space characterizing level crossings has a much richer structure than previously antici-pated. In particular, we present examples for robust level coinci-dences that are preserved while the system parameters are varied within a finite range. Similar to the topological phase transitions characterizing the quantum Hall effect[12], the insulating Z2 topolo-gical phases[1]thus get separated by a gapless (metallic) phase. Such an additional phase was previously predicted in Ref.[13]. Yet it was found that this phase could occur only in 3D, but not in 2D. Also, it was not clear which systems would realize such a phase. Here we take HgTe/CdTe QWs as a realistic example, though many results are relevant also for other quasi-2D systems

Level crossings were studied already in the early days of quantum mechanics [14–16]. They occur, e.g., when atoms are placed in magnetic fields in the transition region between the

weak-field Zeeman effect and the high-field Paschen–Back effect. Also, they occur when molecules and solids are formed from isolated atoms. Hund[14] pointed out that adiabatic changes of 1D systems – unlike multi-dimensional systems – cannot give rise to level crossings. Von Neumann and Wigner[15]quantified how many parameters need to be varied for a level crossing. While levels of different symmetries (i.e., levels transforming according to different irreducible representations, IRs) may cross when a single parameter is varied, to achieve a level crossing among two levels of the same symmetry, it is in general necessary to vary three (two) independent parameters if the underlying eigenvalue problem is Hermitian (orthogonal). Subsequently, this problem was revisited by Herring[16]who found that the analysis by von Neumann and Wigner was not easily transferable to energy bands in a crystal due to the symmetry of the crystal potential. Similar to energy levels in finite systems, levels may coincide in periodic crystals if the levels have different symmetries. Of course, unless the crystal is invariant under inversion, this can occur only for high-symmetry lines or planes in the Brillouin zone (BZ), where the group of the wave vector is different from the trivial group C1. If at one end point k1of

a line of symmetry a band with symmetry

G

i is higher in energy

than the band with symmetry

G

j, while at the other end point

k2the order of

G

iand

G

jis reversed, these levels cross somewhere

in between k1 and k2. Herring classified a level crossing as

‘‘vanishingly improbable’’ if it disappeared upon an infinitesimal perturbation of the crystal potential compatible with all crystal symmetries. In that sense, a level coincidence at a high-symmetry point of the BZ such as the

G

point k¼ 0 becomes vanishingly improbable. For energy levels with the same symmetry, Herring derived several theorems characterizing the conditions under which level crossings may occur. In particular, he found that in Contents lists available atSciVerse ScienceDirect

journal homepage:www.elsevier.com/locate/ssc

Solid State Communications

0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.09.002

n

Corresponding author at: Department of Physics, Northern Illinois University, DeKalb, IL 60115, USA.

E-mail address: rwinkler@niu.edu (R. Winkler).

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the absence of inversion symmetry level crossings that are not vanishingly improbable may occur for isolated points k such that these crossings cannot be destroyed by an infinitesimal change in the crystal potential, but they occur at some point near k. Here we identify several examples for such robust level coincidences. This illustrates that level coincidences in energy bands can be qualita-tively different from level coincidences in other systems[15].

Recently, several studies focusing on topological phase transi-tions recognized the importance of symmetry for level crossings in energy bands[2,8–10]. Murakami et al.[2]studied the phase transition separating spin Hall insulators from the quantum spin Hall regime, focusing on generic low-symmetry configurations with and without inversion symmetry. They found that without inversion symmetry the phase transition is accompanied by a gap closing at points k that are not high-symmetry points. In inver-sion symmetric systems the gap closes only at points k ¼ G=2 where G is a reciprocal lattice vector. Here we show that level crossings in quasi-2D systems can be characterized by a multi-tude of scenarios, taking HgTe/CdTe quantum wells as a specific example for which it is known that the lowest electron-like and the highest hole-like subbands (anti)cross for a critical QW width of about 65 ˚A [5–7,17]. In most semiconductors with a zinc blende structure (point group Td) the s-antibonding orbitals form the conduction band (IR

G

6of Td), whereas the p-bonding orbitals form the valence band (

G

8and

G

7of Td). The curvature of the

G

6

band is thus positive whereas it is negative for the

G

8band. For

finite k, the four-fold degenerate

G

8states (effective spin j ¼ 3=2)

split into the so-called heavy hole (HH, mz¼73=2) and light hole

(LH, mz¼71=2) branches. In HgTe, the order of the

G

8 and

G

6

bands is reversed:

G

6is located below

G

8 and it has a negative

(hole-like) curvature, whereas

G

8 splits into an electron

ðmz¼71=2Þ and a hole ðmz¼73=2Þ branch[18]. HgTe and CdTe

can be combined to form a ternary alloy HgxCd1xTe, where the

fundamental gap E0between the

G

6and

G

8bands can be tuned

continuously from E0¼ þ1:6 eV in CdTe to E0¼ 0:3 eV in HgTe

with a gapless material for x  0:84 [18]. Tuning the material composition x thus allows one to overcome Herring’s conclusion

[16]that a degeneracy at k¼0 between two levels of different symmetries is, in general, vanishingly improbable.

Layers of HgTe and CdTe can also be grown epitaxially on top of each other to form QWs. At the interface the corresponding states need to be matched appropriately. The opposite signs of the effective mass inside and outside the well result in eigenstates localized at the interfaces[19]. We calculate these eigenstates as well as the corresponding subband dispersion EaðkÞ using a

realistic 8  8 multiband Hamiltonian H for the bulk bands

G

6,

G

8, and

G

7, which fully takes into account important details of

EaðkÞ such as anisotropy, nonparabolicity, HH–LH coupling, and

spin–orbit coupling both due to bulk inversion asymmetry (BIA) of the zinc blende structure of HgTe and CdTe as well as structure inversion asymmetry (SIA) of the confining potential V(z). For details concerning H and its numerical solution see Refs.[20,21]. In the following k ¼ ðkx,kyÞdenotes the 2D wave vector.

The symmetry group G of a QW and thus the allowed level crossings depend on the crystallographic orientation of the sur-face used to grow a QW [a (001) sursur-face being the most common in experiments]. It also depends on whether we have a system without or with BIA and/or SIA. The resulting point groups are summarized in Table 1. We show below that these different groups give rise to a rich parameter space for the occurrence of level coincidences. For a proper symmetry classification we project the eigenstates of H onto the IRs of the respective point group[22]. In the following, all IRs are labeled according to Koster et al.[23]. As spin–orbit coupling plays a crucial role for BIA and SIA [20] as well as for topological phase transitions [1–4], all IRs referred to in this work are double-group IRs. For comparison,

Table 1 also lists the point groups if the prevalent axial (or spherical) approximation is used for H. In this approximation, BIA is ignored and different surface orientations become indistinguishable.

First we neglect the small terms in H due to BIA so that the bulk Hamiltonian has the point group Oh. In the absence of SIA, a quasi-2D system grown on a (001) surface has the point group D4h

(which includes inversion) and all electron and hole states throughout the BZ are two-fold degenerate[22]. Subband edges k¼0 in a HgTe/CdTe QW as a function of well width w are shown inFig. 1(a). The HH states transform according to

G

67 of D4h. The electron-like and LH-like subbands transform according

to

G

77. As expected, the

G

67 and

G

77 subbands may cross as a

function of w.

In the presence of SIA we cannot classify the eigenstates anymore according to their behavior under parity. Without BIA the point group becomes C4v. HH states transform according to

G

6 of

C4vand electron- and LH-like states transform according to

G

7. The

level crossings depicted in Fig. 1(a) remain allowed in this case

[8,24].

The situation changes when taking into account BIA. Without SIA the point group becomes D2d. In this case, all subbands transform alternately according to the IRs

G

6 and

G

7 of D2d, irrespective of the dominant spinor components. In particular, both the highest HH state and the lowest conduction band state transform according to

G

6 of D2d so that around wC65 ˚A we obtain an anticrossing between these levels of about 2.9 meV (for k¼0), see Fig. 1(b) [8–10]. With both BIA and SIA the point group becomes C2v. Now we have only one double-group IR

G

5. Thus it follows readily that all subbands anticross as

a function of a continuous parameter such as the well width. While BIA opens a gap at k¼ 0, level coincidences remain possible for some ~ka0 when the well width w is tuned to a critical value ~w[2,16]. Considering a (001) surface with BIA, we find, indeed, that for each direction

f

of k ¼ ðk,

f

Þ, critical values

~

w and ~k exist that give rise to a band crossing. Thus we get a line in k space where the bands cross when w is varied within some finite range. This result holds for QWs on a (001) surface with BIA, without and with SIA (as studied experimentally in Refs.[6,7]). As an example,Fig. 2(a) shows ~k in the presence of a perpendicular electric field Ez¼100 kV=cm.

In general, three independent parameters must be tuned for a level coincidence in a quantum mechanical systems [15]if the underlying eigenvalue problem is Hermitian. While the multi-band Hamiltonian H used here [20] is likewise Hermitian (not orthogonal), only two independent parameters (w and k ¼ 9k9) are necessary to achieve the level degeneracy. We have here an example for the robustness of band coincidences under perturba-tions that was predicted by Herring [16] to occur in systems without a center of inversion (in multiples of four). It shows that level coincidences in energy bands can behave qualitatively different from level coincidences in other quantum mechanical systems[15]. We note that the band coincidences found here are not protected by symmetry in the sense that – unlike the other

Table 1

The point group of a QW for different growth directions starting from a bulk semiconductor with diamond structure (point group Oh) or zinc blende structure

(point group Td) for a system without (‘‘sym.’’) or with (‘‘asym.’’) SIA.

Bulk [001] [111] [110] [mmn] [0mn] [lmn] Axial appr.

Oh sym. D4h D3d D2h C2h C2h Ci D1h

asym. C4v C3v C2v Cs Cs C1 C1v

Td sym. D2d C3v C2v Cs C2 C1 D1h

asym. C2v C3v Cs Cs C1 C1 C1v

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cases discussed above – the group of ~k is the trivial group C1containing only the identity.

The situation is different for quasi-2D systems grown on a (111) surface. In the absence of BIA and SIA, the point group is D3d. HH states at k¼0 transform according to the complex conjugate IRs

G

þ 5 

G

þ 6 or

G

 5 

G



6, where  indicates that these

IRs must be combined due to time reversal symmetry. All other subband edges transform according to

G

47. In the presence of BIA

and/or SIA the point group becomes C3v. Then HH states trans-form according to the complex conjugate IRs

G

5

G

6.

Electron-like and LH-Electron-like states transform according to

G

4. Thus it follows

that on a (111) surface the HH states always cross the other states at k¼0 as a function of w [similar toFig. 1(a)]. The IRs for different geometries starting out from a (001) or (111) surface are summarized inTable 2.

Finally we consider quasi-2D states on a (110) surface. In the absence of BIA and SIA, the point group becomes D2h. Here, all

subbands transform alternately according to

G

þ 5 and

G



5 with the

topmost HH-like subband being

G

þ

5 and the lowest electron-like

subband being

G



5. A level crossing as a function of w is thus again

allowed at k¼0. In the presence of either BIA or SIA the symmetry is reduced to C2v. While the point group in both cases is the same

[25], we obtain a remarkable difference between these cases. With SIA the level crossing occurs for a line in k space, similar to the (001) surface, see Fig. 2(b). With BIA we obtain a level

Fig. 1. (Color online) Subband states in a symmetric HgTe/CdTe quantum well (for k¼0) as a function of well width w calculated with an 8  8 Hamiltonian (a) neglecting BIA (point group D4h) and (b) with BIA (D2d). States transforming according toG67of D4h(G6of D2d) are shown in red; states shown in black transform according toG77of

D4h(G7of D2d).

Fig. 2. Critical wave vectors ~k that give rise to a level coincidence in a HgTe/CdTe QW (a) on a (001) surface taking into account BIA (b) on a (110) surface neglecting BIA. In both cases a perpendicular field Ez¼100 kV=cm was assumed. In (a) the level coincidence requires a well width ~w ¼ 66:1 ˚A for ~kJ½110 and ~w ¼ 66:3 ˚A for ~kJ½110. In

(b) we have ~w ¼ 60:9 ˚A for ~kJ½001 and ~w ¼ 60:7 ˚A for ~kJ½110.

Table 2

Irreducible representations of quasi-2D states ðk ¼ 0Þ on a (001) and (111) surface, starting from a bulk semiconductor with diamond (point group Oh) or zinc blende

(point group Td) structure for a system without (‘‘sym.’’) or with (‘‘asym.’’)

structure inversion asymmetry.

Bulk (001) (111) Group c, LH HH Group c, LH HH Oh sym. D4h G77 G67 D3d G47 G57G67 asym. C4v G7 G6 C3v G4 G5G6 Td sym. D2d G7=6 G6=7 C3v G4 G5G6 asym. C2v G5 G5 C3v G4 G5G6

R. Winkler et al. / Solid State Communications 152 (2012) 2096–2099 2098

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crossing only for kJ½110 with ~k  0:0012 ˚A1 and ~w  62:5 ˚A, thus giving an example for the level crossings occurring for isolated points ~ka0 as discussed by Murakami et al.[2]. These examples illustrate that the occurrence of level crossings at either isolated points or along continuous lines in parameter space is not simply related with the system symmetry[25]. In the presence of both BIA and SIA (group Cs) we have the same situation as with BIA only, i.e., adding SIA changes the values of ~k and ~w, but we keep ~kJ½110.

In conclusion, we have shown that a rich parameter space characterizes the occurrence of level coincidences in the subband structure of quasi-2D systems. In particular, we have identified level coincidences for wave vectors ~ka0 that cannot be removed by a small perturbation of the Hamiltonian compatible with the QW symmetry[16]. Taking into account the full crystal symmetry of real materials is an important difference between the current analysis and previous work that considered only lattice periodi-city, inversion and time reversal symmetry. The full set of symmetries imposes additional constraints on the band Hamiltonian beyond the torus topology of the BZ that reflects the translational symmetry. These additional constraints generally reduce the number of parameters that are required to obtain level crossings

[16]so that robust level coincidences can be achieved even in quasi-2D systems. As quasi-2D systems can be designed and manipulated in various ways not available in 3D this opens new avenues for both experimental and theoretical research of topo-logically nontrivial materials.

As a specific example, we have considered HgTe/CdTe QWs, where a particular level crossing reflects a topological phase transition from spin Hall insulator to a quantum spin Hall regime

[5–7]. The robustness of the level coincidences found here implies that these phases, which are insulating in the bulk, are separated by a gapless phase similar to the metallic phases that separate the insulating quantum Hall phases[12]. While in HgTe/CdTe QWs the range of critical well widths ~w giving rise to the metallic phase is rather small (about 0.1 monolayers), we expect that future research will be able to identify materials showing larger parameter ranges that can be probed more easily in experiments. We note that our symmetry-based classification of level crossings is independent of specific numerical values of the band structure parameters entering the Hamiltonian H. Indeed, our findings are directly applicable also to other quasi-2D systems made of bulk semiconductors with a zinc blende or diamond structure such as hole subbands in GaAs/AlGaAs and SiGe quantum wells. In general, the k  p coupling between the LH1 (

G

þ

7 of D4h) and

HH2 (

G



6) subbands gives rise to an electron-like dispersion of the

LH1 subband for small wave vectors k [26]. If these subbands become (nearly) degenerate at k¼0, the coupling between these subbands becomes the dominant effect. This situation is described by the same effective Hamiltonian that characterizes the subspace consisting of the lowest electron and highest HH subband in a HgTe/CdTe QW [5]. It can be exploited if biaxial strain is used to tune the separation between the LH1 and HH2 subbands[27].

Acknowledgments

R.W. appreciates stimulating discussions with T. Hirahara, A. Hoffmann, L.W. Molenkamp, and S. Murakami. He thanks the Kavli Institute for Theoretical Physics China at the Chinese Academy of Sciences for hospitality and support during the early stage of this work. This work was supported by Taiwan NSC (Contract nos. 99-2112-M-009-006 and 100-2112-M-009-019) and a MOE-ATU Grant. Work at Argonne was supported by DOE BES under Contract no. DE-AC02-06CH11357.

References

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[6] M. K ¨onig, S. Wiedmann, C. Br ¨une, A. Roth, H. Buhmann, L.W. Molenkamp, X.-L. Qi, S.-C. Zhang, Science 318 (2007) 766.

[7] A. Roth, C. Br ¨une, H. Buhmann, L.W. Molenkamp, J. Maciejko, X.-L. Qi, S.-C. Zhang, Science 325 (2009) 294.

[8] M. K ¨onig, H. Buhmann, L.W. Molenkamp, T. Hughes, C.-X. Liu, X.-L. Qi, S.-C. Zhang, J. Phys. Soc. Jpn. 77 (2008) 031007.

[9] C. Liu, T.L. Hughes, X.-L. Qi, K. Wang, S.-C. Zhang, Phys. Rev. Lett. 100 (2008) 236601.

[10] X. Dai, T.L. Hughes, X.-L. Qi, Z. Fang, S.-C. Zhang, Phys. Rev. B 77 (2008) 125319.

[11] J.-W. Luo, A. Zunger, Phys. Rev. Lett. 105 (2010) 176805.

[12] D.J. Thouless, M. Kohmoto, M.P. Nightingale, M. den Nijs, Phys. Rev. Lett. 49 (1982) 405.

[13] S. Murakami, New J. Phys. 9 (2007) 356. [14] F. Hund, Z. Phys. 40 (1927) 742.

[15] J. von Neumann, E. Wigner, Phys. Z. 30 (1929) 467.

[16] C. Herring, On energy coincidences in the theory of Brillouin zones, Ph.D. Thesis, Princeton University, Princeton, NJ, 1937; C. Herring, Phys. Rev. 52 (1937) 365.

[17] A. Pfeuffer-Jeschke, Bandstruktur und Landau-Niveaus quecksilberhaltiger II-VI-Heterostrukturen, Ph.D. Thesis, University of W ¨urzburg, W ¨urzburg, Germany, 2000.

[18] R. Dornhaus, G. Nimtz, B. Schlicht, Narrow-Gap Semiconductors, Springer, Berlin, 1983.

[19] Y.R. Lin-Liu, L.J. Sham, Phys. Rev. B 32 (1985) 5561.

[20] R. Winkler, Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, Springer, Berlin, 2003.

[21] In our calculations for HgTe/CdTe QWs, we use the band parameters of E.G. Novik, et al., Phys. Rev. B 72 (2005) 035321. In theG8andG7bands, BIA

results in k linear terms proportional to Ck, see M. Cardona, et al., Phys. Rev.

Lett. 56 (1986) 2831. In the off-diagonal blocks of H coupling theG6with the

G8andG7bands we also have terms quadratic in k weighted by B8v7and B7v

[20]. We estimate Bþ 8vCB7v¼ 20 eV ˚A 2 and B 8v¼1 eV ˚A 2 .

[22] G.L. Bir, G.E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors, Wiley, New York, 1974.

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[25] 2D systems on a (110) surface with either BIA or SIA have the point group C2v. Yet with SIA the symmetry axis of C2vis perpendicular to the (110) plane,

whereas with BIA this axis is along the in-plane [001] axis. [26] D.A. Broido, L.J. Sham, Phys. Rev. B 31 (1985) 888.

[27] P. Voisin, C. Delalande, M. Voos, L.L. Chang, A. Segmuller, C.A. Chang, L. Esaki, Phys. Rev. B 30 (1984) 2276.

數據

Table 1 also lists the point groups if the prevalent axial (or spherical) approximation is used for H
Fig. 1. (Color online) Subband states in a symmetric HgTe/CdTe quantum well (for k¼0) as a function of well width w calculated with an 8  8 Hamiltonian (a) neglecting BIA (point group D 4h ) and (b) with BIA (D 2d )

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