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Indirect exchange interaction in semiconductors
Solid State Communications, Vol. 35, pp. 187—188. Pergamon Press Ltd. 1980. Printed in Great Britain. INDIRECT EXCHANGE INTERACTION IN SEMICONDUCTORS LLiu Department of Physics and Astronomy, Northwestern University, Evanston, IL 60201, U.S.A. (Received 11 February 1980; in revised for,n 19 March 1980 by F. Bassani)
We study theoretically the nature of the indirect exchange interaction between localized moments in a semiconductor, mediated by the spin-dependent spatial polarization of the valence band electrons. The interaction drops off exponentially with the inter-spin distance and the decay constant depends on the energy gap and carrier effective masses. The sign of the interaction is ferromagnetic within our simplified model but could be reversed for some more complicated band edge structures.
THE POLARIZATION of conduction electrons in a metal by local magnetic moments results in the wellkiiown RKKY indirect exchange interaction [1]. For a dilute magnetic semiconductor alloy, the RKKY mechanism is ineffective as there is only a small number of free carriers present. But there is a corresponding effective interaction between local moments mediated by the spin-dependent spatial polarization of valence band electrons through exchange-induced virtual interband transitions. In this paper we study theoretically the nature of this interband exchange interaction. It is found that the interaction drops off exponentially with the inter-spin distance with a decay constant depending on the energy gap and carrier effective masses. For certain narrow-gap semiconductors with small carrier effective mass, the interaction range can be long, extending over several lattice spacings. Within our simplified model, the sign of the interaction is ferromagnetic but it could be reversed when the interband exchange coupling has some complicated dependence on the angle between the wave vectors of the band edge Bloch electrons. Let us denote the indirect exchange interaction between two local spins, S, and Sj, as H,1S1~S,. Within a two-band approximation, the interband contribution to Jf~jat temperature T = 0 may be obtainedby second order perturbation theory as 1~~ -RU, (1) I-t~~~ k”I~ k~k’E~(k) E~(k) H,~= ‘ , x e where E 0(E~)is the conduction (valence) band energy, and Ru denotes spatial separation of the two local moments. The R-dependent part of the exchange interaction e1~~ ) RU has been factored out, and hence the exchange integral J(k, k’) is to be evaluated with the local moment site as the origin. Explicitly It is given —
by
— ‘
*
J(k, k)
=
1
*
j dr1 dr2 ~~k(rI)’I’(rl)—‘I’ (r~)Ø~,~’(r~),
(2) involving conduction and valence band Bloch functions ~ and ~k~’and local moment orbitals ~,. Since the polarization effect in semiconductors involves all the electronic states in the entire Brillouin zone (BZ), it usually only depends on the gross features of the energy band structure. Contrary to this general expectation, the interaction Hu as given by equation (1) is very sensitive to its details. This can be seen by assuming a constant J and a flat E~or E~.Then the interaction H11 is proportional to i eik. RU. As noted by Sokel and Harrison [2], this sum over the BZ is identically zero because Ru is a lattice vector. In other words, when the electron states in the entire BZ are equally weighted, their contributions to the interband exchange interaction will exactly cancel each other out. Hence the most significant contributions should come from regions in the BZ where the interband exchange integral or the energy gap varies strongly with the wave vector of the Bloch electrons. One possible place to look for a strong variation of energy with wave vector is in the vicinity of band edges. Thus we shall study the nature of the interband transitions involving the two band extrema. We assume that they are both located at the center of the BZ. In its from theboth top of the valence we can explicitly vicinity, E~ and E~are band, parabolic. Measuring energy write them down as follows: 2k2
+h
=
—,
(3)
h2k2 E~=
—
where E, is the energy gap. Normally, the interband
187
188
INDIRECT EXCHANGE INTERACTION IN SEMICONDUCTORS
exchange integral is not expected to depend strongly on the wave vector in the immediate neighborhood of the band edge points. Hence we assume J to be a constant. Now we can calculate the interaction H~in equation (I). In performing the integral over k-space, we avoid introducing any sharp cut-offs which would lead to non-physical oscifiations in r-space. Instead we extend both k and k’ summations to infinity. Because of the k2-dependence of the energy denominator and of the large amount of cancellations expected from an oscillating numerator, the spurious contributions from large k and k’ are made small. We then obtain the following result:
II2iz\~~ ~
j2
Vol. 35, No. 2
mum is still located at the center of the BZ but several conduction band minima are at q~,n being the electron valley index. If electron ellipsoids are approximated by spheres, the result in equation (4) is simply multiplied by an oscillating factor E,, e lQ~ R~j Thus, we may have oscillatory interactions in indirect-gap semiconductors except that the amplitude must be modulated by an exponential factor. In our simplified model, the interband exchange interaction in a direct-gap semiconductor is always ferromagnetic. However, the sign of the interaction could be reversed if the interband exchange integral has some directional dependence on the Bloch wave vectors k and k’. For example, in dilute HgMnTe alloys (- 1% of Mn),
K 1(k0R~1)
3R~ =
—
8ir
1
R
4PE5 V’MeMh K~(koRu)j~
(4)
where K1 is the modified Bessel function [3] of the first order, and K’1 its first derivative. The constant k0 stands for 2 J1/2 (5) k0 = [2(Me + Mh)Eg/h and /1 is the electron—hole reduced mass, i.e. MeMh =
(6)
M~+M~
experiments [5] show that there exists an antiferromagnetic coupling between the moments. The host material HgTe is a zero-gap semiconductor with a degenerate F8 band edge, which is symmetry induced. Bastard and
Lewiner [6] took into account the complicated dependence of the exchange integral Jon the angle between k
and k’ arising from the r8 band edge, and indeed confirmed that the interband exchange interaction should be antiferromagnetic. One could also perceive cases where details of band structure near the band edges or some other points in the BZ could make the behavior of the interaction different from what has been predicted in this paper. In any case, semiconductors are ideal host materials for studying magnetic interactions. Their band
Let us study the limiting behavior of H~
structures are varied but well known. Furthermore their structures may be manipulated by alloying or by applying external fields.
1.Using the
asymptotic expansions for K1 and K~with large arguments [4], we obtain 2E pJ —
54R~ /lrMeMh)h/2 e~oRu ~2koRu
koR~ ~>
~
(7)
2ir~h
On the other hand, the limiting behavior ofK 1 (x) and
Acknowledgements features the interaction [2, 6—8]. I would like Some to thank Drs G.ofBastard and obtained here were also predicted by other authors C. Lewiner for sending me a preprint of their work —
and Dr A. Czachor for helpful discussions.
K~(x)forx 0 gives -~
3h2(Me + Mh )R~’ k H~1
—
27T
0R1~~ 1.
(8)
Comparing the present results with the RKKY interaction, we first note an absence of Friedel oscillations. Instead the interaction stays in one sign, which is negative (ferromagnetic) according to our simplified model. Because of the existence of a gap in the energy spectrum, the interaction damps off exponentially at large inter-spin distances. At small separation, it varies according to a power law which is R~ in our model. The range of the interaction as given by k~ can be long (i.e. several lattice spacings) for narrow gap semiconductors with small electron and hole masses. For zero-gap semiconductors the exponential factor is absent and the interaction behavior is governed by equation (8). We can generalize our model to an indirect-gap semi conductor. Let us assume that the valence band maxi-
1. 2. 3.
4. 5. 6. ~ 8.
REFERENCES M.A. Ruderman & C. Kittel, Phys. Rev. 96, 99 (1954). R. Sokel & W.A. Harrison, Phys. Rev. Lett. 36, 61(1976). The function K1 (x) can be expressed in terms of the Bessel function of the first kind, J1, and Bessel function of the second kipd, N., with imaginary arguments, as K1(x) = (i~/2)e1~~2I [J1(ix) + iiV1 (ix)]. G.N. Watson,A Treatise on the Theory ofBessel Functions. Cambridge Univ. Press, Cambridge, England. M. Jaczynski, J. Kossut & R.R. Galazka, Phys. Status Solidi (b) 88, 73 (1978), and references therein. G. Bastard & C. Lewiner, Phys. Rev. B20, 4256 (1979). J. Ginter, J. Kossut & L Swierkowski, Phys. Status Solidi (to appear). N. Bloembergen & T.J. Rowland, Phys. Rev. 97, 1969 (1955).
We study theoretically the nature of the indirect exchange interaction between localized moments in a semiconductor, mediated by the spin-dependent spatial polarization of the valence band electrons. The interaction drops off exponentially with the inter-spin distance and the decay constant depends on the energy gap and carrier effective masses. The sign of the interaction is ferromagnetic within our simplified model but could be reversed for some more complicated band edge structures.
THE POLARIZATION of conduction electrons in a metal by local magnetic moments results in the wellkiiown RKKY indirect exchange interaction [1]. For a dilute magnetic semiconductor alloy, the RKKY mechanism is ineffective as there is only a small number of free carriers present. But there is a corresponding effective interaction between local moments mediated by the spin-dependent spatial polarization of valence band electrons through exchange-induced virtual interband transitions. In this paper we study theoretically the nature of this interband exchange interaction. It is found that the interaction drops off exponentially with the inter-spin distance with a decay constant depending on the energy gap and carrier effective masses. For certain narrow-gap semiconductors with small carrier effective mass, the interaction range can be long, extending over several lattice spacings. Within our simplified model, the sign of the interaction is ferromagnetic but it could be reversed when the interband exchange coupling has some complicated dependence on the angle between the wave vectors of the band edge Bloch electrons. Let us denote the indirect exchange interaction between two local spins, S, and Sj, as H,1S1~S,. Within a two-band approximation, the interband contribution to Jf~jat temperature T = 0 may be obtainedby second order perturbation theory as 1~~ -RU, (1) I-t~~~ k”I~ k~k’E~(k) E~(k) H,~= ‘ , x e where E 0(E~)is the conduction (valence) band energy, and Ru denotes spatial separation of the two local moments. The R-dependent part of the exchange interaction e1~~ ) RU has been factored out, and hence the exchange integral J(k, k’) is to be evaluated with the local moment site as the origin. Explicitly It is given —
by
— ‘
*
J(k, k)
=
1
*
j dr1 dr2 ~~k(rI)’I’(rl)—‘I’ (r~)Ø~,~’(r~),
(2) involving conduction and valence band Bloch functions ~ and ~k~’and local moment orbitals ~,. Since the polarization effect in semiconductors involves all the electronic states in the entire Brillouin zone (BZ), it usually only depends on the gross features of the energy band structure. Contrary to this general expectation, the interaction Hu as given by equation (1) is very sensitive to its details. This can be seen by assuming a constant J and a flat E~or E~.Then the interaction H11 is proportional to i eik. RU. As noted by Sokel and Harrison [2], this sum over the BZ is identically zero because Ru is a lattice vector. In other words, when the electron states in the entire BZ are equally weighted, their contributions to the interband exchange interaction will exactly cancel each other out. Hence the most significant contributions should come from regions in the BZ where the interband exchange integral or the energy gap varies strongly with the wave vector of the Bloch electrons. One possible place to look for a strong variation of energy with wave vector is in the vicinity of band edges. Thus we shall study the nature of the interband transitions involving the two band extrema. We assume that they are both located at the center of the BZ. In its from theboth top of the valence we can explicitly vicinity, E~ and E~are band, parabolic. Measuring energy write them down as follows: 2k2
+h
=
—,
(3)
h2k2 E~=
—
where E, is the energy gap. Normally, the interband
187
188
INDIRECT EXCHANGE INTERACTION IN SEMICONDUCTORS
exchange integral is not expected to depend strongly on the wave vector in the immediate neighborhood of the band edge points. Hence we assume J to be a constant. Now we can calculate the interaction H~in equation (I). In performing the integral over k-space, we avoid introducing any sharp cut-offs which would lead to non-physical oscifiations in r-space. Instead we extend both k and k’ summations to infinity. Because of the k2-dependence of the energy denominator and of the large amount of cancellations expected from an oscillating numerator, the spurious contributions from large k and k’ are made small. We then obtain the following result:
II2iz\~~ ~
j2
Vol. 35, No. 2
mum is still located at the center of the BZ but several conduction band minima are at q~,n being the electron valley index. If electron ellipsoids are approximated by spheres, the result in equation (4) is simply multiplied by an oscillating factor E,, e lQ~ R~j Thus, we may have oscillatory interactions in indirect-gap semiconductors except that the amplitude must be modulated by an exponential factor. In our simplified model, the interband exchange interaction in a direct-gap semiconductor is always ferromagnetic. However, the sign of the interaction could be reversed if the interband exchange integral has some directional dependence on the Bloch wave vectors k and k’. For example, in dilute HgMnTe alloys (- 1% of Mn),
K 1(k0R~1)
3R~ =
—
8ir
1
R
4PE5 V’MeMh K~(koRu)j~
(4)
where K1 is the modified Bessel function [3] of the first order, and K’1 its first derivative. The constant k0 stands for 2 J1/2 (5) k0 = [2(Me + Mh)Eg/h and /1 is the electron—hole reduced mass, i.e. MeMh =
(6)
M~+M~
experiments [5] show that there exists an antiferromagnetic coupling between the moments. The host material HgTe is a zero-gap semiconductor with a degenerate F8 band edge, which is symmetry induced. Bastard and
Lewiner [6] took into account the complicated dependence of the exchange integral Jon the angle between k
and k’ arising from the r8 band edge, and indeed confirmed that the interband exchange interaction should be antiferromagnetic. One could also perceive cases where details of band structure near the band edges or some other points in the BZ could make the behavior of the interaction different from what has been predicted in this paper. In any case, semiconductors are ideal host materials for studying magnetic interactions. Their band
Let us study the limiting behavior of H~
structures are varied but well known. Furthermore their structures may be manipulated by alloying or by applying external fields.
1.Using the
asymptotic expansions for K1 and K~with large arguments [4], we obtain 2E pJ —
54R~ /lrMeMh)h/2 e~oRu ~2koRu
koR~ ~>
~
(7)
2ir~h
On the other hand, the limiting behavior ofK 1 (x) and
Acknowledgements features the interaction [2, 6—8]. I would like Some to thank Drs G.ofBastard and obtained here were also predicted by other authors C. Lewiner for sending me a preprint of their work —
and Dr A. Czachor for helpful discussions.
K~(x)forx 0 gives -~
3h2(Me + Mh )R~’ k H~1
—
27T
0R1~~ 1.
(8)
Comparing the present results with the RKKY interaction, we first note an absence of Friedel oscillations. Instead the interaction stays in one sign, which is negative (ferromagnetic) according to our simplified model. Because of the existence of a gap in the energy spectrum, the interaction damps off exponentially at large inter-spin distances. At small separation, it varies according to a power law which is R~ in our model. The range of the interaction as given by k~ can be long (i.e. several lattice spacings) for narrow gap semiconductors with small electron and hole masses. For zero-gap semiconductors the exponential factor is absent and the interaction behavior is governed by equation (8). We can generalize our model to an indirect-gap semi conductor. Let us assume that the valence band maxi-
1. 2. 3.
4. 5. 6. ~ 8.
REFERENCES M.A. Ruderman & C. Kittel, Phys. Rev. 96, 99 (1954). R. Sokel & W.A. Harrison, Phys. Rev. Lett. 36, 61(1976). The function K1 (x) can be expressed in terms of the Bessel function of the first kind, J1, and Bessel function of the second kipd, N., with imaginary arguments, as K1(x) = (i~/2)e1~~2I [J1(ix) + iiV1 (ix)]. G.N. Watson,A Treatise on the Theory ofBessel Functions. Cambridge Univ. Press, Cambridge, England. M. Jaczynski, J. Kossut & R.R. Galazka, Phys. Status Solidi (b) 88, 73 (1978), and references therein. G. Bastard & C. Lewiner, Phys. Rev. B20, 4256 (1979). J. Ginter, J. Kossut & L Swierkowski, Phys. Status Solidi (to appear). N. Bloembergen & T.J. Rowland, Phys. Rev. 97, 1969 (1955).