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On the band structure and anisotropy of transport properties of ferromagnetic semiconductors CdCr2Se4 and HgCr2Se4
Solid State Communications, Printed in Great Britain.
0038-1098189 $3.00 + .OO Pergamon Press plc
Vol. 69, No. 7, pp. 761-764, 1989.
ON THE BAND STRUCTURE FERROMAGNETIC
AND ANISOTROPY OF TRANSPORT PROPERTIES SEMICONDUCTORS CdCr, Se, AND HgCr, Se,
OF
M.I. Auslender and N.G. Bebenin Institute of Metal Physics, Academy of Sciences of the USSR, Ural Division, Kovalevskaya 620219 Sverdlovsk, GSP, USSR
18,
(Received 2 May 1988 by M. Balkanski)
The simple model of the band structure is described. The model explains the temperature dependence of electron mass, the anisotropy of magnetoresistance of p-type materials, and the difference of the mobility in n-CdCr,Se, and n-HgCr,Se,. INTRODUCTION THE SPINELS CdCr,Se, and HgCr,Se, are ferromagnetic semiconductors (FMS) with rather high Curie temperatures (T, = 130 K and 110 K respectively) and the electron mobility ( lo’-lo3 cm* V-’ s-l). Although these semiconductors have been investigated during two decades, the experimental data on optical and transport properties are usually discussed without taking into account the peculiarities of the band structure. Moreover, the viewpoint that at the present time the band structure of these compounds cannot be found with any certainty is predominant. The purpose of our paper is to describe the simple model of the band structure which is based on the main features of the results of band calculations [ 1, 21 and the analysis of the experimental data on the polarized light absorption near the fundamental edge [3,4] (the last papers have not yet attracted due attention). The model enables us to explain the following experimental facts: (i) the strong dependence of electron mass in n-HgCr,Se, on temperature; (ii) the anisotropy of magnetoresistance of p-CdCr, Se, and p-HgCr, Se,; (iii) the striking difference between values of the electron mobility in n-CdCr,Se, and n-HgCr,Se,.
are important except for the case T < Tc. As for de-electrons, we treat them to form the Cr local moments which interact with the charge carriers via s-d exchange interaction Hs-d = 1 Z(r - R,,)S,, - s.
(1)
1.K
Here Z(r) is s-d exchange integral, S,, is the spin operator of Cr3+ ion in the site R,, = R, + eK (S = 3/2), eK is the position of Cr3+ ion in the primitive cell, and r, s are position and spin of the carrier. The correctness of s-d model in the case under consideration follows from the results of photoemission experiments [5]. There it was found that for CdCr,Se, the transition Cr3+-Cr4+ takes place at the energy z I .45 eV, so that de-“bands” lie far below the top of the valence band. In the standard Luttinger-Kohn basis 1nk) = u,,o(r) exp (ikr) the matrix elements of & can be written as fZ_,,(nk, n’k’) = 1 Izns(k - k’) * S,(k - k’), K
(2)
where &s(q)
= $ I dr I(r - e,) u,+,(W,dr)
SK(9) =
2 1 S,, exp Gq - R,),
exp (- iqr), (3)
I
THE MODEL OF THE BAND STRUCTURE We suppose that the bottom of s-like conduction band (4s Cr -i- 5s Cd or 4s Cr + 6s Hg) and the top of the p-like valence band (mainly 4p Se) are located at the I point (I, and F,, symmetries respectively). These bands can be described by one-electron Hamiltonian. ‘Near the s-like band there is a group of the empty narrow bands generated by antibonding p-dy orbitals A(4p Se + 3dy Cr) of F,*, I,;, I;, symmetries. For these bands the many-electron effects
Q, stands for primitive cell volume. In the mean field approximation (MFA) we get Q(nk,
n’k’) =
(S) * A,,d(k
A,,. = ; Ztd(O).
- k’), (4)
A,,. # 0 only if u,,,(r) and u,,o(r) belong to the same irreducible representation of 0: group. This means that (in MFA) s-d interaction does not mix s, p, and p-dy bands at I point.
761
762
BAND STRUCTURE
OF CdCr,Se, AND HgCr,Se,
Vol. 69, No. 7
When describing s- and p-like bands, our consideration is quite similar to that for semimagnetic semiconductors (see e.g., [6]). For I, (I,,) band we have A,,. = A,%,,,
(5)
n = (c, c), 0 = f l/2, A, = (Sl C, Z(r - e,) 1S), so the spectrum takes the form EC0 =
Ego + aA,I(S>I
+
‘;, L
where E,, is “unperturbed” energy gap. Let us consider the valence band. Spin-orbit interaction splits I,, into I, and I, bands. Since the value of splitting, A = 0.2-0.4 eV, is appreciably more than s-d exchange splitting, the upper band I, can be treated separately from Iti. Then we have [6] A,,.
=
$A,.&,,,,,~,
n
=
(m, v),
A, =
m
=
-J3/2,
+ l/2,
Fig. 1. Optical transitions which are responsible for absorption of light of “ - ” (1) and “ + ” (2) polarization. The direction of light propagation and quantization axes are parallel to magnetization.
(7) peratures not far from T, are required to determine
WI cZ(r - edl J3, K
A,.
J,,,,,,,are the angular momentum
3/2 matrices. The k-dependence of the hole spectrum is given by (for the case y2 = y3 see [6]) %rn = +mA,I (S) I + T (M<,‘)ijkikj,
T 3(1 - Gv)ygzinj.
(8)
(9)
The y’s are the Luttinger’s parameters of bare I, band, II = (S)/( (S) 1, i, j = x, y, z, the coordinate system is chosen in such a way that Ox )I [lo 01, etc., symbols (h, Z) denotes “heavy” (m = f 3/2) and “light” (m = f l/2) holes. The second term in the right hand side of equation (8) is assumed to be much less than the first one. The experimental data on polarized light absorption near the fundamental edge have been reported in [3] for CdCr,Se, and in [4] for HgCr,Se, single crystals, the values of absorption coefficient k being up to 6 x 103crn’. It has been shown that (i) at low temperatures the absorption of “ - ” polarized light starts at the lower energy than that of “ + ” polarized; (ii) the ratio k, /k_ x 3 (being slightly less than 3 for CdCr,Se, and slightly more than 3 for HgCr,Se,). These facts -__. __._ are_ explained by the optical transition scheme depicted in Fig. 1. This scheme is unique and allows us to find the signs of exchange parameters and the value of A, : A,, A, c 0, A, = - 0.05 eV. The value of A, cannot be obtained because the splitting of band is giant at T < T,. Measurements for tem-
TEMPERATURE DEPENDENCE ELECTRON MASS
OF
When temperature lowering, the effective mass of conduction band m,. in n-HgCr,Se, decreases from w 0.3m, at 300 K to x O.l5m, at 4 K [7], with energy gap E, decreasing from 0.8eV to 0.3eV [8]. We can connect the changes of m, and E, supposing k-p interaction between the conduction and valence bands to be of the most importance. In this case we can use the familiar expression m;’
=
m&’ +
21P12 E, + $A E,(E, + A)’
--i?--
(10) ignoring the valence band exchange splitting which is appreciably less than E,. In MFA E, = E,, l/2 I A,(S) 1, so E, seems to be temperature independent at T > T,. However, the shift of the conduction band is known to take place even in paramagnetic state due to spin fluctuations. We shall take them into account in a simple way considering Eg in (10) as being the true band gap measured in experiment. Then the experimental data for E, together with (10) at A = 0.2 eV give I
m;’ (4K) - rni’ = m;’ (3OOK) - m,’
“N 2*5’
This value is in excellent agreement with the data for m, which lead to ,l x 2.4.
BAND STRUCTURE
Vol. 69, No. 7
OF CdCr,Se, AND HgCr,Se,
MAGNETORESISTANCE OF p-TYPE FERROSPINELS We restrict ourselves to the case of low temperatures when 1A, (S) 1 is much more than k,T, the characteristic nondegenerate holes energy. Then equation (8) holds and the main contribution to the transport comes from m = + 312 subband. Taking for the relaxation time tensor rii = q,(m,/m,)3’Z6~ where md = (det IM)“~ is density-ofstates effective mass, for tensor of resistivities we get eii =
e. exp (-&lkJ)
M;‘M,.
(11)
Here fi2/2M, = yI + y2, e. = M, /(e2NPoro), NPo = (2nmok,T)312/(2nh)3, &, is the chemical potential for holes. In experiment one usually measures the resistivity at the fixed direction of the electric current j, eol, which connects with eii by the relation em = Ceijvivj, ij
v = j/lj].
(12)
Obviously, antisymmetric part of eii which is responsible for the Hall effect (whatever its nature) does not contribute to eel. The analysis of the experimental data shows the quantities 6 = (y2 - y3)/(y, + y2) and p = y,/ (y, + y2 - 3y3) to be less than unity. Expanding eol in a power series of 6 and p, and keeping only the linear terms in the small parameters, we obtain etil =
e. exp (-&,/k,T)
1 + 3~ cos2 (n, v) (
+
3iqr+vf r
.
(13)
>
Thus the resistivity of the p-type crystals is anisotropic. The anisotropy comes from (i) evident dependence of eii on Mii which leads to the dependence of resistivity not only on orientation of n and j relative to the cubic axes but also on the orientation of these vectors relative to each other (expression in parentheses in (13)); (ii) dependence of [, on orientation of magnetization vector relative to the cubic axes (but not to j). It follows from the orientation dependence of acceptor activation energy which is given by [lo] &* = P(n)
=
e
When magnetic field is high enough, the magnetization is along H all over the sample and we may employ eq. (13) in order to calculate @a](H).If H = 0, there is a domain structure. In every domain the resistivity is again given by (13) but, in general, the resistivity of the sample does not coincide with that of any domain. In fact one measures the resistivity averaged over the magnetization distribution. It means that the magnetoresistance of p-type FMS is due to the change of magnetic structure caused by magnetic field and, hence, must tend to saturation when magnetic field is being increased. Equation (13) shows that the magnitude of Ae/e is closely connected with the anisotropy of eoI: the more anisotropy, the greater magnetoresistance. The present theory allow us to understand the main features of the results of early experiments on Ae/e as well as the data on the magnetoresistance anisotropy. The anisotropy of de/q in p-CdCr, Se, was recently studied in [ll, 121. When the current was directed along [ 1 1 l] or [10 0] axes, the magnetoresistance did not change under rotation of field in the plane perpendicular to the current. On the other hand, when the field was rotated from being parallel to the current to a direction in the plane perpendicular to that, the 180” oscillations of Ae/e were observed [l 11. Further, when j II [ 1 T 0] and the field was rotated in the plane (1 1 0), the 180” oscillations of Ae/e were also found in heavily doped CdCr, Se, : Ag [ 121. It is easy to see that equation (13) explains all the features of the 180” oscillations as well as the absence of those for the configurations j 11[ 1 1 I], [ 10 01, H I j if we assume E, 4 kBT. When the band conduction dominates, eqs. (13) and (14) predict also the 90” oscillations to superimpose the 180” ones at low temperature. It was really observed in heavily doped p-HgCr,Se, (for details see [lo]). In lightly doped CdCr,Se, : Ag the authors of [ 121 observed the oscillations of Ae/e, the angular dependence of which can be described by the function P(n). Thus these oscillations are caused mainly by the n-dependence of the chemical potential. It may indicate that the band conduction does not prevail in this case.
Eg - slP(n), &z~ + rz$~ + $,Pz~
(14)
where so and E, (E, < et,) are positive constants. Usually experimental data refer to not em itself but to so called magnetoresistance
!!!&l=
763
4d-V
- ef.i@)
e&O
’
(15)
ON THE p-dy BANDS LOCATION The band calculations show that the bottom of every p-dy band is at X point whereas at I point there are the tops of the bands [ 1, 21. The direct allowed transitions I,, + I,2, I-;,, I& lead to the sharp rising of light absorption which is observed at ho 2 1.4eV in CdCr, Se, and fiw 2 0.9 eV in HgCr, Se, (see, for
764
BAND
STRUCTURE
OF CdCr,Se,
example, [3, 41). Considering the temperature dependence of this transitions together with that of the absorption edge, we can conclude that in HgCr,Se, the bottom of s-like band (F,) always lies lower than the tops ofp-dy bands. In CdCr,Se, it is true only at T < 200K while at T > 200K the s-band bottom is slightly higher than the p-dy bands maxima at r point. The more difficult problem is to determine where the bottoms of p-dy bands locate. For the time being there are no experimental data which enable one to do this unambiguously. Nevertheless, some tentative conclusions can be made. In HgCr,Se, the electron mobility is rather high, being up to 1800 cm2V ss’ at 4 K and 30 cm* V’ ss’ at 300 K, and the electron mass is not more than 0.3m, [9]. This stands to reason that the charge carriers move mostly in the broad band, so the bottom of any p-dy band lies higher than that of the s-band at all temperatures. On the contrary, in n-CdCr,Se, the mobility does not exceed 1 cm2V-’ s-’ whatever doping (see e.g., [13]) (the data on electron mass have not been reported). Thus we may take that in CdCr, Se, the bottom of a p-dy band is lower than the bottom of s-like band. DISCUSSION Of course, the present work is not the first one where the band structure of CdCr,Se, and HgCr,Se, have been discussed. For example the transport, magnetic and optical properties of HgCr,Se, were recently studied by Selmi, Mauger, and Heritier [9]. To interpret the experiments, they supposed that in CdCr,Se, and HgCr,Se, the lowest conduction band was of d-like, the s-like conduction band lies higher and the p-like valence band was assumed to be nondegenerate parabolic one. The extremums of all the bands were supposed to be at r point. This band scheme contradicts the band calculations. Also it fails to explain the values of mobility and effective mass in n-HgCr,Se,. Indeed, the dy-orbitals do not overlap directly, so p-dy bands are formed only due to p-d covalency
AND
HgCr,Se,
Vol. 69, No. 7
effects and, consequently, they are rather narrow. It is unlikely the electrons in these bands to have such small effective mass and high mobility. As regards the valence band, if it had been nondegenerate at r point, all anisotropy effects of AQ/Q (as well as the magnetoresistance itself) and circular dichroism would have been absent at T < Tc in p-type CdCr,Se, and HgCr, Se,.
REFERENCES 1.
2. 3.
4. 5. 6. 7. 8.
9. 10.
11. 12. 13.
T. Kambara, T. Oguchi & K.I. Gondaira, J. Phys. C13, 1493 (1980); T. Oguchi, T. Kambara & K.I. Gondaira, Phys. Rev. B22, 872 (1980). T. Oguchi, T. Kambara & K.I. Gondaira, Phys. Rev. B24, 3441 (1981). L.L. Golik, Z.E. Kun’kova, T.G. Aminov & V.T. Kalinnikov, Fiz. TV. Tela 22, 877 (1980); L.L. Golik, Z.E. Kun’kova, V.E. Pakseev, T.G. Aminov & G.G. Shabunina, Fiz. TV. Tela 26 3080 (1984). Z.E. Kun’kova, L.L. Golik & V.E. Pakseev, Fiz. TV. Tela 25, 1877 (1983). W.J. Miniscalco, B.C. Collum, N.G. Stoffel & G. Margaritondo, Phys. Rev. B25, 2497 (1982). J.A. Gaj, J. Ginter & R.R. Galazka, Phys. Status Solidi (6) 89, 655 (1978). A. Selmi, R. Le Toullec & R. Faymonville, Phys. Status Solidi (6) 114, K97 (1982). T. Arai, M. Wakaki, S. Onari, K. Kudo, T. Satoh & T. Tsushima, J. Phys. Sot. Japan %I,68 (1973). A. Selmi, A. Mauger & M. Heritier, JMMM 66, 295 (1987). M.I. Auslender, N.G. Bebenin, B.A. Gizhevskii, V.A. Kostylev, N.M. Chebotaev, S.V. Naumov & A.A. Samokhvalov. Preprint of Institute of Metal Physics No. 87/2 (1987) [in Russian]. K. Kodama & T. Niimi, J. Appl. Phys. 49, 679 (1978). V.N. Berzhanskii & V.K. Chernov, Fiz. TV. Tela 24, 1895 (1982). A. Amith & G.L. Gunsalus, J. Appl. Phys. 40, 1020 (1969); A. Amith & L. Friedman, Phys. Rev. B2, 434 (1970).
0038-1098189 $3.00 + .OO Pergamon Press plc
Vol. 69, No. 7, pp. 761-764, 1989.
ON THE BAND STRUCTURE FERROMAGNETIC
AND ANISOTROPY OF TRANSPORT PROPERTIES SEMICONDUCTORS CdCr, Se, AND HgCr, Se,
OF
M.I. Auslender and N.G. Bebenin Institute of Metal Physics, Academy of Sciences of the USSR, Ural Division, Kovalevskaya 620219 Sverdlovsk, GSP, USSR
18,
(Received 2 May 1988 by M. Balkanski)
The simple model of the band structure is described. The model explains the temperature dependence of electron mass, the anisotropy of magnetoresistance of p-type materials, and the difference of the mobility in n-CdCr,Se, and n-HgCr,Se,. INTRODUCTION THE SPINELS CdCr,Se, and HgCr,Se, are ferromagnetic semiconductors (FMS) with rather high Curie temperatures (T, = 130 K and 110 K respectively) and the electron mobility ( lo’-lo3 cm* V-’ s-l). Although these semiconductors have been investigated during two decades, the experimental data on optical and transport properties are usually discussed without taking into account the peculiarities of the band structure. Moreover, the viewpoint that at the present time the band structure of these compounds cannot be found with any certainty is predominant. The purpose of our paper is to describe the simple model of the band structure which is based on the main features of the results of band calculations [ 1, 21 and the analysis of the experimental data on the polarized light absorption near the fundamental edge [3,4] (the last papers have not yet attracted due attention). The model enables us to explain the following experimental facts: (i) the strong dependence of electron mass in n-HgCr,Se, on temperature; (ii) the anisotropy of magnetoresistance of p-CdCr, Se, and p-HgCr, Se,; (iii) the striking difference between values of the electron mobility in n-CdCr,Se, and n-HgCr,Se,.
are important except for the case T < Tc. As for de-electrons, we treat them to form the Cr local moments which interact with the charge carriers via s-d exchange interaction Hs-d = 1 Z(r - R,,)S,, - s.
(1)
1.K
Here Z(r) is s-d exchange integral, S,, is the spin operator of Cr3+ ion in the site R,, = R, + eK (S = 3/2), eK is the position of Cr3+ ion in the primitive cell, and r, s are position and spin of the carrier. The correctness of s-d model in the case under consideration follows from the results of photoemission experiments [5]. There it was found that for CdCr,Se, the transition Cr3+-Cr4+ takes place at the energy z I .45 eV, so that de-“bands” lie far below the top of the valence band. In the standard Luttinger-Kohn basis 1nk) = u,,o(r) exp (ikr) the matrix elements of & can be written as fZ_,,(nk, n’k’) = 1 Izns(k - k’) * S,(k - k’), K
(2)
where &s(q)
= $ I dr I(r - e,) u,+,(W,dr)
SK(9) =
2 1 S,, exp Gq - R,),
exp (- iqr), (3)
I
THE MODEL OF THE BAND STRUCTURE We suppose that the bottom of s-like conduction band (4s Cr -i- 5s Cd or 4s Cr + 6s Hg) and the top of the p-like valence band (mainly 4p Se) are located at the I point (I, and F,, symmetries respectively). These bands can be described by one-electron Hamiltonian. ‘Near the s-like band there is a group of the empty narrow bands generated by antibonding p-dy orbitals A(4p Se + 3dy Cr) of F,*, I,;, I;, symmetries. For these bands the many-electron effects
Q, stands for primitive cell volume. In the mean field approximation (MFA) we get Q(nk,
n’k’) =
(S) * A,,d(k
A,,. = ; Ztd(O).
- k’), (4)
A,,. # 0 only if u,,,(r) and u,,o(r) belong to the same irreducible representation of 0: group. This means that (in MFA) s-d interaction does not mix s, p, and p-dy bands at I point.
761
762
BAND STRUCTURE
OF CdCr,Se, AND HgCr,Se,
Vol. 69, No. 7
When describing s- and p-like bands, our consideration is quite similar to that for semimagnetic semiconductors (see e.g., [6]). For I, (I,,) band we have A,,. = A,%,,,
(5)
n = (c, c), 0 = f l/2, A, = (Sl C, Z(r - e,) 1S), so the spectrum takes the form EC0 =
Ego + aA,I(S>I
+
‘;, L
where E,, is “unperturbed” energy gap. Let us consider the valence band. Spin-orbit interaction splits I,, into I, and I, bands. Since the value of splitting, A = 0.2-0.4 eV, is appreciably more than s-d exchange splitting, the upper band I, can be treated separately from Iti. Then we have [6] A,,.
=
$A,.&,,,,,~,
n
=
(m, v),
A, =
m
=
-J3/2,
+ l/2,
Fig. 1. Optical transitions which are responsible for absorption of light of “ - ” (1) and “ + ” (2) polarization. The direction of light propagation and quantization axes are parallel to magnetization.
(7) peratures not far from T, are required to determine
WI cZ(r - edl J3, K
A,.
J,,,,,,,are the angular momentum
3/2 matrices. The k-dependence of the hole spectrum is given by (for the case y2 = y3 see [6]) %rn = +mA,I (S) I + T (M<,‘)ijkikj,
T 3(1 - Gv)ygzinj.
(8)
(9)
The y’s are the Luttinger’s parameters of bare I, band, II = (S)/( (S) 1, i, j = x, y, z, the coordinate system is chosen in such a way that Ox )I [lo 01, etc., symbols (h, Z) denotes “heavy” (m = f 3/2) and “light” (m = f l/2) holes. The second term in the right hand side of equation (8) is assumed to be much less than the first one. The experimental data on polarized light absorption near the fundamental edge have been reported in [3] for CdCr,Se, and in [4] for HgCr,Se, single crystals, the values of absorption coefficient k being up to 6 x 103crn’. It has been shown that (i) at low temperatures the absorption of “ - ” polarized light starts at the lower energy than that of “ + ” polarized; (ii) the ratio k, /k_ x 3 (being slightly less than 3 for CdCr,Se, and slightly more than 3 for HgCr,Se,). These facts -__. __._ are_ explained by the optical transition scheme depicted in Fig. 1. This scheme is unique and allows us to find the signs of exchange parameters and the value of A, : A,, A, c 0, A, = - 0.05 eV. The value of A, cannot be obtained because the splitting of band is giant at T < T,. Measurements for tem-
TEMPERATURE DEPENDENCE ELECTRON MASS
OF
When temperature lowering, the effective mass of conduction band m,. in n-HgCr,Se, decreases from w 0.3m, at 300 K to x O.l5m, at 4 K [7], with energy gap E, decreasing from 0.8eV to 0.3eV [8]. We can connect the changes of m, and E, supposing k-p interaction between the conduction and valence bands to be of the most importance. In this case we can use the familiar expression m;’
=
m&’ +
21P12 E, + $A E,(E, + A)’
--i?--
(10) ignoring the valence band exchange splitting which is appreciably less than E,. In MFA E, = E,, l/2 I A,(S) 1, so E, seems to be temperature independent at T > T,. However, the shift of the conduction band is known to take place even in paramagnetic state due to spin fluctuations. We shall take them into account in a simple way considering Eg in (10) as being the true band gap measured in experiment. Then the experimental data for E, together with (10) at A = 0.2 eV give I
m;’ (4K) - rni’ = m;’ (3OOK) - m,’
“N 2*5’
This value is in excellent agreement with the data for m, which lead to ,l x 2.4.
BAND STRUCTURE
Vol. 69, No. 7
OF CdCr,Se, AND HgCr,Se,
MAGNETORESISTANCE OF p-TYPE FERROSPINELS We restrict ourselves to the case of low temperatures when 1A, (S) 1 is much more than k,T, the characteristic nondegenerate holes energy. Then equation (8) holds and the main contribution to the transport comes from m = + 312 subband. Taking for the relaxation time tensor rii = q,(m,/m,)3’Z6~ where md = (det IM)“~ is density-ofstates effective mass, for tensor of resistivities we get eii =
e. exp (-&lkJ)
M;‘M,.
(11)
Here fi2/2M, = yI + y2, e. = M, /(e2NPoro), NPo = (2nmok,T)312/(2nh)3, &, is the chemical potential for holes. In experiment one usually measures the resistivity at the fixed direction of the electric current j, eol, which connects with eii by the relation em = Ceijvivj, ij
v = j/lj].
(12)
Obviously, antisymmetric part of eii which is responsible for the Hall effect (whatever its nature) does not contribute to eel. The analysis of the experimental data shows the quantities 6 = (y2 - y3)/(y, + y2) and p = y,/ (y, + y2 - 3y3) to be less than unity. Expanding eol in a power series of 6 and p, and keeping only the linear terms in the small parameters, we obtain etil =
e. exp (-&,/k,T)
1 + 3~ cos2 (n, v) (
+
3iqr+vf r
.
(13)
>
Thus the resistivity of the p-type crystals is anisotropic. The anisotropy comes from (i) evident dependence of eii on Mii which leads to the dependence of resistivity not only on orientation of n and j relative to the cubic axes but also on the orientation of these vectors relative to each other (expression in parentheses in (13)); (ii) dependence of [, on orientation of magnetization vector relative to the cubic axes (but not to j). It follows from the orientation dependence of acceptor activation energy which is given by [lo] &* = P(n)
=
e
When magnetic field is high enough, the magnetization is along H all over the sample and we may employ eq. (13) in order to calculate @a](H).If H = 0, there is a domain structure. In every domain the resistivity is again given by (13) but, in general, the resistivity of the sample does not coincide with that of any domain. In fact one measures the resistivity averaged over the magnetization distribution. It means that the magnetoresistance of p-type FMS is due to the change of magnetic structure caused by magnetic field and, hence, must tend to saturation when magnetic field is being increased. Equation (13) shows that the magnitude of Ae/e is closely connected with the anisotropy of eoI: the more anisotropy, the greater magnetoresistance. The present theory allow us to understand the main features of the results of early experiments on Ae/e as well as the data on the magnetoresistance anisotropy. The anisotropy of de/q in p-CdCr, Se, was recently studied in [ll, 121. When the current was directed along [ 1 1 l] or [10 0] axes, the magnetoresistance did not change under rotation of field in the plane perpendicular to the current. On the other hand, when the field was rotated from being parallel to the current to a direction in the plane perpendicular to that, the 180” oscillations of Ae/e were observed [l 11. Further, when j II [ 1 T 0] and the field was rotated in the plane (1 1 0), the 180” oscillations of Ae/e were also found in heavily doped CdCr, Se, : Ag [ 121. It is easy to see that equation (13) explains all the features of the 180” oscillations as well as the absence of those for the configurations j 11[ 1 1 I], [ 10 01, H I j if we assume E, 4 kBT. When the band conduction dominates, eqs. (13) and (14) predict also the 90” oscillations to superimpose the 180” ones at low temperature. It was really observed in heavily doped p-HgCr,Se, (for details see [lo]). In lightly doped CdCr,Se, : Ag the authors of [ 121 observed the oscillations of Ae/e, the angular dependence of which can be described by the function P(n). Thus these oscillations are caused mainly by the n-dependence of the chemical potential. It may indicate that the band conduction does not prevail in this case.
Eg - slP(n), &z~ + rz$~ + $,Pz~
(14)
where so and E, (E, < et,) are positive constants. Usually experimental data refer to not em itself but to so called magnetoresistance
!!!&l=
763
4d-V
- ef.i@)
e&O
’
(15)
ON THE p-dy BANDS LOCATION The band calculations show that the bottom of every p-dy band is at X point whereas at I point there are the tops of the bands [ 1, 21. The direct allowed transitions I,, + I,2, I-;,, I& lead to the sharp rising of light absorption which is observed at ho 2 1.4eV in CdCr, Se, and fiw 2 0.9 eV in HgCr, Se, (see, for
764
BAND
STRUCTURE
OF CdCr,Se,
example, [3, 41). Considering the temperature dependence of this transitions together with that of the absorption edge, we can conclude that in HgCr,Se, the bottom of s-like band (F,) always lies lower than the tops ofp-dy bands. In CdCr,Se, it is true only at T < 200K while at T > 200K the s-band bottom is slightly higher than the p-dy bands maxima at r point. The more difficult problem is to determine where the bottoms of p-dy bands locate. For the time being there are no experimental data which enable one to do this unambiguously. Nevertheless, some tentative conclusions can be made. In HgCr,Se, the electron mobility is rather high, being up to 1800 cm2V ss’ at 4 K and 30 cm* V’ ss’ at 300 K, and the electron mass is not more than 0.3m, [9]. This stands to reason that the charge carriers move mostly in the broad band, so the bottom of any p-dy band lies higher than that of the s-band at all temperatures. On the contrary, in n-CdCr,Se, the mobility does not exceed 1 cm2V-’ s-’ whatever doping (see e.g., [13]) (the data on electron mass have not been reported). Thus we may take that in CdCr, Se, the bottom of a p-dy band is lower than the bottom of s-like band. DISCUSSION Of course, the present work is not the first one where the band structure of CdCr,Se, and HgCr,Se, have been discussed. For example the transport, magnetic and optical properties of HgCr,Se, were recently studied by Selmi, Mauger, and Heritier [9]. To interpret the experiments, they supposed that in CdCr,Se, and HgCr,Se, the lowest conduction band was of d-like, the s-like conduction band lies higher and the p-like valence band was assumed to be nondegenerate parabolic one. The extremums of all the bands were supposed to be at r point. This band scheme contradicts the band calculations. Also it fails to explain the values of mobility and effective mass in n-HgCr,Se,. Indeed, the dy-orbitals do not overlap directly, so p-dy bands are formed only due to p-d covalency
AND
HgCr,Se,
Vol. 69, No. 7
effects and, consequently, they are rather narrow. It is unlikely the electrons in these bands to have such small effective mass and high mobility. As regards the valence band, if it had been nondegenerate at r point, all anisotropy effects of AQ/Q (as well as the magnetoresistance itself) and circular dichroism would have been absent at T < Tc in p-type CdCr,Se, and HgCr, Se,.
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