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Anomalous hopping magnetoresistance in semiconductors with complex magnetic structure: Application to lightly doped La2CuO4
Solid State’Communications, Printed in Great Britain.
Vol. 78, No. 3, pp. 205-210,
ANOMALOUS HOPPING MAGNETORESISTANCE MAGNETIC STRUCTURE: APPLICATION Alexander P.N. Lebedev
Physical
1991.
00381098/91 $3.00 + .OO Pergamon Press plc
IN SEMICONDUCTORS TO LIGHTLY DOPED
WITH COMPLEX La,CuO,
0. Gogolin
Institute, Academy of Sciences of the USSR, Leninskii prospect 53, USSR
117924, Moscow,
and Alexey S. Ioselevich L.D. Landau Institute Kosygina str. 2, USSR*
for Theoretical Physics, Academy of Sciences of the USSR, 117940, Moscow, and Institute for Scientific Interchange, Villa Gualino, Viale Settimio Sever0 65, 10133 Torino, Italy (Received
10 December
1990 by A.L. Efros)
A mechanism of hopping magnetoresistance (MR) due to the interaction of localized carriers’ spins and and spins of magnetic subsystem in magnetic semiconductors is proposed. The external magnetic field H affects resistance R indirectly: H governs the order in magnetic subsystem, which in its turn affects the hopping probabilities. The arising magnetoresistance is unusual: it may be negative and may have jumps or kinks at magnetic phase transitions. The experimental data on MR of La,CuO, is interpreted. It is shown that for the quantitative agreement the assumption of a polaronic effect in this substance is required.
1. INTRODUCTION THIS WORK was inspired by the experiments [1, 21 on MR of La,CuO,. In these experiments a correlation between the characteristic features of MR and magnetic phase transitions was observed, although the mechanism of MR was unclear. The idea of the explanation of results of [1] was proposed in our letter [3]. Here we generalize the approach of paper [3]. We develop a simplest model in which a proposed MR mechanism appears. This model can be easily modified in order to take into account the details of specific substances structure. In particular, we explain the results of [2], which requires a somewhat deeper insight in the structure of acceptor states. We believe that the proposed mechanism can dominate not only in La,CuO, but also in other magnetic semiconductors with hopping transport. 2. THE MODEL In order to demonstrate the basic MR spin mechanism we shall consider the two-sublattice 3D Ising model with exchange J > 0 (spins Si are directed
* Permanent
address.
along the z-axis). It’s ground state is antiferromagnetic (AF). In an external field H = Hc a phase transition occurs from AF-state (Fig. la) to a ferromagnetic one (F-state, Fig. lb). Suppose that some of the magnetic sites are donors (i.e. besides the spins S, there are also the localized electronic levels si associated with these sites). At low temperatures T the electrons are bound to the donors and a process of the conductivity proceeds via variable range hopping (VRH). The spin rri of an electron, localized at the ith donor interacts with a spin of a donor site Sj. The corresponding Hamiltonian is H = K&a, where K is an effective exchange. In the mean field approximation the molecular field affecting the electrons spin rr, is Ai
=
KM,,
(1)
where M, is the sublattice magnetization (S;) = M,, CLis a number of sublattice. In AF-state M, = - M2 = (0, 0, S) and in F-state M, = M2 = (0, 0, S). For the sake of simplicity we have neglected a “direct” Zeeman interaction: gpBHui. This approximation is well founded if K ti J, but, in any case, taking the direct interaction into account does not change the physical results. If the lifetime rhop of an electron on a donor is
205
206
ANOMALOUS
HOPPING
MAGNETORESISTANCE
IN SEMICONDUCTORS
Vol. 78, No. 3
multiphonon polaronic hops. The theory of such hops was developed in [4] where it was shown that W(E) cc exp (- E/2T) for T > T* = hw,,/2 In is a characteristic phonon fre( W,IE), where Oph quency and W, is a polaronic shift. We shall restrict ourselves to relatively high T > T*. Then it can be easily obtained from (3) that (al
6 5 In (B$“/~j~‘)
lb)
=
In [ch (A/2T)].
(4)
Fig. 1. Spin configurations and electronic hops in a simple model. Spins of magnetic ions are shown as arrows, donors as circles, even hops are solid lines, odd hops are broken lines. (a) AF-state (at H < H,). One half of the hops are even, another half is odd. (b) F-state (at H > Hc). All hops are even.
The quantity 6 characterizes the distinction between even and odd transitions, the sign of 6 determines the sign of MR (see below). The theory of hopping conductivity is based on the idea of a percolation through the random resistances network [5]. The resistance
much larger than the relaxation time for its spin, then the distribution function for an electron spin will be of the Gibbs form
R,, =
g(o,)
=
exp (-Aai)/2
ch (A/2T).
(2)
To obtain the average probability I#$ of a hop (i -+ j) one has to sum over the initial electron states with a weight g(o,). Suppose that the hopping electron cannot reverse its spin during the hop. Then it is clear that B$ depends strongly on whether the spins of initial and final states are parallel (wj+’ - an even transition) or antiparallel (I+ ) - an odd one, Fig. 2) l$;+’ cc B’(A), W!,-’ cc {ema”rW(A
-
A)
+ ea12’ W(A + A))/2 ch (A/2T).
(3)
Here W(E) is a probability of a hop between two specific spin states in the splitted doublets (corresponding to any line in Fig. 2) as a function of the difference between their energies. The shape of W(E) function depends on the character of transitions. Having in mind the applications to La,CuO, (and other perovskites) where there are some evidences of polaronic effect, we shall consider here only
T/e2J;(l
- fi)K,
=
B exp (&,),
(5)
corresponds to each pair of donors (zj). Here f; = [ 1 + exp (E,/T)] ’ is the probability of i th donor to be occupied by an electron; si is the position of the energy level (the centre of the spin doublet) accounted for the Fermi level; B is a preexponential factor; tij is a so called “connectivity function”. If both i andj donors belong to the same sublattice (i.e. for an even transition) one can obtain [4, 51
2(ri,/a)
+ (Is,1 + Is/l)/2T,
(6)
where the first term arises from the squared overlap integral (ri, is the distance between i andj donors and a is the radius of the localized state), while the second term arises from the different T-dependent factors in equations (3) and (5). For the odd transitions according to the definition of 6 we have
4(f) + 6. ‘I
(7)
In a strong magnetic field (H > H, i.e. in the F-state) there is only one sublattice and all the hops are even (Fig. 1b). In this case for the calculation of the resistance in VRH regime a standard percolation method can be applied. The resistance is R,
=
B exp (ti!)),
(8)
where cf? = (4n,./gTu3)“” corresponds to a percolation threshold, determined by an equation
;J;‘A’ J IAi (l/4)gTa3(tjo’)4
Ai,
Ai,
(4
lb)
Fig. 2. Energetical diagram of spin-conserving electronic hops. (a) even hop (Ai is parallel to A,), (b) odd hop (A, is antiparallel to A,).
=
n,,
(9)
where g is a density of electronic states at a Fermi level, n, g 5 is a universal critical concentration for the dimensionless percolation problem [5]. In a weak field (H < H,, Fig. 1a) there are two sublattices and one half of the hops are even while another half is odd. So all the donors are divided in two classes (according to their belonging to magnetic
Vol. 78, No. 3
ANOMALOUS
HOPPING
MAGNETORESISTANCE
sublattices) and for intraclass hops and interclass ones the connectivity function is determined by equations (6) and (7) respectively. In order to obtain & for such an “extended percolation problem” we shall generalize equation (9) calculating the total number of sites j (of both classes) connected with a given site i according to a connectivity criterion <,, < <, . Then we have
where 9(x) = 1 for x > 0 and 9(x) = 0 for x < 0; g/2 is a density of states belonging to one sublattice. The first term in equation (10) comes from the sites of the class to which the site i belongs and the second one comes from the sites of another class. The equation (10) is not rigorous is one aspect. Strictly speaking, the value of n,. can depend on the shape of “a percolation figure” in r - E space, and this figure changes its form when 6 is varied. But the numerical simulations (c.f. [5]) show that the variations of n,. are very small (of order of 1Oh) in all the cases studied. So, we hope, that considering n, = const in (10) is a good approximation. Then, solving equation (lo), we obtain d/2 + {[h4/2 + ([10))4]‘/2 4‘(S)
=
21/4rjo)
36/4}“*
(lo’ + 6/2.
AR’R
(4) and (12): = =
(Ro - RH)‘RH [ch (A/2T)]“2 -
In lightly
for
6 > 2”45y).
Note, that this result coincides with the first order perturbation theory result [5]. Indeed, according to [5] the linear in 6 correction to 5, must have a form 5, - t$‘) = (6, ). But since 6, = 0 for even hops and 6, = 6 for odd ones, one has (6,) = 612. For large 6 (6 > 2’145!?) all the connections between sites of different classes are broken, so the percolation occurs via sites of each class independently and the lower result in (11) corresponds to the usual hopping but with a two times less density of states. So, in both limiting cases equation (10) turns to give a correct result and there are good chances that it works well in the intermediate region 6 ? 5!?. The theory may be generalized also for the case of more than two sublattices and the percolation threshold may be found [6]. Summing up the results of this section, the resistance R does not depend on H for H -c H, and for H = Hc it undergoes a jump. The relative magnitude of this jump for 6 @ ty’ can be obtained from
exp (d/2) -
1.
1 (13)
3. APPLICATION TO La,CuO,: THEORY VS EXPERIMENT
6 < 2”45jo)
(12)
=
The main features of anomalous MR were described above for a simplest model of 3D Ising antiferromagnet. For the application of a method to more sophisticated substances one has: (i) to study a behaviour of magnetic sublattices in an external field, (ii) to examine a structure of impurity states in order to determine the molecular field affecting the spin of localized carrier, (iii) to calculate the hopping probabilities B$ and (iv) to formulate and to solve an extended percolation problem with a number of classes equal to the number of different possible configurations of molecular fields affecting the spins of carriers localized on different impurities.
for
Discuss the limiting cases of equation (11). When 6 = 0 the two classes merge and 4, = [i!’ as in a conventional percolation problem. For 6 4 5:? we have & =
equations
207
IN SEMICONDUCTORS
doped
or slightly
nonstoichiometric
(11)
La,CuO, the 3D VRH conductivity was observed: R(T) cc exp [(To/T)“4], To N 105-106K [7, 81. The carriers are holes localized presumably at the acceptors. The H-dependence of the resistance R (obtained in [1, 21) is shown in Fig. 3. In order to explain this dependence we shall fulfil the program listed at the end of the previous section. The magnetic structure of La-system is well known. It’s distinction from the simple antiferromagnet is due to the orthorhombic distortions, which produce the anisotropic antisymmetric exchange [9, lo] and out-of-plane spin canting (by a small angle 9 N 3 x lo3 [l]). As a result, each Cu-0 plane acquires a small ferromagnetic moment, the moments of subsequent planes are ordered antiferromagnetitally along the b-axis at H = 0. In the field H I(b this antiferromagnetic order is changed to a ferromagnetic one (weak ferromagnetism) at H = H, N 4T. The magnetic behaviour in H J_ b is more complicated [2]. Thus at H I(c there is a magnetic phase transition at H = H, N 10 T accompanied by the reorientation of a staggered moments M, from bc to ab plane. Index c( = +,distinguishes even and odd Cu-0 planes (with opposite orientations of ferromagnetic moments at zero field). At H = H2 N 20T the second transition occurs: M, become parallel to b axis. For details c.f. [2, 61.
208
ANOMALOUS
HOPPING
MAGNETORESISTANCE
R0-I) R(O)
R(H)
R(O)
I
I
I
5
Hl
“2
HO
h
(cl
1 F
0.5
IN SEMICONDUCTORS
Vol. 78. No. 3
interpretation of the experimental data (see below). It should be noted that strictly speaking bi is not just a holes spin, but rather a total spin of a compound system including a hole and the neighbouring coppers. The spin of such a dressed electron appears to stay l/2 (i.e. unchanged) for all the cases of interest [6], so the ‘magnetic dressing’ effects do not change the results of the present theory. The vector o is proportional to the orthorhombic rotation angle in the vicinity of impurity. If we would take for that the bulk value, known from the neutron experiments [I] we obtain w = w0 2: 10 K which is too small for the interpretation of strong MR. However, the local distortion has not to coincide necessarily with a bulk one; it may be enhanced substantially if to admit the polaronic effect [12-141. If the polaronic effect is strong (i.e. distortions are of order of unity) then o - 100 K, which fits well the data of [l, 21 (see below). In [14] we have studied the interaction or a localized hole and the low lying phonon modes and have shown that there may arise JahnTeller (or pseudo Jahn-Teller) enhancement of local orthorhombic distortion. Note also [6] that besides the a-component (the bulk vector o0 is parallel to the a-axis) the vector w can acquire the c-component
(Iwul= Id>.
I
100
T (K) Fig. 3. (a) The field-dependence of the resistivity for H 11b (according to [l]). (b) The same for H )Ic: solid line - experimental data [2] at T = 24 K, broken line - the theoretical result (equation (15)), where H, = 10 T and H, = 17 T were taken from [2] and the parameter oM/2T = 2.19 was obtained from the magnitude of the overall resistance drop AR/R = (R(0) - R(co))/R(oo). (c) Temperature dependence of AR/R. It is the same for H 11b and H IIc cases [2]. As to acceptor state, by now its structure cannot be extracted directly from experimental data. We shall suppose here that the symmetry group of acceptor includes the permutation of magnetic sublattices. It is true in particular for a simplest model of a hole localized on a single 0 atom [ 111, as well as for some other models [6]. For such a symmetry the molecular field affecting a holes spin bi is non-zero only due to the canting of spins:
and an exchange term (of (1) type) is forbidden by a symmetry. The last fact is extremely important for the
The molecular fields A, are shown in Fig. 4 for the case H II6. The directions of A, are opposite for impurities situated in the even and odd planes respectively. So, there are two classes of sites in the percolation problem and the consideration of Section 2 can be applied to this case without any modifications. The
(a)
Fig. 4. The configurations of molecular fields A, (for even planes) and A (for odd ones) at H II6: (a) AF-state (H < H,.), (b) F-state (H > Hc).
Vol. 78. No. 3
ANOMALOUS
HOPPING
MAGNETORESISTANCE
shape of R(H) dependence in Fig. 3a coincides completely with that predicted by the theory developed above. The magnitude of a jump is given by equation (13) with A = lo,lM. From the experimental temperature dependence of AR (Fig. 3c) and equation (13) we obtain the T-dependence of A (Fig. 5). If the polaronic shift W, $ T then o is almost T-independent and Fig. 5 must reproduce the T-dependence of staggered magnetization M(T). Since M ? 1, the magnitude of o must be as large as u - 200K, which may be due only to a polaronic enhancement of the local orthorhombicity in the vicinity of an acceptor. We think that this fact provides an additional evidence for a substantial polaronic effect in La,CuO,. Some other aspects of polaronic effect we have discussed in [ 151. The most striking qualitative feature of M(T) (Fig. 5) is the drop of M at low T. Such a drop (though less pronounced) was obtained also in neutron scattering studies [16] and we believe that it is an inherent property of a magnetic system (the theoretical discussion see in [ 171). We do not discuss here the quantitative comparison of Fig. 5 with experimental data since there were no experiments (to our knowledge) in which MR and the staggered magnetization were measured on the same samples.
The case H _Lb is much more complicated. It is considered in our paper [6] in details. Here we shall point out only the main peculiarities of this situation. First of all the crystals of La,CuO, are usually twinned and the field orientation with respect to the crystal axes a and c is different in different domains. Besides that importance is the possible presence of c-component of o vector. The four classes may arise in the extended percolation problem since each acceptor is characterized not only by the number (even or odd) of the corresponding plane but also by the sign of w,. Generally, these signs for different acceptors may be either random or correlated, which affects MR. The magnetic phase transition at H = H, (H 11c) is the first order transition. It means that this transition would have been accompanied by the jump in R, (just as in H IIb case). Butexperimentally such a jump is absent and there are ody kinks in R(H) at both H, and H,. Besides that the overall relative drop of resistivity (R(0) - R(co))/R(co) turned out to be identical for H IIb and HI b. Both these facts can be explained if we suppose some specific type of 0,. correlation (for details see [6]). The result for R(H) is R(H)
=
B exp (5!“‘){(1 + x)/[2x
(H/Hz )’ >
1 1,
P
2 *O’
x)
+ H,)
-
H21j2,
(15)
H < H, H, < H < H2 H, < H.
This dependence is presented in Fig. 3b and it fits the experimental curve satisfactorily. In conclusion, the proposed mechanism explains well the effect of MR in La,CuO,. The analysis of experimental data [ 1,2] enables one to conclude about the symmetry of the acceptor state (i.e. the point symmetry group must include a magnetic sublattice permutation) and about a local orthorhombicity enhancement in the vicinity of acceptor. This enhancement apparently has a cooperative character which is manifested by the correlations of c-components of vector of orthorhombic rotations for different impurities. Acknowledgements
Fig. 5. The temperature dependence of A = 1w,lM obtained by fitting of equation (13) to the data from Fig. 3c.
+ (1 -
x sch (oM( 1 + x)“*/2T)]} -I”,
{HH,I]H,(H,
Besides that, all our calculations were carried out in the framework of the mean field approximation and the neglecting of magnetic fluctuations can lead to the under-estimation of M(T) at low T (c.f. [6]).
209
IN SEMICONDUCTORS
We are indebted to R.J. Birgeneau, M.A. Kastner, N.W. Preyer & T. Thio for the discussions concerning the experiments and also to S.A. Brazovskii, B.I. Halperin, M.E. Raikh, D.E. Khmelnitskii, E.G. Tsitsishvili and B.I. Shklovskii for valuable comments. One of us (A.S.I.) is grateful to ISI Foundation for the hospitality and support during his stay at the Institute of Scientific Interchange.
210
ANOMALOUS
HOPPING
MAGNETORESISTANCE
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
T. Thio et al., Phys. Rev. B38, 905 (1988). T. Thio et al., Phys. Rev. B41, 231 (1990). A.O. Gogolin & A.S. Ioselevich, JETP Lett. 51, 174(1990). H. Sher & T. Holstein, Phil Msg. B44, 343 (1981). B.I. Shklovskii & A.L. Efros, Electronic Properties of Doped Semiconductors. Springer (1984). A.O. Gogolin & A.S. Ioselevich, JETP 98, 681 (1990) (in Russian). M.A. Kastner et al., Phys. Rev. B37, 111 (1988). N.W. Preyer et al., Phys. Rev. B39,11563 (1989). I. Dzyaloshinsky, J. Phys. Chem. Sol. 4, 241 (1958).
10. 11. 12. 13. 14. 15. 16. 17.
IN SEMICONDUCTORS
Vol. 78, No. 3
T. Moriya, Phys. Rev. 120, 91 (1960). A. Aharony et al., Phys. Rev. Lett. 60, 1330 (1988). S. Barisic & I. Batistic, Physica Scripta. 27, 78 (1989). C.M. Foster et al., Physica C. C162-164 (1989); Synthetic Metals. 33, 171 (1989). A.O. Gogolin & A.S. Ioselevich, JETP Lett. 50, 502 (1989). A.O. Gogolin & A.S. Ioselevich, JETP Lett. 51, 532 (1990). Y. Endoh et al., Phys. Rev. B37, 7443 (1988). L.I. Glazman & A.S. Ioselevich, JETP Lett. 49, 579 (1989); Z. Phys. B80, 133 (1990).
Vol. 78, No. 3, pp. 205-210,
ANOMALOUS HOPPING MAGNETORESISTANCE MAGNETIC STRUCTURE: APPLICATION Alexander P.N. Lebedev
Physical
1991.
00381098/91 $3.00 + .OO Pergamon Press plc
IN SEMICONDUCTORS TO LIGHTLY DOPED
WITH COMPLEX La,CuO,
0. Gogolin
Institute, Academy of Sciences of the USSR, Leninskii prospect 53, USSR
117924, Moscow,
and Alexey S. Ioselevich L.D. Landau Institute Kosygina str. 2, USSR*
for Theoretical Physics, Academy of Sciences of the USSR, 117940, Moscow, and Institute for Scientific Interchange, Villa Gualino, Viale Settimio Sever0 65, 10133 Torino, Italy (Received
10 December
1990 by A.L. Efros)
A mechanism of hopping magnetoresistance (MR) due to the interaction of localized carriers’ spins and and spins of magnetic subsystem in magnetic semiconductors is proposed. The external magnetic field H affects resistance R indirectly: H governs the order in magnetic subsystem, which in its turn affects the hopping probabilities. The arising magnetoresistance is unusual: it may be negative and may have jumps or kinks at magnetic phase transitions. The experimental data on MR of La,CuO, is interpreted. It is shown that for the quantitative agreement the assumption of a polaronic effect in this substance is required.
1. INTRODUCTION THIS WORK was inspired by the experiments [1, 21 on MR of La,CuO,. In these experiments a correlation between the characteristic features of MR and magnetic phase transitions was observed, although the mechanism of MR was unclear. The idea of the explanation of results of [1] was proposed in our letter [3]. Here we generalize the approach of paper [3]. We develop a simplest model in which a proposed MR mechanism appears. This model can be easily modified in order to take into account the details of specific substances structure. In particular, we explain the results of [2], which requires a somewhat deeper insight in the structure of acceptor states. We believe that the proposed mechanism can dominate not only in La,CuO, but also in other magnetic semiconductors with hopping transport. 2. THE MODEL In order to demonstrate the basic MR spin mechanism we shall consider the two-sublattice 3D Ising model with exchange J > 0 (spins Si are directed
* Permanent
address.
along the z-axis). It’s ground state is antiferromagnetic (AF). In an external field H = Hc a phase transition occurs from AF-state (Fig. la) to a ferromagnetic one (F-state, Fig. lb). Suppose that some of the magnetic sites are donors (i.e. besides the spins S, there are also the localized electronic levels si associated with these sites). At low temperatures T the electrons are bound to the donors and a process of the conductivity proceeds via variable range hopping (VRH). The spin rri of an electron, localized at the ith donor interacts with a spin of a donor site Sj. The corresponding Hamiltonian is H = K&a, where K is an effective exchange. In the mean field approximation the molecular field affecting the electrons spin rr, is Ai
=
KM,,
(1)
where M, is the sublattice magnetization (S;) = M,, CLis a number of sublattice. In AF-state M, = - M2 = (0, 0, S) and in F-state M, = M2 = (0, 0, S). For the sake of simplicity we have neglected a “direct” Zeeman interaction: gpBHui. This approximation is well founded if K ti J, but, in any case, taking the direct interaction into account does not change the physical results. If the lifetime rhop of an electron on a donor is
205
206
ANOMALOUS
HOPPING
MAGNETORESISTANCE
IN SEMICONDUCTORS
Vol. 78, No. 3
multiphonon polaronic hops. The theory of such hops was developed in [4] where it was shown that W(E) cc exp (- E/2T) for T > T* = hw,,/2 In is a characteristic phonon fre( W,IE), where Oph quency and W, is a polaronic shift. We shall restrict ourselves to relatively high T > T*. Then it can be easily obtained from (3) that (al
6 5 In (B$“/~j~‘)
lb)
=
In [ch (A/2T)].
(4)
Fig. 1. Spin configurations and electronic hops in a simple model. Spins of magnetic ions are shown as arrows, donors as circles, even hops are solid lines, odd hops are broken lines. (a) AF-state (at H < H,). One half of the hops are even, another half is odd. (b) F-state (at H > Hc). All hops are even.
The quantity 6 characterizes the distinction between even and odd transitions, the sign of 6 determines the sign of MR (see below). The theory of hopping conductivity is based on the idea of a percolation through the random resistances network [5]. The resistance
much larger than the relaxation time for its spin, then the distribution function for an electron spin will be of the Gibbs form
R,, =
g(o,)
=
exp (-Aai)/2
ch (A/2T).
(2)
To obtain the average probability I#$ of a hop (i -+ j) one has to sum over the initial electron states with a weight g(o,). Suppose that the hopping electron cannot reverse its spin during the hop. Then it is clear that B$ depends strongly on whether the spins of initial and final states are parallel (wj+’ - an even transition) or antiparallel (I+ ) - an odd one, Fig. 2) l$;+’ cc B’(A), W!,-’ cc {ema”rW(A
-
A)
+ ea12’ W(A + A))/2 ch (A/2T).
(3)
Here W(E) is a probability of a hop between two specific spin states in the splitted doublets (corresponding to any line in Fig. 2) as a function of the difference between their energies. The shape of W(E) function depends on the character of transitions. Having in mind the applications to La,CuO, (and other perovskites) where there are some evidences of polaronic effect, we shall consider here only
T/e2J;(l
- fi)K,
=
B exp (&,),
(5)
corresponds to each pair of donors (zj). Here f; = [ 1 + exp (E,/T)] ’ is the probability of i th donor to be occupied by an electron; si is the position of the energy level (the centre of the spin doublet) accounted for the Fermi level; B is a preexponential factor; tij is a so called “connectivity function”. If both i andj donors belong to the same sublattice (i.e. for an even transition) one can obtain [4, 51
2(ri,/a)
+ (Is,1 + Is/l)/2T,
(6)
where the first term arises from the squared overlap integral (ri, is the distance between i andj donors and a is the radius of the localized state), while the second term arises from the different T-dependent factors in equations (3) and (5). For the odd transitions according to the definition of 6 we have
4(f) + 6. ‘I
(7)
In a strong magnetic field (H > H, i.e. in the F-state) there is only one sublattice and all the hops are even (Fig. 1b). In this case for the calculation of the resistance in VRH regime a standard percolation method can be applied. The resistance is R,
=
B exp (ti!)),
(8)
where cf? = (4n,./gTu3)“” corresponds to a percolation threshold, determined by an equation
;J;‘A’ J IAi (l/4)gTa3(tjo’)4
Ai,
Ai,
(4
lb)
Fig. 2. Energetical diagram of spin-conserving electronic hops. (a) even hop (Ai is parallel to A,), (b) odd hop (A, is antiparallel to A,).
=
n,,
(9)
where g is a density of electronic states at a Fermi level, n, g 5 is a universal critical concentration for the dimensionless percolation problem [5]. In a weak field (H < H,, Fig. 1a) there are two sublattices and one half of the hops are even while another half is odd. So all the donors are divided in two classes (according to their belonging to magnetic
Vol. 78, No. 3
ANOMALOUS
HOPPING
MAGNETORESISTANCE
sublattices) and for intraclass hops and interclass ones the connectivity function is determined by equations (6) and (7) respectively. In order to obtain & for such an “extended percolation problem” we shall generalize equation (9) calculating the total number of sites j (of both classes) connected with a given site i according to a connectivity criterion <,, < <, . Then we have
where 9(x) = 1 for x > 0 and 9(x) = 0 for x < 0; g/2 is a density of states belonging to one sublattice. The first term in equation (10) comes from the sites of the class to which the site i belongs and the second one comes from the sites of another class. The equation (10) is not rigorous is one aspect. Strictly speaking, the value of n,. can depend on the shape of “a percolation figure” in r - E space, and this figure changes its form when 6 is varied. But the numerical simulations (c.f. [5]) show that the variations of n,. are very small (of order of 1Oh) in all the cases studied. So, we hope, that considering n, = const in (10) is a good approximation. Then, solving equation (lo), we obtain d/2 + {[h4/2 + ([10))4]‘/2 4‘(S)
=
21/4rjo)
36/4}“*
(lo’ + 6/2.
AR’R
(4) and (12): = =
(Ro - RH)‘RH [ch (A/2T)]“2 -
In lightly
for
6 > 2”45y).
Note, that this result coincides with the first order perturbation theory result [5]. Indeed, according to [5] the linear in 6 correction to 5, must have a form 5, - t$‘) = (6, ). But since 6, = 0 for even hops and 6, = 6 for odd ones, one has (6,) = 612. For large 6 (6 > 2’145!?) all the connections between sites of different classes are broken, so the percolation occurs via sites of each class independently and the lower result in (11) corresponds to the usual hopping but with a two times less density of states. So, in both limiting cases equation (10) turns to give a correct result and there are good chances that it works well in the intermediate region 6 ? 5!?. The theory may be generalized also for the case of more than two sublattices and the percolation threshold may be found [6]. Summing up the results of this section, the resistance R does not depend on H for H -c H, and for H = Hc it undergoes a jump. The relative magnitude of this jump for 6 @ ty’ can be obtained from
exp (d/2) -
1.
1 (13)
3. APPLICATION TO La,CuO,: THEORY VS EXPERIMENT
6 < 2”45jo)
(12)
=
The main features of anomalous MR were described above for a simplest model of 3D Ising antiferromagnet. For the application of a method to more sophisticated substances one has: (i) to study a behaviour of magnetic sublattices in an external field, (ii) to examine a structure of impurity states in order to determine the molecular field affecting the spin of localized carrier, (iii) to calculate the hopping probabilities B$ and (iv) to formulate and to solve an extended percolation problem with a number of classes equal to the number of different possible configurations of molecular fields affecting the spins of carriers localized on different impurities.
for
Discuss the limiting cases of equation (11). When 6 = 0 the two classes merge and 4, = [i!’ as in a conventional percolation problem. For 6 4 5:? we have & =
equations
207
IN SEMICONDUCTORS
doped
or slightly
nonstoichiometric
(11)
La,CuO, the 3D VRH conductivity was observed: R(T) cc exp [(To/T)“4], To N 105-106K [7, 81. The carriers are holes localized presumably at the acceptors. The H-dependence of the resistance R (obtained in [1, 21) is shown in Fig. 3. In order to explain this dependence we shall fulfil the program listed at the end of the previous section. The magnetic structure of La-system is well known. It’s distinction from the simple antiferromagnet is due to the orthorhombic distortions, which produce the anisotropic antisymmetric exchange [9, lo] and out-of-plane spin canting (by a small angle 9 N 3 x lo3 [l]). As a result, each Cu-0 plane acquires a small ferromagnetic moment, the moments of subsequent planes are ordered antiferromagnetitally along the b-axis at H = 0. In the field H I(b this antiferromagnetic order is changed to a ferromagnetic one (weak ferromagnetism) at H = H, N 4T. The magnetic behaviour in H J_ b is more complicated [2]. Thus at H I(c there is a magnetic phase transition at H = H, N 10 T accompanied by the reorientation of a staggered moments M, from bc to ab plane. Index c( = +,distinguishes even and odd Cu-0 planes (with opposite orientations of ferromagnetic moments at zero field). At H = H2 N 20T the second transition occurs: M, become parallel to b axis. For details c.f. [2, 61.
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ANOMALOUS
HOPPING
MAGNETORESISTANCE
R0-I) R(O)
R(H)
R(O)
I
I
I
5
Hl
“2
HO
h
(cl
1 F
0.5
IN SEMICONDUCTORS
Vol. 78. No. 3
interpretation of the experimental data (see below). It should be noted that strictly speaking bi is not just a holes spin, but rather a total spin of a compound system including a hole and the neighbouring coppers. The spin of such a dressed electron appears to stay l/2 (i.e. unchanged) for all the cases of interest [6], so the ‘magnetic dressing’ effects do not change the results of the present theory. The vector o is proportional to the orthorhombic rotation angle in the vicinity of impurity. If we would take for that the bulk value, known from the neutron experiments [I] we obtain w = w0 2: 10 K which is too small for the interpretation of strong MR. However, the local distortion has not to coincide necessarily with a bulk one; it may be enhanced substantially if to admit the polaronic effect [12-141. If the polaronic effect is strong (i.e. distortions are of order of unity) then o - 100 K, which fits well the data of [l, 21 (see below). In [14] we have studied the interaction or a localized hole and the low lying phonon modes and have shown that there may arise JahnTeller (or pseudo Jahn-Teller) enhancement of local orthorhombic distortion. Note also [6] that besides the a-component (the bulk vector o0 is parallel to the a-axis) the vector w can acquire the c-component
(Iwul= Id>.
I
100
T (K) Fig. 3. (a) The field-dependence of the resistivity for H 11b (according to [l]). (b) The same for H )Ic: solid line - experimental data [2] at T = 24 K, broken line - the theoretical result (equation (15)), where H, = 10 T and H, = 17 T were taken from [2] and the parameter oM/2T = 2.19 was obtained from the magnitude of the overall resistance drop AR/R = (R(0) - R(co))/R(oo). (c) Temperature dependence of AR/R. It is the same for H 11b and H IIc cases [2]. As to acceptor state, by now its structure cannot be extracted directly from experimental data. We shall suppose here that the symmetry group of acceptor includes the permutation of magnetic sublattices. It is true in particular for a simplest model of a hole localized on a single 0 atom [ 111, as well as for some other models [6]. For such a symmetry the molecular field affecting a holes spin bi is non-zero only due to the canting of spins:
and an exchange term (of (1) type) is forbidden by a symmetry. The last fact is extremely important for the
The molecular fields A, are shown in Fig. 4 for the case H II6. The directions of A, are opposite for impurities situated in the even and odd planes respectively. So, there are two classes of sites in the percolation problem and the consideration of Section 2 can be applied to this case without any modifications. The
(a)
Fig. 4. The configurations of molecular fields A, (for even planes) and A (for odd ones) at H II6: (a) AF-state (H < H,.), (b) F-state (H > Hc).
Vol. 78. No. 3
ANOMALOUS
HOPPING
MAGNETORESISTANCE
shape of R(H) dependence in Fig. 3a coincides completely with that predicted by the theory developed above. The magnitude of a jump is given by equation (13) with A = lo,lM. From the experimental temperature dependence of AR (Fig. 3c) and equation (13) we obtain the T-dependence of A (Fig. 5). If the polaronic shift W, $ T then o is almost T-independent and Fig. 5 must reproduce the T-dependence of staggered magnetization M(T). Since M ? 1, the magnitude of o must be as large as u - 200K, which may be due only to a polaronic enhancement of the local orthorhombicity in the vicinity of an acceptor. We think that this fact provides an additional evidence for a substantial polaronic effect in La,CuO,. Some other aspects of polaronic effect we have discussed in [ 151. The most striking qualitative feature of M(T) (Fig. 5) is the drop of M at low T. Such a drop (though less pronounced) was obtained also in neutron scattering studies [16] and we believe that it is an inherent property of a magnetic system (the theoretical discussion see in [ 171). We do not discuss here the quantitative comparison of Fig. 5 with experimental data since there were no experiments (to our knowledge) in which MR and the staggered magnetization were measured on the same samples.
The case H _Lb is much more complicated. It is considered in our paper [6] in details. Here we shall point out only the main peculiarities of this situation. First of all the crystals of La,CuO, are usually twinned and the field orientation with respect to the crystal axes a and c is different in different domains. Besides that importance is the possible presence of c-component of o vector. The four classes may arise in the extended percolation problem since each acceptor is characterized not only by the number (even or odd) of the corresponding plane but also by the sign of w,. Generally, these signs for different acceptors may be either random or correlated, which affects MR. The magnetic phase transition at H = H, (H 11c) is the first order transition. It means that this transition would have been accompanied by the jump in R, (just as in H IIb case). Butexperimentally such a jump is absent and there are ody kinks in R(H) at both H, and H,. Besides that the overall relative drop of resistivity (R(0) - R(co))/R(co) turned out to be identical for H IIb and HI b. Both these facts can be explained if we suppose some specific type of 0,. correlation (for details see [6]). The result for R(H) is R(H)
=
B exp (5!“‘){(1 + x)/[2x
(H/Hz )’ >
1 1,
P
2 *O’
x)
+ H,)
-
H21j2,
(15)
H < H, H, < H < H2 H, < H.
This dependence is presented in Fig. 3b and it fits the experimental curve satisfactorily. In conclusion, the proposed mechanism explains well the effect of MR in La,CuO,. The analysis of experimental data [ 1,2] enables one to conclude about the symmetry of the acceptor state (i.e. the point symmetry group must include a magnetic sublattice permutation) and about a local orthorhombicity enhancement in the vicinity of acceptor. This enhancement apparently has a cooperative character which is manifested by the correlations of c-components of vector of orthorhombic rotations for different impurities. Acknowledgements
Fig. 5. The temperature dependence of A = 1w,lM obtained by fitting of equation (13) to the data from Fig. 3c.
+ (1 -
x sch (oM( 1 + x)“*/2T)]} -I”,
{HH,I]H,(H,
Besides that, all our calculations were carried out in the framework of the mean field approximation and the neglecting of magnetic fluctuations can lead to the under-estimation of M(T) at low T (c.f. [6]).
209
IN SEMICONDUCTORS
We are indebted to R.J. Birgeneau, M.A. Kastner, N.W. Preyer & T. Thio for the discussions concerning the experiments and also to S.A. Brazovskii, B.I. Halperin, M.E. Raikh, D.E. Khmelnitskii, E.G. Tsitsishvili and B.I. Shklovskii for valuable comments. One of us (A.S.I.) is grateful to ISI Foundation for the hospitality and support during his stay at the Institute of Scientific Interchange.
210
ANOMALOUS
HOPPING
MAGNETORESISTANCE
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
T. Thio et al., Phys. Rev. B38, 905 (1988). T. Thio et al., Phys. Rev. B41, 231 (1990). A.O. Gogolin & A.S. Ioselevich, JETP Lett. 51, 174(1990). H. Sher & T. Holstein, Phil Msg. B44, 343 (1981). B.I. Shklovskii & A.L. Efros, Electronic Properties of Doped Semiconductors. Springer (1984). A.O. Gogolin & A.S. Ioselevich, JETP 98, 681 (1990) (in Russian). M.A. Kastner et al., Phys. Rev. B37, 111 (1988). N.W. Preyer et al., Phys. Rev. B39,11563 (1989). I. Dzyaloshinsky, J. Phys. Chem. Sol. 4, 241 (1958).
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Vol. 78, No. 3
T. Moriya, Phys. Rev. 120, 91 (1960). A. Aharony et al., Phys. Rev. Lett. 60, 1330 (1988). S. Barisic & I. Batistic, Physica Scripta. 27, 78 (1989). C.M. Foster et al., Physica C. C162-164 (1989); Synthetic Metals. 33, 171 (1989). A.O. Gogolin & A.S. Ioselevich, JETP Lett. 50, 502 (1989). A.O. Gogolin & A.S. Ioselevich, JETP Lett. 51, 532 (1990). Y. Endoh et al., Phys. Rev. B37, 7443 (1988). L.I. Glazman & A.S. Ioselevich, JETP Lett. 49, 579 (1989); Z. Phys. B80, 133 (1990).