Negative polarizability of 2D electrons in HgTe quantum well
V. Ya. Aleshkin, A. V. Germanenko, G. M. Minkov, A. A. Sherstobitov
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Negative polarizability of 2D electrons in HgTe quantum well
V. Ya. Aleshkin,
1, 2
A. V. Germanenko, G. M. Minkov,
3, 4 and A. A. Sherstobitov
3, 4 Institute for Physics of Microstructures RAS, 603087 Nizhny Novgorod, Russia Lobachevsky University of Nizhny Novgorod, 603950 Nizhny Novgorod, Russia School of Natural Sciences and Mathematics, Ural Federal University, 620002 Ekaterinburg, Russia M.N. Miheev Institute of Metal Physicsx of Ural Branch of Russian Academy of Sciences, 620108 Ekaterinburg, Russia (Dated: March 6, 2019)The polarizability of electrons occupying the lowest subband of spatial quantization inCdTe/Cd x Hg − x Te/CdTe quantum wells is calculated. It is shown that polarizability in the quan-tum well without cadmium is negative, i.e., the displacement of an electron in an electric field appliedperpendicularly to the quantum well plane is opposite to the force acting on it. The negative po-larizability of 2D electrons can reduce the dielectric constant of quantum wells by up to (10 − I. INTRODUCTION
Two-dimensional (2D) systems based on gapless semi-conductors HgTe are unique object. Mercury telluride issemiconductor with inverted ordering of Γ and Γ bands.The Γ band, which is the conduction band in conven-tional semiconductor, is located in HgTe lower in the en-ergy than the Γ band. Unusual positioning of the bandsleads to crucial features of the electron spectrum underspace confinement [1–4]. So at some critical width of theHgTe/CdTe quantum well (QW), d = d c ≃ . d > d c , the lowest electron subband ismainly formed from the Γ states at small quasimomen-tum value ( k ), while the Γ states form the hole statesin the depth of the valence band. Such a band struc-ture is referred to as inverted structure. At d < d c , theband ordering is normal; the highest valence subband atzero quasimomentum is formed from the heavy hole Γ states, while the lowest electron subband is formed bothfrom the Γ states and from the light Γ states.These peculiarities manifest themselves in a new andinteresting physical phenomena as exemplified by topo-logical states arising in such systems (see, review paper[6]). In the present paper we report one further anoma-lous effect originated from the specific of the electronspectrum in the HgTe QWs.It is well known that electrons occupying the lowestsubband of spatial quantization in a quantum well ofsemiconductors with a normal band structure possesspositive polarizability with respect to an electric fieldapplying along the normal to the quantum well. For in-stance, an electron in a GaAs quantum well shifts in thedirection of force under influence of an electric field. Re-call the polarizability ( α ) determines the response of abound system to external field. It is the coupling coef-ficient between the dipole moment P arising in the elec-tric field E ; P = α E . Polarizability is often used todescribe the behavior of molecules and atoms in an elec-tric field [7]. But it can also be used to describe electronsin quantum wells [8]. As will be shown below, electronsin the lowest subband in HgTe/CdTe quantum well sub- ene r g y ( e V ) k (nm -1 ) (001)-QW(013)-QW Figure 1. The E vs k dependences for the (013)- and (001)-HgTe/CdTe QWs of 12 nm width. The k vector is directedalong [100]. The solid and dashed curves are two branchessplit due to the presence of electric field and interface inversionasymmetry. jected to an electric field are displaced in the directionopposite to the acting force. This means that 2D elec-tron gas in the HgTe/CdTe wells has a negative polar-izability. This effect is predicted in the HgTe QWs ofdifferent width both with inverted and with normal en-ergy spectrum. Therewith, the polarizability increases inthe Cd x Hg − x Te/CdTe quantum wells with the increas-ing cadmium content and becomes positive at x > . − x Cd c Te becomesnormal
II. RESULTS AND DISCUSSION
To calculate the polarizability, the ensemble averagedisplacement of the mean electron coordinate was calcu-lated. The calculations were carried out in the frameworkof the Kane model taking into account the deformationeffects and the absence of inversion symmetry of the in-terfaces forming the quantum well. As known an addi- -10 -5 0 5 10
E E = 50 kV/cm E=0 z (nm) (b)
HgTe/CdTe ene r g y ( e V ) -10 -5 0 5 100.00.10.20.3 E = 50 kV/cm E=0 || ( n m - ) z (nm) (a) GaAs/AlAs E Figure 2. (Color online) The dependence | ψ ( z ) | in the pres-ence and absence of electric field for the GaAs/AlAs (a) andHgTe/CdTe (b) quantum wells, k = 0. The run of the conduc-tion band bottom in the presence of electric field are shownas well. tional term to the Hamiltonian describing symmetry low-ering on heterointerfaces was suggested by E.L.Ivchenko(see Ref. [9]). This term is given in the Appendix. Theparameters of materials and the method of solving theSchr¨odinger equation were taken from [10]. As an exam-ple, Fig. 1 shows the electron spectrum in HgTe/CdTeQW of 12 nm width grown on the (001) and (013) planesin the presence of a uniform electric field E = 10 kV/cmapplied along the normal to the quantum well, which co-incides with z -axis. The interface inversion asymmetryand the electric field lead to a small spin splitting of thesubbands. Note the conduction band spectrum is almostisotropic and close to each other for both substrate ori-entations.In order to demonstrate an unusual response of elec-trons in the lowest subband in HgTe/CdTe QW to theelectric field, let us compare this effect in GaAs and HgTebased quantum wells. Figure 2 shows the probabilitydensity | ψ ( z ) | for the electron on the bottom of con-duction band in GaAs/AlAs and HgTe/CdTe QWs inthe presence and absence of the electric field. As clearlyseen the mean electron coordinate shifts in opposite di-rections in GaAs/AlAs and HgTe/CdTe quantum wellswhen the electric field is applied. In the GaAs/AlAsquantum well the electron density shifts against the elec-tric field [Fig. 2(a)], i.e., in the direction of the force − e E ( e is the elementary charge) which exerts the electron. Inthe HgTe/CdTe quantum well, the electron shifts againstthe force [Fig. 2(b)] that is in contradiction with intuitiveexpectations.Our calculations show that the shift of electron densityin the HgTe QWs is strongly dependent on the quasimo-mentum vector. This is illustrated by Fig. 3. It showsthe k dependences of the change in the electron meancoordinate calculated as δ h z ± k i = h z ± k ( E ) i − h z ± k (0) i , (1) -0.2 0.0 0.2-0.20.00.2 k [100] (nm -1 ) k [ ] ( n m - ) -0.2 0.0 0.2 k [100] (nm -1 ) -1.5-1.2-0.9-0.6-0.30.00.30.60.91.21.5
23 nm − (the left panel in Fig. 4), while for k & .
23 nm − it demonstrate normal behavior charac-terized by δ h z k i < ε ) of the quantum well. Ina bulk material, the dielectric constant can be defined asthe ratio of the electric induction to the electric field. Theelectric induction is sum of the electric field and polariza-tion of a unit volume multiplied by 4 π [11]. In a quantumwell additional polarization occurs due to electron pres-ence in the conduction band. Average over quantum wellvalue of this polarization is αEn/d , where E is the elec-tric field value, n is the sheet electron concentration, d is the quantum well width, α is the electron polarizabil-ity. Therefore we obtain the following expression for theaverage dielectric constant ε = ε i + 4 παnd , (3)where ε i is the dielectric constant of the quantum wellwithout electrons (equal to 20 . . α = − eδ hh z ii /E. (4)Here, δ hh z ii is change in the electron mean coordinate,Eq. (2), averaged over the ensemble of 2D electrons.Fig. 5 shows the electron density dependences of thepolarizability contribution 4 παn/d for (001)- and (013)-HgTe/CdTe QWs of two different widths, d = 12 nm and4 nm, which correspond to the inverted and normal en-ergy spectrum, respectively. As seen the electrons havea negative polarizability in the quantum wells both withthe inverted and with the normal band structure. It isalso seen that the absolute value of the electron contri-bution to the dielectric constant of 12 nm QW exceedsthat for the 4 nm quantum well. Note, the calculationsgives approximately the same results when the barriersare the solid solution Cd . Hg . Te, i.e., the change ofthe height of the barriers within reasonable limits doesnot affect the polarizability.In order to clarify the physical cause of negative elec-tron polarizability, we note three important arguments.The first argument is that the wave functions near k = 0in 12 nm and 4 nm quantum wells are formed mainly bythe states of the Γ band. The proportion of these statesin the electron wave function decreases with increasingwave vector, similarly to a decrease in the polarizationmagnitude (see Fig. 4). The second argument is that theelectron polarizability is positive in such quantum wells inwhich the electron wave function is formed mainly by thestates of the Γ band. For examples, 1 nm-HgTe/CdTequantum well or quantum wells with a cadmium frac-tion greater than 0 .
168 can be given. The third argu-ment is that the Γ -band states form the electron wave -3-2-10123 / d n (cm -2 ) d=12 nm, (013) d=12 nm, (001) d=4 nm, (013) d=4 nm, (001) Figure 5. (Color online) The electron density dependence ofthe electron polarizability contribution to the dielectric con-stant of the quantum wells with d = 4 . functions of the valence band in the GaAs/AlAs quan-tum wells mentioned above. The electron polarizabilityof the valence band states in such quantum wells is neg-ative , which corresponds to a positive hole polarizability.Taking into account these arguments, we can concludethat the cause of the negative electron polarizability inHgTe/CdTe QWs is the presence of a significant fractionof the Γ band states in the electron wave function.If the QW heterostructure is doped asymmetrically,an intrinsic electric field exists in the well even in theabsence of an external field due to the spatial separationof electrons and ionized donors. In order to estimate theeffect in this case, we have calculated the polarizability ofthe electron gas in the quantum well subjected to uniformfield of 50 kV/cm. It turns out that the n dependence ofthe electron polarizability contribution to the dielectricconstant remains qualitatively the same, but its absolutevalue is approximately 10 percent less as compared withthe rectangular quantum well. III. CONCLUSION
Electrons in the conduction band of the HgTe quan-tum well exhibit an amazing behavior in an electric fieldnormal to the quantum well plane. The electron den-sity is shifted in the direction opposite to the actingforce. As a result, the electron polarizability of 2D elec-tron gas is negative over a wide range of electron densi-ties. The negative polarizability of a 2D electrons reducesthe dielectric constant of the rectangular HgTe quantumwell on the maximal value of about (10 −
15) percent at n ≃ cm − . The effect weakens with increasing Cdcontent in the quantum well and the polarizability be-comes positive at x ≃ .
168 when the bulk Cd x Hg − x Tespectrum transforms from the gapless with inverted bandstructure to the spectrum with open gap and normalband ordering. The anomalous negative polarizabilitycan be detected through the Kerr and/or Pockels effects,it can manifests itself in the capacitance experiments per-formed on the gated heterostructures with HgTe/CdTequantum wells.
ACKNOWLEDGMENTS
The work has been supported in part by the RussianFoundation for Basic Research (Grant
Appendix: The Ivchenko term
We use the Ivchenko term responsible for the interfaceinversion asymmetry in the following form H I = ± γ δ ( z ± d/ × − i i √
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