Inversion asymmetry spin level splitting in zero-gap semimagnetic semiconductors

Solid State Comupications, Vo1.61,No.4, pp.231-235, 1987. Printed in Great Britain.

INVERSION -Y

SPIN LEVEL SPLITTING IN ZERO-GAP SEMIMAGNETICSEMICONDUCTORS H.M.A.

Department of Physics.

0038-1098/87 $3.00 + .OO Pergamon Journals Ltd.

Schleijpen,

F.A.P.

Blom and J.H.

Eindhoven University of Technology, The Netherlands (Received

Wolter

P.O. Box 513. 5600 MB Eindhoven,

October 6 1986 by M. Balkanski)

We present a new model to interpret recent measurements on Hgi-.Mn,Se which show an angular dependence of the nodes in the Shubnikov-de Haas amplitude. The model is based on an infinite Hamiltonian matrix including inversion asynmretry terms. We additionally took into account the exchange interaction, typical for Semimagnetic Semiconductors. By truncating the matrix we can calculate the electron energy levels, allowing us to describe the experimentally observed nodal field positions.

Recent papers report on experimental results of Shubnikov-de Haas (SdH) oscillations in oriented siz31e crystals of Hg,_.Mn,Sel and belong to the Hgi-,Fe.Se . These materials class of Semi Magnetic Semiconductors (SMSC). It has been found that these ternary semiconductors exhibit nodes in the amplitude of the SdH oscillations. The field positions of these nodes strongly depend on the angle between the magnetic field and the crystal axes. This angular dependence resembles the anisotropy of the nodal positions observed in the SdH amplitude in HgSe4.5. The appearance of the nodes in HgSe is related to the splitting of the energy bands due to the lack of inversion symmetry in the zincblende lattice. In SMSC, however, this splitting is enhanced by the exchange interaction between the spins of the mobile electrons and the localized spins of the Mn or Fe atoms. In this communication we present a new model to calculate the splitting of the electron energy levels in a taking into account both magnetic field, inversion asymmetry and exchange interaction. From this splitting we can derive the angular dependence of the nodal field positions in the SdH amplitude. Under which conditions the splitting leads to a node in the SdH amplitude can be seen from the theoretical expression for the SdH amplitude. Because the higher harmonics are strongly damped we have to consider only the first harmonic of the SdH oscillations. Its amplitude is given by a monotonic function of magnetic field and temperature which is modulated by a factor cos(ru). This factor differs from unity when the spin degeneracy of the landau levels is lifted. u is the ratio between this spin splitting and the landau splitting. At the nodal positions cos(au) equals zero, whenever u = k + l/2. with k an integer value. The spin splitting of the landau levels due to inversion asymmetry has been first treated theoretically by Roth et al.6.7. Their model examines the splitting only for three magnetic field orientations. The field positions of the nodes for the magnetic field oriented parallel to the [111] and [oOl]-axis and the absence of nodes for the field parallel to the [llO] axis could be explained. It is not possible to use this model

for arbitrary orientations. To obtain the complete angular dependence of the inversion asymmetry splitting we base our model on the model by Weiler et a1.s. In zero they describe the band structure magnetic field, of a narrow-gap semiconductor by the usual four band model. With this model the energies of the conduction band, the light and heavy hole bands and the split off band can be calculated. The parameters used in this model are the energy gap E s, the momentummatrix element P and the spin orbit split off energy A. The warping of the Fermi surface due to the influence of higher bends is described by the parameters ai, az. 13, K, F, q and N1. The inversion asymmetry introduces four additional parameters C. G, Nz and NJ. In the presence of a magnetic field each of the four bands splits into an infinite series of landau levels. Each landau level is split into two spin levels. In this case the Hamiltonian matrix becomes infinite. The inversion asymmetry matrix elements couple adjacent landau levels via raising and lowering operators. The energy levels corresponding to landau number n are coupled to the levels numbered n-l, n+l, n-2, n+2. n-3 and n+3. Due to this coupling it is not possible any more to factorize the Hamiltonian matrix in a series of independent submatrices as has been done by Pidgeon and Brown’. They neglected the inversion asynunetry terms and took into account only part of the warping terms in order to remove the coupling between the adjacent landau levels. In this way they were able to factorize the infinite Hamiltonian to a series of 4x4 matrices. Solving each of these independent matrices the eigenenergies can be obtained. numerically, However, since we cannot neglect the inversion asymmetry matrix elements in our case we have to choose another way to obtain the energy eigenvalues. To obtain a finite matrix which can be solved numerically, we truncate the infinite Hamiltonian matrix above and below certain landau numbers. Doing this. we neglect the coupling between the landau levels on the edges of the truncated matrix and the landau levels just outside this matrix. Neglecting this coupling will strongly influence the energy eigenvalues obtained for the Landau levels on the edges.

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INVERSION ASYMMETRYSPIN LEVEL SPLITTING

However. because

this influence gradually decreases while approaching the central landau levels, good approximations for the energies of the central Landau levels can be obtained. The accuracy of this method will be discussed below. So far the Hamiltonian matrix is valid only for non magnetic semiconductors. In the case of SMSC. the exchange interaction has to be included in the Hamiltonian. The exchange Hamiltonian has the usual Heisenberg form”. HeX =

z B,

J(r-B.)o*S,

Vol.

In our calculations we use the parameter values given in table 2. We keep the warping parameters q and NI equal to zero, because q gives only a very small end almost isotropic contribution to the spin splitting of the Landau levels and NI only affects the l-6 band. which is a valence band in zero gap materials. The values of the inversion asynsnetry parameters will be discussed below. We tested the accuracy of the energy eigenvalues obtained from our truncated matrix, taking typical values for the magnetic field B = 1 T. the Fermi energy EF = 100 meV and the

inversion asymmetry C = 3 x lo-" eVm. Since we

where u is the electron spin operator, S, is the spin operator of the magnetic ion at site R, and J(r) is the corresponding exchange integral. For the set of basis functions used by Weiler et al. we calculated the exchange interaction matrix elements which are independent of the landau number. These additional matrix elements are given in table 1. The exchange interaction is described by two parameters a and j3. representing the exchange interaction strength for electrons with wave functions with s and p like symmetry respectively. x is the fraction of magnetic ions in the material. S’ = SX + is, and S, are the components of the spin of the magnetic ion. Taking the thermal average of these components this matrix is considerably simplified: = = 0. Only the spin component parallel to the magnetic field does not vanish. depends on magnetic field and temperature but is isotropic with respect to the magnetic field orientation. Therefore the exchange contribution to the energy level splittings, which is almost proportional to . depends on field and temperature. The value of as a function of field and temperature can be obtained from magnetization measurements’.

are only interested in the energy levels close to the Fermi energy we choose the Landau number of the central level of the truncated matrix in such a way that the corresponding energy level is as close as possible to the Fermi energy. Then we increase the number of Landau levels in our matrix until the energies corresponding to the five central levels change less than 0.1 percent. To fulfill this convergency criterion we have to take into account eleven landau levels (i.e. five above and five below the central level). Eleven

Table

2. Bandparameters

of Hgo.s~sMno.ozsSe

Energy gap

E, = P = A = _a1= 72 = 33 = K.= F= Ekchange parameters a = p = Inversion asymmetry parameter C = Momentum matrix element Spin orbit splitting Higher band parameters

-165 meV 7.2 x lo-" eVm 0.39 eV 5.77 -1.17 0.57 0.98 0 -0.8 eV 1.3 eV 2.3 x lo-" eVm

Table 1. Exchange interactionmatrix elements

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61, No. 4

13)

15)

17)

12)

16)

14)

18)

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INVERSION ASYMMETRY SPIN LEVEL SPLITTING

levels are also sufficient to obtain this accuracy when the magnetic field is increased to 2.5 T. keeping the other parameters constant. To obtain the energy eigenvalues for these eleven levels we have to solve a 88 x 88 matrix numerically. The total spin splitting can be separated in an exchange part and an inversion asymmetry part. In fig. 1 the angular dependence of the inversion asymmetry contribution is shown. For each curve only one of the inversion asymmetry parameters differs from zero. The splittings in Fig. 1 are calculated for B = 1 T and a Fermi energy of 80 meV. Following Weiler et al. we keep Ns equal to NJ. From Fig. 1 we can see that all inversion asymmetry parameters cause a similar angular dependence of the splitting: an increase of the splitting for B//[lll] and a decrease of the splitting for &r/[OOl]. For B//[llO] there is no inversion asymmetry contribution to the spin splitting. Because all the parameters Cl. G and Ns = Ns cause the same shape of the spin splitting anisotropy, it is possible to obtain the same energy splitting, as a function of the field orientation, while increasing one parameter and reducing another at the same time. Thus it is

7

I CZ 2.5 10-l’ eVm

,_ I_

9o” 60” Angle between Fig.

3o” 0” B and CO.O.11 axis

1. Angular dependence of the inversion asyrmnetry contribution to the spin splitting for the three inversion asymmetry parameters C. G and Ns = NJ.

233

impossible to determine the parameter values only from the angular dependence of the spin splitting. In principle the different k-vector dependence of the matrix elements corresponding to the different inversion asymmetry parameters allows to determine these parameter values separately. The C matrix elements vary linearly with k and the G and Ns = NJ elements quadratically with k. However, the range of carrier concentrations and consequently the range of k-values for which experimental data are available5 is too small to distinguish between a linear or a quadratic k-dependance of the inversion asyrmnetry splitting. Therefore we will concentrate only on one parameter, for which we choose C. Keeping G and Ns = Ns equal to zero the value obtained for C will be an upper limit for this parameter. We will now apply our model to the interpretation of our data on Bgo.s&ino.ossSe. In ref. 1 we pointed out that in our experiment the spin splitting was about 2.5 times as large as the Landau splitting. This large splitting arises mainly from the exchange interaction. As can be seen in Fig. 1 the spin splitting is not affected by the inversion asymmetry when the magnetic field is oriented parallel to the [llO]axis. Thus for this particular orientation the spin splitting is only influenced by the exchange interaction which allows us to determine the exchange parameters a and /3. Following the procedure given in ref. 1 and using the bandparameters given in table 2 we obtain a = -0.8 eV and S = 1.3 eV. With these values we calculated the spin splitting for the experimental conditions where the nodes occur in the SdB signal for B//[OOl] and B//[lll]. This is shown in Fig. 2. For each field orientation we start on the left with the energy levels without exchange interaction and without inversion asymmetry. Then we increase a to -0.8 eV and afterwards p to 1.3 eV, the values obtained for B//[llO]. We now obtain the energy levels in the presence of exchange interaction but still without inversion asymmetry. Finally the inversion asymmetry parameter C is increased from 0 to 5 x lo-” eVm. The nodes will occur when the spin splitting equals 2.5 times the landau splitting. This way we can determine the value of C. This resulted in C = 2.4 x lo-‘I eVm in the B//[OOl] orientation and C = 2.3 x lo-‘I eVm for B//[lll]. As we expect for the B#[llO] orientation, variation of C causes only a slight shift of the energy levels, but the splitting remains constant. Choosing C equal to 2.3 x 10-l’ eVm we can now calculate for the first time the complete angular dependence of the nodal positions in the SdB amplitude. In Fig. 3 we compare the calculated positions (solid curve) to the experimental data obtained in ref. 1 (closed circles). We use the data obtained at T = 2.03 K for a carrier concentration of 1.2 x 1O24 m-s. To test the influence of C we also calculated some nodal positions for C = 2 x 10-l’ eVm (open circles). Our value for the C parameter. which is an upper limit, is 2.3 x 10-l’ eVm. We expect the value of C to be close to the value for pure since the manganese concentration of the &Se. crystal for which the data were taken was only 2.5 percent. Recently Cardona et al.” published

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INVERSION ASYMliRTRYSPIN LEVEL SPLITTING

234

a(eV) 0 -0.8 p(eV) 0

0

0 -0.8 0 1.3

.' 0 -0.8 0 1.3

1.3

0

2.10‘1' 4xJ"

Fig.

0

2.1@' 1~10“'

2. Energy levels of the conduction band for three magnetic field orientations. Further explanation is given in the text.

18 r 1.6_

I

1

t

9o"

60°

I

3o"

I

O0

Anglebetween B and CO,O,ll axis Fig.

61, No. 4

3. Angular dependence of the nodes for Hgo.s7sMno,ossSe at 2.03 K with a carrier concentration n = 1.2 x 10” rn-=. The solid curve is calculated using the parameters in Table 2. o = -0.8 eV, p = 1.3 eV and C = 2.3 x lo-” eVm. The closed circles represent the measurements. The open circles give the calculated nodal positions for the same parameters except C = 2 x 10-l’ eVm.

2.10“'4.10-"

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INVERSION ASVIMETR'ISPIN LEVEL SPLITTING

a value C = 0.8 x lo-” eVm for HgSe. Keeping in mind the scatter over two orders of magnitude for similar results for the inversion asynsnetry parameters in InSbs~‘2, these values for HgSe and Hgi_,M.n,Se are in reasonable agreement.

235

Although we tested our new model only on data on Hg~_,Mr&e this model should also be applicable to Hgi_.Fe,Se, but unfortunately the necessary magnetization data are not yet available for this material.

References

F.A.P. Blom, Phys. Stat. 1. H.M.A. Schleijpen, Sol. (b) 135. 605 (1986). R. Reifenberger, Phys. Rev. B 2, 2. M. Vaziri. 6585 (1986). I.M. 3. N.G. Gluzman. L.D. Sabirzyanova. Tsidil’kovski. L.D. Paranchich and S.Yu. Paranchich, Sov. Phys. Semicond. 20. 55 (1986). Phys. Rev. 138. A829 (1965). 4. C.R. Whitsett, R.R. Galazka. W.M. Becker, Phys. 5. D.G. Seiler, Rev. B 2, 4274 (1971). 6. L.M. Roth, S.H. Groves. P.W. Wyatt. Phys.Rev. Lett. l9. 576 (1967).

7. L.M. Roth, Phys. Rev. 173, 755 (1968). R.L. Aggarwal, B. Lax. F'hys. 8. M.H. Weiler. Rev. B lJ, 3269 (1978). 9. C. Pidgeon, R. Brown, Phys. Rev. 146, 575 (1966). J. Kossut. R.R. Galazka, Phys. 10. M. Jaczynski. Stat. Sol. (b) 88, 73 (1978). G. Fasol. Phys. 11. M. Cardona. N.E. Christensen, Rev. Lett. g, 2831 (1986). 12. C. Pidgeon. S. Groves, Phys. Rev. 186, 824 (1969).