Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry

The peculiarities of quantum oscillations in bulk semiconductors with Cnv symmetry caused by the lack of their symmetry centre are considered. A quasi-qubic model is used for finding the magnetic levels. The algorithm for numerical calculating the levels in the presence of the tilted magnetic field...

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Опубліковано в: :	Semiconductor Physics Quantum Electronics & Optoelectronics
Дата:	2005
Автори:	Ivchenko, V.V., Sergeev, A.N., Elnik, V.S., Chuiko, G.P.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2005
Онлайн доступ:	https://nasplib.isofts.kiev.ua/handle/123456789/120658
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry / V.V. Ivchenko, A.N. Sergeev, V.S. Elnik, G.P. Chuiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 22-27. — Бібліогр.: 13 назв. — англ.

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spelling	Ivchenko, V.V. Sergeev, A.N. Elnik, V.S. Chuiko, G.P. 2017-06-12T15:44:14Z 2017-06-12T15:44:14Z 2005 Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry / V.V. Ivchenko, A.N. Sergeev, V.S. Elnik, G.P. Chuiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 22-27. — Бібліогр.: 13 назв. — англ. 1560-8034 PACS: 71.28.+d, 71.18.+y https://nasplib.isofts.kiev.ua/handle/123456789/120658 The peculiarities of quantum oscillations in bulk semiconductors with Cnv symmetry caused by the lack of their symmetry centre are considered. A quasi-qubic model is used for finding the magnetic levels. The algorithm for numerical calculating the levels in the presence of the tilted magnetic field is suggested. Numerical estimations are performed for Сd₃As₂ and Cd₃₋xZnxAs₂. It is shown that the most suitable conditions for observing the beating effect in these compounds are the high values of the electron concentrations and small angles between the direction of the magnetic field and crystal symmetry axis. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
title	Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry
spellingShingle	Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry Ivchenko, V.V. Sergeev, A.N. Elnik, V.S. Chuiko, G.P.
title_short	Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry
title_full	Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry
title_fullStr	Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry
title_full_unstemmed	Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry
title_sort	inversion asymmetry effect on quantum oscillations in 3-d crystals with cnv symmetry
author	Ivchenko, V.V. Sergeev, A.N. Elnik, V.S. Chuiko, G.P.
author_facet	Ivchenko, V.V. Sergeev, A.N. Elnik, V.S. Chuiko, G.P.
publishDate	2005
language	English
container_title	Semiconductor Physics Quantum Electronics & Optoelectronics
publisher	Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
format	Article
description	The peculiarities of quantum oscillations in bulk semiconductors with Cnv symmetry caused by the lack of their symmetry centre are considered. A quasi-qubic model is used for finding the magnetic levels. The algorithm for numerical calculating the levels in the presence of the tilted magnetic field is suggested. Numerical estimations are performed for Сd₃As₂ and Cd₃₋xZnxAs₂. It is shown that the most suitable conditions for observing the beating effect in these compounds are the high values of the electron concentrations and small angles between the direction of the magnetic field and crystal symmetry axis.
issn	1560-8034
url	https://nasplib.isofts.kiev.ua/handle/123456789/120658
citation_txt	Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry / V.V. Ivchenko, A.N. Sergeev, V.S. Elnik, G.P. Chuiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2005. — Т. 8, № 1. — С. 22-27. — Бібліогр.: 13 назв. — англ.
work_keys_str_mv	AT ivchenkovv inversionasymmetryeffectonquantumoscillationsin3dcrystalswithcnvsymmetry AT sergeevan inversionasymmetryeffectonquantumoscillationsin3dcrystalswithcnvsymmetry AT elnikvs inversionasymmetryeffectonquantumoscillationsin3dcrystalswithcnvsymmetry AT chuikogp inversionasymmetryeffectonquantumoscillationsin3dcrystalswithcnvsymmetry
first_indexed	2025-11-26T06:25:15Z
last_indexed	2025-11-26T06:25:15Z
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fulltext	Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 22-27. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 22 PACS: 71.28.+d, 71.18.+y Inversion asymmetry effect on quantum oscillations in 3D crystals with Cnv symmetry V.V. Ivchenko1, A.N. Sergeev1, V.S. Elnik1, G.P. Chuiko2 1Kherson State University, Chair of Physics, 27, 40 Rokiv Zhovtnya str., 73000 Kherson, Ukraine Phone: +380-0552-326768, e-mail: ivchenko@ksu.kherson.ua 2Kherson State Technical University, Department of Cybernetics, 24, Beryslavske shosse, 73008 Kherson, Ukraine Phone: +380-0552-326922, e-mail: chuiko@public.kherson.ua Abstract. The peculiarities of quantum oscillations in bulk semiconductors with Cnv symmetry caused by the lack of their symmetry centre are considered. A quasi-qubic model is used for finding the magnetic levels. The algorithm for numerical calculating the levels in the presence of the tilted magnetic field is suggested. Numerical estimations are performed for Сd3As2 and Cd3-xZnxAs2. It is shown that the most suitable conditions for observing the beating effect in these compounds are the high values of the electron concentrations and small angles between the direction of the magnetic field and crystal symmetry axis. Keywords: spin splitting of bands, Fermi surface, quantum oscillations, beating effect, node, cadmium arsenide. Manuscript received 31.03.05, accepted for publications 18.05.05. 1. Introduction In the crystals without the symmetry center while straying from the Brillouin zone center, complete removing the energy band degeneracy may occur. The opposite signs of the electron spin projections onto the quantization axis correspond to the splitted states. For the crystals with υnC symmetry class, such a splitting is linear in the wavevector k. The appropriate isoenergetic surface is described by the following equation in the cylindrical coordinate system [1]: 1 )( 2 2 2 2 0 =+ ±⊥ b k a kk z , (1) where a , b are the semiaxes of the ellipse in which axial rotation along zk produces the surface; 0k is the nonzero (as a result of the symmetry center lack) value of the ellipse center shift from the origin of coordinates within the plane perpendicular to the crystal main axis. Since 0,, kba depend on the band parameters as well as on the electron energy ε, the alteration of the latter one may lead to changes in the shape and topology of isoenergetic surface. When ak >0 , the surface is topologically equivalent to the torus while for ak <0 it is the fourth order surface self-crossing in two conical points (Fig. 1). The possibility to manifest peculiarities of isoenergetic surfaces (1) in quantum-oscillatory phenomena (Shubnikov – de Haas (SdH) and de Haas – van Alphen effects (dHvA)) was discussed in [2, 3]. However, these works dealt only with the cases when the magnetic field was parallel or perpendicular to the zk - axis and the simple one-band parabolic model was used. In this paper, we present a more comprehensive study of this problem. In particular, we take into account the possible nonparabolicity of the energy spectrum and using the effective mass method consider the conditions for beating appearing in the case of different electron concentrations and orientations of magnetic field. Our theoretical analysis was accompanied by numerical calculations performed for Cd3As2 and 2xx-3 AsZnCd that have 12 4υC space group symmetry. 2. Theory and numerical calculations It is known [4] that, in the quasi-classical approximation, the period of quantum oscillations is defined by the extreme in zk ′ ( zk ′ is a component of the wavevector along the magnetic field B) cross-section area mS of the Fermi surface (FS) normal to the external magnetic field direction: T = 2πe/ћSm. For FS which has more than one extreme cross-section area normal to a given magnetic field direction, there will be oscillatory effects associated with each such area. Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 22-27. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 23 Fig. 1. Shape of the isoenergetic surfaces described by equation (1): ak <0 (a), ak >0 (b). For the majority of semiconductors, the characteristic of the deviation of FS (1) from an ellipsoidal shape r = k0/a << 1. The quasi-classical theory developed for this case [2, 3] predicts the existence of regular “beats” in quantum oscillations connected with the existence of two close central cross-section areas of FS. It is easy to derive the expression for finding the beating period in the form )(80 κπ ErTTB = (2) where F baabeT F εεθθε =+= )sincos2()( 22222 0 h is the oscillatory period obtained by ignoring the inversion asymmetry; εF is the Fermi energy; )(κE is the complete elliptic integral of the second kind with the parameter θκ 222 cot)(11 ab+= ; θ is the angle between the field direction and zk -axis. It should be noted, however, that within the limits of quasi-classical approach the interaction between the electron spins and magnetic field (Zeeman’s term) is not taken into account. Very often such an interaction complicates the beating pattern [5, 6]. So, to correctly describe this effect one should use the quantum-mechanical approach. It is known [4] that the harmonics amplitudes of the SdH(dHvA)-signal contain the multiplicative factor cos(jπν) (j is the harmonic number; cnn ωεεν h)( −+ −= , where ± nε is the energy of the Landau magnetic subband with the extreme value zk ′ , cω is the cyclotron frequency). When the ν-factor magnetic field depen- dence is available, the harmonics amplitudes may turn into zero for certain values of the magnetic field – a “node” appears in the oscillatory curve. As a rule, these peculiarities manifest themselves in the range of the magnetic fields where it is possible to detect only the first harmonic. The condition of the node appearing for it is written as 5.0−= iν , where i is the node number. To determine the subband energies, we use the quasi- cubic band model [7], which properly describes the effects of spin splitting, anisotropy and nonparabolicity of bands energy. In the eight-band effective mass approximation, the electronic wavefunction can be chosen in the following form: ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +−+ +++ =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ↓ ↑ = −+ +− RRZS RRZS N 7542 86311 Ψ Ψ ϕϕϕϕ ϕϕϕϕ Ψ , where N is the number of a unit cell in crystal; ( ) 2iYXR ±=± and the symbols ↑ and ↓ mean the spin-up and spin-down functions, respectively. ZYXS ,,, are the periodic Bloch amplitudes transfor- med like atomic s- and p-functions under the operations of the tetrahedral group at Γ point. ϕ i are envelope functions satisfying the set of eight coupled differential equations, which in the case of quasi-cubic approxima- tion can be represented in the following matrix form [8]: .0 000000ˆ 00000ˆ0 00000ˆ 0000ˆ0 0000ˆ0 00000ˆ̂ˆ 0ˆ0ˆˆ00 ˆ0ˆ00ˆ0 8 7 6 5 4 3 2 1 31 31 12 12 1 1 = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ −− −− ∆− ∆ ∆+− ∆+− −+ + − + + − −+ +− ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ε ε ε ε εξ εξ ξε ξε kiP kiP kiP kiP kiP kPi kiPkiPkiP kiPkiPkiP z z z z (3) Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 22-27. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 24 ,0cosˆ)ˆˆ(sin)ˆˆ(2 sin)ˆˆ(2 cosˆ)ˆˆ( 2 1 2 \|\|2 2 22 123 232 \|\|2 2 22 1 =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ −−+−++ +−−+−+− ⊥ ⊥ ϕ ϕ θ γθ θθ γ l f kfkkf l fkikif l fkikif l f kfkkf zyxyx yxzyx (5) Here εεε −= g1 , )3(2 ∆++−= δεε , =3ε )32( ∆+−= ε , 321 ∆=∆ ; )ˆˆ)(21(ˆ yx kikk ±=± , where zyx kkk ˆ,ˆ,ˆ are the components of the operator A)(ˆ heik +∇−= , A is the vector-potential of field. Hence the model along with the optical energy gap gε needs to know the values of spin-orbit splitting of the valence band ∆ and interband momentum matrix element P, crystal field splitting the valence band at Γ point δ , and ξ is the parameter defined by ”interaction” between S and Z states via the asymmetric part of the crystal potential [8, 9]. Let us introduce the axis transformation zxx θ−θ=′ sincos , yy =′ , zxz θθ cossin +=′ , and choose the “Landau” gauge in the form ).0,0,( yB ′−=′A In this case, we have the next commutation relations [ ] ,cosˆ,ˆ 2 θ l ikk yx −= [ ] ,sinˆ,ˆ 2 θ l ikk zy −= [ ] 0ˆ,ˆ =zx kk , (4) where eBl h= is the cyclotron radius. Let us express 83 ϕϕ K in terms of 1ϕ , 2ϕ and eliminate them by substituting in the first two equations of the set (3). Then using (4) we obtain where [ ])32()()()( ∆++∆+−= εδεεεεεεγ g , 3)(2 \|\| ∆+= δεPf , 32 ∆=⊥ εPf , [ ])3()32()( 2 1 ∆++∆+= εδεεε Pf , )32()( 2 2 ∆+= εεε Pf , 3)(3 ∆= ξεε Pf . From (5) it follows that 0,, kba may be expressed in terms of polynomials ,γ if as 2 1 2 31 2 )( fffa += γ , 21 2 31 2 )( ffffb += γ , 130 ffk = . It is convenient to perform a unitary transformation 1ˆ −UUH , ⎟⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎝ ⎛ − = 2 cos 2 sin 2 sin 2 cos ββ ββ U ,tan \|\|ff⊥=β in order to make a new two-component quasi- Hamiltonian to be a diagonal one up to the terms proportional to 3f . Expand now the new components 1 ~ϕ , 2 ~ϕ into an infinite series of linear harmonic oscillatory functions: ∑′′+′′= n nnzx azkxki φϕ ))(exp(~ 1 , ∑′′+′′= n nnzx bzkxki φϕ ))(exp(2 , and introduce the creation and annihilation operators according to [10]: )ˆˆ)(ˆ( 4 )ˆ(ˆ 4 † zyx kCkBiikA AB laa ′+′−+′= , [ ] 1ˆ,ˆ † =aa , where θθ 2 2 2 1 sincos ffA += , 1fB = , θθ θθ 2 2 2 1 21 sincos cossin)( ff ffC + − = . It is possible to express the components of the wavevector operator via these operators as follows: zx k A Caa Al ABk ′⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −++= ˆcossin)ˆˆ( 2 cos4ˆ † 4 θθθ , )ˆˆ( 2 4ˆ † 4 aa Bl ABik y −−= , zz k A Caa Al ABk ′⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ +++−= ˆsincos)ˆˆ( 2 sin4ˆ † 4 θθθ . Taking into account these remarks, it is easy to show that the system (5) is reduced to the infinite system of algebraical equations, a fragment of which has the form: [ ] [ ] 0 L0 LL 0L 1 1 1 1 1 T 1 1 T 1 = ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎜ ⎝ ⎛ Λ −Λ −Λ + + − − ++ + − n n n n n n nn nnn nn b a b a b a i ii i , (6) where ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ −⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +− +⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +− =Λ 22 110 0 22 11 0 0 0 0 ν ν n BT n BT n , 3 2 2 3 4 0 1 1 2 1 \|\| 1 0 \|\| 1 0 2L (4 ( cos sin )) sin ( )cos ( )cos sin n f l n f f f f f f f f f f ν θ θ θ ν θ ν θ θ ⊥ ⊥ = × + +⎛ ⎞ ×⎜ ⎟− −⎝ ⎠ , Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 22-27. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 25 F fff ff F εε θθ θθ εν = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ + + = ⊥ )sincos( )sincos( )( 2 2 2 11 2222 \|\| 0 is the value of ν -factor in the limit of strong magnetic fields (in (6) we put 0=′zk , because it corresponds to the extreme FS cross-section). It is possible to derive the analytical expression for finding the energy levels from (6) only in the case when 0=ξθ . With 0=ξ ( 03 =f ), we have the result first obtained in [11]. If 0=θ one gets 2 0 2 0 2 0 )1(8 2 11 ν+++= knlnBT , (7a) 2 0 2 0 2 0 )1()1(8 2 111 ν+++−+= klnnBT (7b) (while obtaining the expressions (7), we performed 1+→ nn shift for the system of “–” levels, because this numeration is more preferable from the physical point of view). Using the formulae (7) and applying the method for ν -factor calculation suggested in [11] for this direction of the magnetic field, one obtains F BTrBF εε νεν =⎥⎦ ⎤ ⎢⎣ ⎡ −++= 1)1()/4(),( 2 0 2 0 (8) The dependence of electron ν -factor on the magnetic field can be explained in this case by complicated nature of the spin-orbit interaction and as a result of it, the spin splitting of the Landau levels is no longer linear dependent on the magnetic field. When 1>>i (i.e., for relatively weak fields), from (8) it follows that rTBB ii 40 11 1 =− −− + . Hence, the nodes with large numbers are periodically located along the scale 1−B with the period (2). As the magnetic field increases, the location of the nodes turns to be irregular (the Zeeman subbands splitting reveals itself). If we take into account the fact that with the lowering the node number the difference between the magnetic field values corresponding to two neighbour nodes grows rapidly, it becomes obvious that only some first nodes can be observed in the fields where quantum oscillations start to manifest themselves. Let us perform the specific calculations for Cd3As2 and Cd3–xZnxAs2. For this purpose, we use the rank values of their band parameters suggested in [8]. Since the value of the parameter ξ is known only by its order [8], all the calculations are made for several values of this parameter. Fig. 2 shows the concentration dependences of the first nodes’ location calculated for three different values of the parameter ξ . The characteristic feature of all these curves is the increase (which is more or less smooth depending on ξ value) of the field value corresponding to the node when increasing the electron concentration. Let us now examine the behaviour of ν -factor in the tilted magnetic field. The difficulty of this task is caused by the fact that it is impossible to apply the standard perturbation theory to the parameter 3f because the combination nf3 being a part of (4.7) is not small [12]. Besides, the additional difficulties in comparison with 2D systems are connected with the fact that only the magnetic field component normal to the 2D layer effects on the orbital moment, while in the bulk crystal such an interaction is determined as a whole by the field. For defining the energy spectrum when 0, ≠ξθ , let us use the technique of folding the determinants of the block matrixes. Applying the well-known Schur formula it is possible to show that ,0)L)(LL)(Ldet( 1II 1 TT 1 1I 11 =Λ−Λ−Λ − −+ − ++ nnn nnnn where the matrixes ,I 1+Λ n and II 1−Λn are determined using the following recurrent relations: ,L)(L T 2 1I 221 I 1 + − ++++ Λ−Λ=Λ nnnnn .L)(L 1 1II 2 T 11 II 1 − − −−−− Λ−Λ=Λ nnnnn Fig. 2. Theoretical dependence of the first (a), second (b) and third (c) node location on the electron concentration for Cd3As2: meV30=ξ (1); meV15=ξ (2); meV7=ξ (3). Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 22-27. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 26 Fig. 3. Theoretical dependence of the maximal magnetic field corresponding to the nodes of oscillation on the azimuth angle for Cd3-xZnxAs2: eV1.0−=gε (a-c), eV1.0=gε (d-f); 324 m103 −×=en (a),(d); 324 m10 −=en (b), (e); 323 m104 −×=en (c), (f); meV30=ξ (1), meV15=ξ (2), meV7=ξ (3). While performing the specific numerical calculations, this algorithm is realized through the truncating the infinite matrix on the both sides of the fixed block 0nΛ . In our calculations, the convergence for the eigenvalues was attained for the truncated matrix of 20×20 size. Fig. 3 shows the results of numerical calculations for the dependence of the maximal magnetic field corresponding to the nodes on the azimuth angle. The following features are clearly seen: 1. With the increasing θ the magnetic field decreases monotonously. 2. Beating disappears in the case of large values θ ; the critical angle ( o70~ ) at which the beating starts to disappear has a weak dependence on the band parameters and the carrier concentration. 3. With the decreasing ξ and rising gε , the values of the magnetic field reduce and locate within the narrower range. 4. Conclusions The investigations carried out in this work show that the beating patterns in crystals with υnC symmetry have their own specific character. Particularly, for V 2 II 3 BA compounds one should expect the decrease of the maximal magnetic field corresponding to the nodes of oscillation when the electron concentration decreases and the field tilting about the symmetry axis increases. Moreover, starting from a certain azimuth angle the beatings should disappear. The mentioned peculiarities are supposed to help in specifying the value of the parameter ξ which defines the magnitude of zero spin splitting of bands when studying the beating effect in V 2 II 3 BA semiconductors (in which it has already been observed experimentally [13]). References 1. E.I. Rashba, On various cases of origination of toroidal isoenergetic surfaces in semiconductors// Ukr. Fiz. Zhurn. 8(10), p. 1064-1074 (1963) (in Ukrainian). 2. I.I. Boiko, E.I. Rashba, Properties of semiconduc- tors with the loop of extremums. II. Magnetic susceptibility in the field normal to the loop’s plane // Fiz. Tverd. Tela 2(8), p. 1874-1883 (1960) (in Russian). 3. E.I. Rashba, I.I. Boiko, Properties of the magnetic field with the loop of extremums. III. Behaviour in the field parallel to the loop’s plane // Ibid. 3(4), p. 1277-1289 (1961) (in Russian). 4. D. Shoenberg, Magnetic oscillations in metals, Mir, Moscow (1986) (in Russian). 5. L.M. Roth, Magnetic energy levels in an energy band with inversion asymmetry splitting// Phys. Rev. 173(3), p. 755-762 (1968). 6. D.G. Seiler, W.M. Becker, L.M. Roth, Inversion- asymmetry splitting of the conduction band in GaSb from the Shubnikov-de Haas oscillations measu- rements// Phys. Rev. B1(2), p. 764-775 (1970). Semiconductor Physics, Quantum Electronics & Optoelectronics. 2005. V. 8, N 2. P. 22-27. © 2005, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 27 7. H. Kildal, Band structure of CdGeAs2 near 0=k r // Ibid. B10(12), p. 5082-5087 (1974). 8. V.V. Ivchenko, A.N. Sergeev, V.S. Elnik, and N.M. Chuiko, Absorption edge of Cadmium Arsenide// Physics and Chemistry of Solid State 4(4), p. 673-680 (2003) (in Ukrainian). 9. G. Chuiko, O. Dvornik, V. Ivchenko, Generalized Kildal-Bodnar’s dispersion law for 4 mm ( υ4C ) ordered crystals // Ukr. Fiz. Zhurn. 45(10), p. 1188- 1192 (2000). 10. M. Singh, P.R. Wallace, S. Askenazy, Effect of “free-electron” terms on the g-factor of some semi- conductors and semimetals // J. Phys. C: Solid State Phys. 15, p. 6731-6739 (1982). 11. P.R. Wallace, Electronic g-factor in Cd3As2// Phys. status solidi (b) 92(1), p. 49-55 (1979). 12. Yu.A. Bychkov, V.I. Melnikov, E.I. Rashba, Influence of the spin-orbit coupling on the spectrum of 2D-electrons in a tilted magnetic field // Journ. Exp. and Theor. Phys. 98(2(8)), p. 717-725 (1990) (in Russian). 13. M.J. Aubin, J.C. Portal, Shubnikov – de Haas oscillations // Solid State Communs 38, p. 695-702 (1981).

Inversion asymmetry effect on quantum oscillations in 3-D crystals with Cnv symmetry

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