Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition

The band structure and dependences of the intrinsic concentration in the mercury-cadmium-telluride (MCT) Hg₀.₃₂Cd₀.₆₈Te/Hg₀ Cd₀ Te/Hg₀.₃₂Cd₀.₆₈Te quantum wells in the framework of the 8x8 k.p envelope function method on the well width L and composition x were calculated. Modeling of the energy sp...

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Datum:	2014
Hauptverfasser:	Melezhik, E.O., Gumenjuk-Sichevska, J.V., Dvoretskii, S.A.
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Sprache:	English
Veröffentlicht:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2014
Schriftenreihe:	Semiconductor Physics Quantum Electronics & Optoelectronics
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spelling	irk-123456789-1183762017-05-31T03:07:20Z Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition Melezhik, E.O. Gumenjuk-Sichevska, J.V. Dvoretskii, S.A. The band structure and dependences of the intrinsic concentration in the mercury-cadmium-telluride (MCT) Hg₀.₃₂Cd₀.₆₈Te/Hg₀ Cd₀ Te/Hg₀.₃₂Cd₀.₆₈Te quantum wells in the framework of the 8x8 k.p envelope function method on the well width L and composition x were calculated. Modeling of the energy spectra showed that the intrinsic concentration can vary about an order of magnitude with variation of the well width and chemical composition in the range of x < 0.16 and well width L < 20 nm at the liquid nitrogen temperature. These strong variations of the carrier concentration are caused by the insulator-semimetal topological transition. 2014 Article Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition / E.O. Melezhik, J.V. Gumenjuk-Sichevska, S.A. Dvoretskii // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 2. — С. 179-183. — Бібліогр.: 7 назв. — англ. 1560-8034 PACS 73.21.Fg, 84.40.-x http://dspace.nbuv.gov.ua/handle/123456789/118376 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The band structure and dependences of the intrinsic concentration in the mercury-cadmium-telluride (MCT) Hg₀.₃₂Cd₀.₆₈Te/Hg₀ Cd₀ Te/Hg₀.₃₂Cd₀.₆₈Te quantum wells in the framework of the 8x8 k.p envelope function method on the well width L and composition x were calculated. Modeling of the energy spectra showed that the intrinsic concentration can vary about an order of magnitude with variation of the well width and chemical composition in the range of x < 0.16 and well width L < 20 nm at the liquid nitrogen temperature. These strong variations of the carrier concentration are caused by the insulator-semimetal topological transition.
format	Article
author	Melezhik, E.O. Gumenjuk-Sichevska, J.V. Dvoretskii, S.A.
spellingShingle	Melezhik, E.O. Gumenjuk-Sichevska, J.V. Dvoretskii, S.A. Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet	Melezhik, E.O. Gumenjuk-Sichevska, J.V. Dvoretskii, S.A.
author_sort	Melezhik, E.O.
title	Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition
title_short	Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition
title_full	Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition
title_fullStr	Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition
title_full_unstemmed	Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition
title_sort	intrinsic concentration dependences in the hgcdte quantum well in the range of the insulator-semimetal topological transition
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate	2014
url	http://dspace.nbuv.gov.ua/handle/123456789/118376
citation_txt	Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition / E.O. Melezhik, J.V. Gumenjuk-Sichevska, S.A. Dvoretskii // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2014. — Т. 17, № 2. — С. 179-183. — Бібліогр.: 7 назв. — англ.
series	Semiconductor Physics Quantum Electronics & Optoelectronics
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first_indexed	2025-07-08T13:52:12Z
last_indexed	2025-07-08T13:52:12Z
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fulltext	Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 179-183. © 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 179 PACS 73.21.Fg, 84.40.-x Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition E.O. Melezhik1, J.V. Gumenjuk-Sichevska2, S.A. Dvoretskii3 1V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine 41, prospect Nauky, 03028 Kyiv, Ukraine Corresponding author e-mail: emelezhik@gmail.com; e-mail gumenjuk@gmail.com 2Institute of Semiconductor Physics of SB RAS, 13, pr. Lavrentieva, 630090 Novosibirsk, Russia Abstract. The band structure and dependences of the intrinsic concentration in the mercury-cadmium-telluride (MCT) Hg0.32Cd0.68Te/ TeCdHg xx1 /Hg0.32Cd0.68Te quantum wells in the framework of the 88 k.p envelope function method on the well width L and composition x were calculated. Modeling of the energy spectra showed that the intrinsic concentration can vary about an order of magnitude with variation of the well width and chemical composition in the range of x  0.16 and well width L  20 nm at the liquid nitrogen temperature. These strong variations of the carrier concentration are caused by the insulator-semimetal topological transition. Keywords: mercury cadmium telluride HgCdTe, quantum well, insulator-semimetal topological transition, hot electron bolometer. Manuscript received 17.01.14; revised version received 25.04.14; accepted for publication 12.06.14; published online 30.06.14. 1. Introduction Recent studies [1] have shown that mercury cadmium telluride (MCT) can be promising for the development of THz/sub-THz detectors operating at moderate cooling. Whereas MCT bolometer detector noise equivalent power (NEP) is sufficiently dependent on the intrinsic concentration value [2]. We aimed to find proper conditions to reach better NEP conditions for the MCT bolometric structure based on quantum well. The purpose of our investigation was to outline the chemical composition of QW and its width in the insulator- semimetal topological transition region to choose their proper values for growing such structures as THz/sub- THz detectors. The insulator-semimetal topological transition in CdTe/HgTe/CdTe quantum wells takes place at the well thickness L  8 nm and temperature T = 77 K [3]. At L < 8 nm, the band order in the well and inside barriers is equal, and at L > 8 nm the band structure of the well is inverted and the conduction band has Γ8 symmetry. In our work, we investigate the topological transition conditions for the nonzero composition parameter x in the well and inside barriers. Calculations of the band structure and intrinsic concentrations were made at the temperature of liquid nitrogen (T = 77 K) where these structures can be efficiently applied as THz/sub-THz detectors. The energy spectrum of Hg0.32Cd0.68Te/Hg1–x CdxTe quantum wells was modeled. The composition x  0.68 of the barrier layer was chosen, as it is often used in real heterostructures [4-6] to minimize misfit strains between the well and barrier layers. Moreover, in real structures there can be two sources of strains: from the lattice misfit between the well and barrier layer as well as from the lattice misfit between the barrier and substrate. These strains are often of opposite sign, and the resulting strain in the well is small. Since the strains can be eliminated technologically, they are neglected in numerical modeling. To account precisely these complex strains is a rather complicated procedure. The calculations of energy spectra are provided in the framework of the envelope functions approach, when the carrier wave-function is expanded on the basis of Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 179-183. © 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 180 eight Bloch band-edge (in-plane k = 0) functions [7]. The system is assumed to be periodical, the barrier well width is chosen to correspond to the isolated quantum well. In the calculations, there was used the barrier of 100-nm width, which is good approximation for the infinite width barrier. The calculations were made for the [001] oriented quantum well, thus, the z-axis coincides with the growth direction of the well. The carrier wave-function can be written as:      rurFr n n n , (1) where Fn(r) are the envelope functions, n varies from 1 up to 8. Assuming translation invariance in the plane perpendicular to the growth direction (z-axis), Fn(r) can be represented as:       zfykxkirF nyxn  exp . (2) The functions fn(z) in the well (A) and inside the barrier (B) are expanded on the full basis of the Legendre polynomials:                  L z Lc L z Lczf i i i i na i i i na A n 22 0 11 , (3)                    B i i i i nb B i i i nb B n L z Lc L z Lczf 22 0 11 . where Li are the Legendre polynomials of i-th order. The number i0 of Legendre polynomials in the expansion (3), necessary to obtain accurate solutions, is taken to be 16 [7]. Energy levels and envelope-function coefficients can be found from the simultaneous solution of the system of coupled differential equations – Hamiltonian equation and boundary conditions system [7]. The Hamiltonian of the system in the 8-band basis is written as [7]: where 222 \|\| yx kkk  is the in-plane momentum, yx ikkk  , z i kz    ,       zzc kFkkF m zET 1212 2 2 \|\| 0 2   ,    zzv kkk m zEU 1 2 \|\|1 0 2 2   ,  zz kkk m V 2 2 \|\|1 0 2 2 2   ,  22 0 2 33 2   kk m R  ,     zz kkk m S ,,3 2 3 0 2    ,            zz kkk m S , 3 1 ,3 2 ~ 3 0 2 ,  zkk m C , 0 2    . Here, [A, B] = AB – BA is the usual commutator, while {A, B} = AB + BA is the anti-commutator for two operators A and B; P is the Kane matrix element, P, as a rule, is taken to be equal in the well and inside the barrier (see e.g. [7]). Ec(z) and Ev(z) are the respective edges of the conduction and valence bands; Δ is the spin-orbit splitting. The Luttinger parameters γ1, γ2, γ3, κ and F describe coupling with remote bands, the additional parameters μ and  are defined as   2/23  ,   2/23  [7]. The Luttinger parameters of HgTe and CdTe were taken from [7], they were assumed to vary linearly with concentration. As a rule, they are assumed to be temperature-independent.                                                                   UCSVSRPkPk CURSVSPkPk SRVUSRPk VSSVUCRPkPk SVRCVUSPkPk RSRSVUPk PkPkPkPkPkT PkPkPkPkPkT H z z z z zz zz †††† † †† ††† † 2 1 2 ~ 2 3 2 3 1 3 1 2 ~ 2 3 2 2 1 3 1 3 1 2 1 20 2 1 0 2 ~ 2 3 3 2 6 1 ~ 3 2 2 6 1 3 2 2 2 1 00 2 1 3 1 3 1 2 1 3 2 6 1 00 3 1 3 1 0 6 1 3 2 2 1 0 (4) Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 179-183. © 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 181 Boundary conditions appear due to two physical requirements. First, the probability density to find electron near the defined coordinate should be continuous at the interface. It imposes the requirement of continuity of the wave-function (1). Band-edge Bloch functions are linearly independent, in-plane momentum is conserved when crossing the interface. Thus, the requirement of continuity of the wave-function (1) is equal to the requirement of continuity of each of the envelope wave-functions (3). Second, the Luttinger parameters γ1, γ2, γ3, κ and F are different in the well and inside the barrier. They are assumed to change its step-like character at the interface [7]. In the Hamiltonian (4), these parameters are used inside the z  operator. Consequently, at the interface, one can face the situation when the functions inside z  also change step-like character. Applying z  to the step-like function leads to the infinite Hamiltonian matrix elements at the interface, which is certainly unphysical. Thus, the second set of the boundary conditions requires continuity of all the equations inside the z  operators. It can be written mathematically as the requirement of continuity of the integrals from each element of Eq. (4) over the z-coordinate. It is written as the requirement of continuity of the matrix product nfD  at the interface: where elements containing P vanish because P is taken to be the same in the well and inside the barriers. Here,   z F m t    12 2 0 2 , zm u    1 0 2 2  , zm v    2 0 2 ,     ki m s 3 0 2 3 2  ,     ki m s 3 0 2 3 2  ,     ki m s 3/3 2 ~ 3 0 2 ,     ki m s 3/3 2 ~ 3 0 2 ,  ki m c 0 2 . 2. Results The intrinsic concentrations and Fermi level were calculated by fitting the concentrations of electrons and heavy holes. The calculations are separated in two parts: (i) dependence of the carrier concentration on the well width and (ii) dependence of the carrier concentration on the composition x of the well.                                                                ucsvsPi cusvsPi svus vssvucPi svcvusPi ssvu t t D 2 1 2~ 2 3 0 3 1 0 0~ 2 3 2 2 1 0 3 1 2 1 00000 2~ 2 3 0 3 2 0 ~ 2 3 200 3 2 0 2 1 0000 0000000 0000000 (5) Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 179-183. © 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 182 Fig. 3. Dependence of the energy spectrum and Fermi level of QW with the width L = 20 nm on the composition x. Curves on the graph are marked as follows: Fermi level – (1), E1 level – (2), E0 level – (3), HH0 level – (4), HH1 level – (5), HH2 level – (6), HH3 level – (7), LH0 level – (8). Fig. 1. Dependence of energy spectrum and Fermi level of QW with the composition x = 0.06 on the well width L. Curves on the graph are marked as follows: Fermi level (1), E1 level – (2), E0 level – (3), HH0 level – (4), HH1 level – (5), HH2 level – (6), HH3 level – (7). The dependence of the carrier concentration on L (Fig. 2) is calculated for the composition of the well x = 0.06. For this composition, the critical thickness of the insulator-semimetal topological transition is around L = 12 nm (see Fig. 1) and its influence on the intrinsic concentration can be examined. From Fig. 2, one can see that the localized intrinsic concentration grows along with the well width. This growth is explained by the presence of the topological phase transition, which opens the band gap for widths less than 12 nm. For larger well widths, the system is semimetal and the band gap is absent, thus, the intrinsic concentration demonstrates plateau at larger well widths. Another important feature is that the local minimum of the Fermi level is situated near the topological transition. One should note that for higher concentrations like x = 0.12, the critical thickness is around L ~ 20 nm, and the growing section in Fig. 2 will be shifted to higher well widths. 8 10 12 14 16 18 20 0 5 10 15 20 25 30 35 E le ct ro n co nc en tr at io n, 1 015 c m -3 Well width L, nm Fig. 2. Dependence of intrinsic concentration on the well width L in QW with the composition x = 0.06. Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 179-183. © 2014, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 183 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0 10 20 30 40 50 60 E le ct ro n co nc en tr at io n n, 1 015 c m -3 QW composition x Fig. 4. Dependence of the intrinsic concentration on the QW composition x, well width L = 20 nm. The dependence of the electron concentration on the QW composition x is presented in Fig. 4. The corresponding energy spectrum is presented in Fig. 3. It should be noted that composition changes also lead to the topological phase transition. For x = 0, the critical thickness for the Hg0.32Cd0.68Te/ Hg1–x CdxTe quantum well is L = 6.7 nm, while for higher concentrations, this thickness grows and approaches more than L  20 nm at x = 0.12, which should be accounted when growing MCT QWs for THz bolometer detectors. This effect can be explained by stronger influence of the semiconducting barrier layers, which enhances when the well band gap decreases. The carrier concentration in the well (Fig. 4) decreases as the composition grows. This dependence is almost linear. 3. Conclusions The maximal intrinsic concentration of carriers of the order of n ~ 51016 cm–3 can be achieved for compositions x close to 0 and well widths that are greater than the critical thickness of the topological transition. For bolometer type detectors, the optimal intrinsic concentration seems to be one order lower [2]. It should be noted that the increase of the composition x leads to decrease of the electron effective mass and, thus, to increase of the electron mobility, but the carrier concentration becomes smaller. Thus, for n- type samples the carrier conductivity will be influenced by two concurring trends which should be taken into account when designing QWs for THz/sub-THz detectors. References 1. F. Sizov, V. Petriakov, V. Zabudsky, D. Krasilnikov, M. Smoliy, and S. Dvoretski, Millimeter-wave hybrid un-cooled narrow-gap hot- carrier and Schottky diodes direct detectors // Appl. Phys. Lett. 101, 082108 (2012). 2. V. Dobrovolski, F. Sizov, THz/sub-THz bolometer based on electron heating in a semiconductor waveguide // Optoelectronics Review, 18, p. 250- 258 (2010). 3. Jun Li and Kai Chang, Electric field driven quantum phase transition between band insulator and topological insulator // Appl. Phys. Lett. 95, 222110 (2009). 4. M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, Xiao-Liang Qi, Shou- Cheng Zhang, Quantum spin Hall insulator state in HgTe quantum wells // Science, 318 (5851), p. 766- 770 (2007). 5. B.A. Bernevig, T.L. Hughes, Shou-Cheng Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells // Science, 314 (5806), p. 1757-1761 (2006). 6. V. Latussek, Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades “Elektronische Zustande in Typ-III- Halbleiterheterostrukturen“, Julius-Maximilians- Universitat, Wurzburg, 2004. 7. E.G. Novik, A. Pfeuffer-Jeschke, T. Jungwirth, V. Latussek, C.R. Becker, G. Landwehr, H. Buhmann, and L.W. Molenkamp, Band structure of semimagnetic Hg1–yMnyTe quantum wells // Phys. Rev. B, 72, 035321 (2005). Semiconductor Physics, Quantum Electronics & Optoelectronics, 2014. V. 17, N 2. P. 179-183. PACS 73.21.Fg, 84.40.-x Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition E.O. Melezhik1, J.V. Gumenjuk-Sichevska2, S.A. Dvoretskii3 1V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine 41, prospect Nauky, 03028 Kyiv, Ukraine Corresponding author e-mail: emelezhik@gmail.com; e-mail gumenjuk@gmail.com 2Institute of Semiconductor Physics of SB RAS, 13, pr. Lavrentieva, 630090 Novosibirsk, Russia Abstract. The band structure and dependences of the intrinsic concentration in the mercury-cadmium-telluride (MCT) Hg0.32Cd0.68Te/ Te Cd Hg x x 1 - /Hg0.32Cd0.68Te quantum wells in the framework of the 8(8 k.p envelope function method on the well width L and composition x were calculated. Modeling of the energy spectra showed that the intrinsic concentration can vary about an order of magnitude with variation of the well width and chemical composition in the range of x ( 0.16 and well width L ( 20 nm at the liquid nitrogen temperature. These strong variations of the carrier concentration are caused by the insulator-semimetal topological transition. Keywords: mercury cadmium telluride HgCdTe, quantum well, insulator-semimetal topological transition, hot electron bolometer. Manuscript received 17.01.14; revised version received 25.04.14; accepted for publication 12.06.14; published online 30.06.14. 1. Introduction Recent studies [1] have shown that mercury cadmium telluride (MCT) can be promising for the development of THz/sub-THz detectors operating at moderate cooling. Whereas MCT bolometer detector noise equivalent power (NEP) is sufficiently dependent on the intrinsic concentration value [2]. We aimed to find proper conditions to reach better NEP conditions for the MCT bolometric structure based on quantum well. The purpose of our investigation was to outline the chemical composition of QW and its width in the insulator-semimetal topological transition region to choose their proper values for growing such structures as THz/sub-THz detectors. The insulator-semimetal topological transition in CdTe/HgTe/CdTe quantum wells takes place at the well thickness L ( 8 nm and temperature T = 77 K [3]. At L < 8 nm, the band order in the well and inside barriers is equal, and at L > 8 nm the band structure of the well is inverted and the conduction band has Γ8 symmetry. In our work, we investigate the topological transition conditions for the nonzero composition parameter x in the well and inside barriers. Calculations of the band structure and intrinsic concentrations were made at the temperature of liquid nitrogen (T = 77 K) where these structures can be efficiently applied as THz/sub-THz detectors. The energy spectrum of Hg0.32Cd0.68Te/Hg1–x CdxTe quantum wells was modeled. The composition x ( 0.68 of the barrier layer was chosen, as it is often used in real heterostructures [4-6] to minimize misfit strains between the well and barrier layers. Moreover, in real structures there can be two sources of strains: from the lattice misfit between the well and barrier layer as well as from the lattice misfit between the barrier and substrate. These strains are often of opposite sign, and the resulting strain in the well is small. Since the strains can be eliminated technologically, they are neglected in numerical modeling. To account precisely these complex strains is a rather complicated procedure. The calculations of energy spectra are provided in the framework of the envelope functions approach, when the carrier wave-function is expanded on the basis of eight Bloch band-edge (in-plane k = 0) functions [7]. The system is assumed to be periodical, the barrier well width is chosen to correspond to the isolated quantum well. In the calculations, there was used the barrier of 100-nm width, which is good approximation for the infinite width barrier. The calculations were made for the [001] oriented quantum well, thus, the z-axis coincides with the growth direction of the well. The carrier wave-function can be written as: ( ) ( ) ( ) r u r F r n n n å = y , (1) where Fn(r) are the envelope functions, n varies from 1 up to 8. Assuming translation invariance in the plane perpendicular to the growth direction (z-axis), Fn(r) can be represented as: ( ) ( ) [ ] ( ) z f y k x k i r F n y x n + = exp . (2) The functions fn(z) in the well (A) and inside the barrier (B) are expanded on the full basis of the Legendre polynomials: ( ) ÷ ø ö ç è æ » ÷ ø ö ç è æ = å å = ¥ = L z L c L z L c z f i i i i na i i i na A n 2 2 0 1 1 , (3) ( ) ÷ ÷ ø ö ç ç è æ » ÷ ÷ ø ö ç ç è æ = å å = ¥ = B i i i i nb B i i i nb B n L z L c L z L c z f 2 2 0 1 1 . where Li are the Legendre polynomials of i-th order. The number i0 of Legendre polynomials in the expansion (3), necessary to obtain accurate solutions, is taken to be 16 [7]. Energy levels and envelope-function coefficients can be found from the simultaneous solution of the system of coupled differential equations – Hamiltonian equation and boundary conditions system [7]. The Hamiltonian of the system in the 8-band basis is written as [7]: where is the in-plane momentum, y x ik k k ± = ± , z i k z ¶ ¶ - = , ( ) ( ) ( ) [ ] z z c k F k k F m z E T 1 2 1 2 2 2 \|\| 0 2 + + + + = h QUOTE , ( ) ( ) z z v k k k m z E U 1 2 \|\| 1 0 2 2 g + g - = h QUOTE , ( ) z z k k k m V 2 2 \|\| 1 0 2 2 2 g - g - = h , ( ) 2 2 0 2 3 3 2 - + g - m - = k k m R h , { } [ ] ( ) z z k k k m S , , 3 2 3 0 2 k + g - = ± ± h QUOTE , { } [ ] ÷ ø ö ç è æ k - g - = ± ± z z k k k m S , 3 1 , 3 2 ~ 3 0 2 h QUOTE , [ ] z k k m C , 0 2 k = - h . 8 10 12 14 16 18 20 0 5 10 15 20 25 30 35 Electron concentration, 10 15 cm -3 Well width L, nm Here, [A, B] = AB – BA is the usual commutator, while {A, B} = AB + BA is the anti-commutator for two operators A and B; P is the Kane matrix element, P, as a rule, is taken to be equal in the well and inside the barrier (see e.g. [7]). Ec(z) and F describe coupling with remote bands, the additional parameters μ and and Ev(z) are the respective edges of the conduction and valence bands; Δ is the spin-orbit splitting. The Luttinger parameters γ1, γ2, γ3, κ are defined as , ( ) 2 / 2 3 g + g = g [7]. The Luttinger parameters of HgTe and CdTe were taken from [7], they were assumed to vary linearly with concentration. As a rule, they are assumed to be temperature-independent. Boundary conditions appear due to two physical requirements. First, the probability density to find electron near the defined coordinate should be continuous at the interface. It imposes the requirement of continuity of the wave-function (1). Band-edge Bloch functions are linearly independent, in-plane momentum is conserved when crossing the interface. Thus, the requirement of continuity of the wave-function (1) is equal to the requirement of continuity of each of the envelope wave-functions (3). Second, the Luttinger parameters γ1, γ2, γ3, κz ¶ ¶ to the step-like function leads to the infinite Hamiltonian matrix elements at the interface, which is certainly unphysical. z ¶ ¶ also change step-like character. Applying QUOTE operator. Consequently, at the interface, one can face the situation when the functions inside and F are different in the well and inside the barrier. They are assumed to change its step-like character at the interface [7]. In the Hamiltonian (4), these parameters are used inside the ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ø ö ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç è æ ¢ - ¢ - - ¢ - ¢ + - - ¢ - - - - ¢ - - - + = * - + - + + + + - * - + - - u c s v s P i c u s v s P i s v u s v s s v u c P i s v c v u s P i s s v u t t D 2 1 2 ~ 2 3 0 3 1 0 0 ~ 2 3 2 2 1 0 3 1 2 1 0 0 0 0 0 2 ~ 2 3 0 3 2 0 ~ 2 3 2 0 0 3 2 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Thus, the second set of the boundary conditions requires continuity of all the equations inside the QUOTE at the interface: z ¶ ¶ operators. It can be written mathematically as the requirement of continuity of the integrals from each element of Eq. (4) over the z-coordinate. It is written as the requirement of continuity of the matrix product where elements containing P vanish because P is taken to be the same in the well and inside the barriers. Here, ( ) z F m t ¶ ¶ + - = 1 2 2 0 2 h , z m u ¶ ¶ g = 1 0 2 2 h , z m v ¶ ¶ g - = 2 0 2 h , ( ) ± ± k - g = k i m s 3 0 2 3 2 h , ( ) ± ± k + g - = ¢ k i m s 3 0 2 3 2 h , ( ) ± ± k + g = k i m s 3 / 3 2 ~ 3 0 2 h , ( ) ± ± k - g - = ¢ k i m s 3 / 3 2 ~ 3 0 2 h , - k = k i m c 0 2 h . 2. Results The intrinsic concentrations and Fermi level were calculated by fitting the concentrations of electrons and heavy holes. The calculations are separated in two parts: (i) dependence of the carrier concentration on the well width and (ii) dependence of the carrier concentration on the composition x of the well. ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ø ö ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç è æ D - - - - - D - - - - + - - - - - - - - - + - - - - - - = + - + + - - + + + + + + - - - - - - + - + - - + U C S V S R P k P k C U R S V S P k P k S R V U S R P k V S S V U C R P k P k S V R C V U S P k P k R S R S V U P k Pk Pk Pk Pk Pk T Pk Pk Pk Pk Pk T H z z z z z z z z † † † † † † † † † † † 2 1 2 ~ 2 3 2 3 1 3 1 2 ~ 2 3 2 2 1 3 1 3 1 2 1 2 0 2 1 0 2 ~ 2 3 3 2 6 1 ~ 3 2 2 6 1 3 2 2 2 1 0 0 2 1 3 1 3 1 2 1 3 2 6 1 0 0 3 1 3 1 0 6 1 3 2 2 1 0 The dependence of the carrier concentration on L (Fig. 2) is calculated for the composition of the well x = 0.06. For this composition, the critical thickness of the insulator-semimetal topological transition is around L = 12 nm (see Fig. 1) and its influence on the intrinsic concentration can be examined. From Fig. 2, one can see that the localized intrinsic concentration grows along with the well width. This growth is explained by the presence of the topological phase transition, which opens the band gap for widths less than 12 nm. For larger well widths, the system is semimetal and the band gap is absent, thus, the intrinsic concentration demonstrates plateau at larger well widths. Another important feature is that the local minimum of the Fermi level is situated near the topological transition. One should note that for higher concentrations like x = 0.12, the critical thickness is around L ~ 20 nm, and the growing section in Fig. 2 will be shifted to higher well widths. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0 10 20 30 40 50 60 Electron concentration n, 10 15 cm -3 QW composition x Fig. 4. Dependence of the intrinsic concentration on the QW composition x, well width L = 20 nm. The dependence of the electron concentration on the QW composition x is presented in Fig. 4. The corresponding energy spectrum is presented in Fig. 3. It should be noted that composition changes also lead to the topological phase transition. For x = 0, the critical thickness for the Hg0.32Cd0.68Te/ Hg1–x CdxTe quantum well is L = 6.7 nm, while for higher concentrations, this thickness grows and approaches more than L ( 20 nm at x = 0.12, which should be accounted when growing MCT QWs for THz bolometer detectors. This effect can be explained by stronger influence of the semiconducting barrier layers, which enhances when the well band gap decreases. The carrier concentration in the well (Fig. 4) decreases as the composition grows. This dependence is almost linear. 3. Conclusions The maximal intrinsic concentration of carriers of the order of n ~ 5(1016 cm–3 can be achieved for compositions x close to 0 and well widths that are greater than the critical thickness of the topological transition. For bolometer type detectors, the optimal intrinsic concentration seems to be one order lower [2]. It should be noted that the increase of the composition x leads to decrease of the electron effective mass and, thus, to increase of the electron mobility, but the carrier concentration becomes smaller. Thus, for n-type samples the carrier conductivity will be influenced by two concurring trends which should be taken into account when designing QWs for THz/sub-THz detectors. References 1. F. Sizov, V. Petriakov, V. Zabudsky, D. Krasilnikov, M. Smoliy, and S. Dvoretski, Millimeter-wave hybrid un-cooled narrow-gap hot-carrier and Schottky diodes direct detectors // Appl. Phys. Lett. 101, 082108 (2012). 2. V. Dobrovolski, F. Sizov, THz/sub-THz bolometer based on electron heating in a semiconductor waveguide // Optoelectronics Review, 18, p. 250-258 (2010). 3. Jun Li and Kai Chang, Electric field driven quantum phase transition between band insulator and topological insulator // Appl. Phys. Lett. 95, 222110 (2009). 4. M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, Xiao-Liang Qi, Shou-Cheng Zhang, Quantum spin Hall insulator state in HgTe quantum wells // Science, 318 (5851), p. 766-770 (2007). 5. B.A. Bernevig, T.L. Hughes, Shou-Cheng Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells // Science, 314 (5806), p. 1757-1761 (2006). 6. V. Latussek, Dissertation zur Erlangung des naturwissenschaftlichen Doktorgrades “Elektronische Zustande in Typ-III-Halbleiterheterostrukturen“, Julius-Maximilians-Universitat, Wurzburg, 2004. 7. E.G. Novik, A. Pfeuffer-Jeschke, T. Jungwirth, V. Latussek, C.R. Becker, G. Landwehr, H. Buhmann, and L.W. Molenkamp, Band structure of semimagnetic Hg1–yMnyTe quantum wells // Phys. Rev. B, 72, 035321 (2005). � EMBED Origin50.Graph �� Fig. 2. Dependence of intrinsic concentration on the well width L in QW with the composition x = 0.06. � EMBED PBrush �� Fig. 3. Dependence of the energy spectrum and Fermi level of QW with the width L = 20 nm on the composition x. Curves on the graph are marked as follows: Fermi level – (1), E1 level – (2), E0 level – (3), HH0 level – (4), HH1 level – (5), HH2 level – (6), HH3 level – (7), LH0 level – (8). � EMBED PBrush �� Fig. 1. Dependence of energy spectrum and Fermi level of QW with the composition x = 0.06 on the well width L. Curves on the graph are marked as follows: Fermi level (1), E1 level – (2), E0 level – (3), HH0 level – (4), HH1 level – (5), HH2 level – (6), HH3 level – (7). � EMBED Equation.3 �� QUOTE � �� (5) � EMBED Equation.3 �� (4) © 2014, V. 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Intrinsic concentration dependences in the HgCdTe quantum well in the range of the insulator-semimetal topological transition

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