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Magnetoresistance related to on-site spin correlations in the nearest neighbor hopping conductivity
Solid State Communications, Vol. 108, No. 6, pp. 355–359, 1998 c 1998 Published by Elsevier Science Ltd
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Pergamon
PII: S0038–1098(98)00358-5
MAGNETORESISTANCE RELATED TO ON-SITE SPIN CORRELATIONS IN THE NEAREST NEIGHBOR HOPPING CONDUCTIVITY N.V. Agrinskaya and V.I. Kozub ∗ A.F.Ioffe Physical–Technical Institute, 194021, St. Petersburg, Russia (Received 28 July 1998; accepted 4 August 1998 by A.L. Efros)
We consider the magnetoresistance in the nearest neighbor hopping conductivity with an account of both Hubbard and spin correlations. It is shown that no spin magnetoresistance exists for the conductivity within the lower Hubbard band. In contrast, the ε2 conductivity involving the upper Hubbard band states exhibits an increase of activation energy which in high magnetic fields has a universal form µ0 gH. The results obtained are in good agreement with existing experimental data for temperature dependent magnetoresistance in different doped matec 1998 Published by Elsevier Science Ltd. All rights reserved rials. Keywords: A. Semiconductors, C. Impurities in semiconductors, D. Electronic transport
The hopping conductivity over states within the second Hubbard band (involving double-occupied impurity sites) has been considered already in early years of studies of the conductivity over localized states [1, 2]; it is controlled by a specific activation energy ε2 and thus is often told about as the ε2 conductivity. A presence of this channel of the hopping conductivity was never disputed; its importance is expected to be the most for intermediate impurity concentrations when the Hubbard energy (U = e2 /κa, a being the Bohr radius, κ being the dielectric constant) is of the same order as a width of the impurity band (ε3 ∝ (e2 N 1/3 /κ), N being the impurity concentration). Note however that the very temperature dependence does not allow to discriminate clearly between the different exponential laws especially when the exponents are not strongly different. This fact and obvious difficulties of analytical calculations of the ε2 value lead to the situation when during last years the nearest neighbor hopping was typically considered as related to the “standard” hopping within the impurity band (ε3 conductivity) [3]. At the same time one expects that the on-site Hubbard correlation can give rise to specific physical behavior which can allow to discriminate between the ∗
E-mail: [email protected]
channels in question. In particular, we would like to mention the paper [4] where a specific role of the ε2 conductivity in thermoelectric effect was emphasized. In what follows we will discuss another aspect of the ε2 conductivity related to on-site spin correlations which lead to positive spin magnetoresistance for the ε2 conductivity. Until now a lot of papers reported a magnetic field induced enhancement of the activation energy for the hopping conductivity (see e.g. references given in Table ); the corresponding positive magnetoresistance was an addition to positive hopping magnetoresistance of the temperature-independent factor (related to wave functions shrinkage). As early as at the beginning of sixties Yamanouchi [5] had ascribed this magnetoresistance to spin effects in the ε2 conductivity. However the author had considered a simplified model dealing with impurity level rather than with impurity band. Then, the paper had not considered analytically different limiting cases and was mainly concentrated on the numerical analysis for the region of relatively weak fields and high temperatures. The analysis had given an apparent quadratic field dependence of the ε2 value: ε2 = ε2 (H = 0) + γH 2 which was extensively cited. However the citations were not typically accompanied by any discussion of the physical picture. Then, the magnetic field induced enhancement of
355
356
MAGNETORESISTANCE RELATED TO ON-SITE SPIN CORRELATIONS
Vol. 108, No. 6
Table 1. Mater.
g-factor
fields, (T)
gµ0 × 1020 (exper.)
K
ND , (cm−3)
Ref.
n-Ge:Sb n-Ge:P; H k [001] ¯ ] n-Ge:P; H k [110 n-InP n-GaAs n-GaAs p-Ge:Ga p-InSb
1.6∗ 1.7∗∗ 1.25∗∗ 1.3 0.45 0.45 — —
5–18 10–14.6 10–14.6 4–7 20–35 8–14 5–10 2.5–4
1.6 1.8 1.1 1.3 0.4 0.6 0.5 3
0.1 0.08 0.08 0.35 0.7 — 0.4 0.65
1.2 ×1017 2 ×1016 2 ×1016 8 ×1015 5 ×1016 5 ×1015 3–5 ×1016 1.5 ×1016
[12] [14] [14] [13] [16] [18] [7] [15]
* g-factor for shallow donors in Ge was taken from [17]. ** effective g-factors estimated on the base of the values of g⊥ and gk given in [14].
the activation energy was sometimes ascribed to an admixture of excited impurity state due to the quadratic Zeeman effect [6]. Other authors, however, ascribed the field dependence of the activation energy to orbital effects (including in particular correlated hopping [7]), related to a magnetic field effect on overlapping integrals [3]. Some later a role of the double occupied centers leading to giant positive magnetoresistance was considered for the case of variable range hopping by Kurobe and Kamimura [8]; in the situation considered the double occupied states were expected to exist at the Fermi level. In what follows the corresponding behavior was studied extensively (see e.g. [9,10]) However possible effects on the nearest neighbor hopping were not discussed. Then, we should also mention the recent paper by Shchamkhalova and Tkach [11] suggesting a spin mechanism leading to giant negative magnetoresistance for the ε3 channel. Thus one notes that the situation with magnetic field effect on the activation energies in the nearest neighbor hopping is still not a conventional one. In what follows we will give a short analysis of the problem. We will show that the ε3 conductivity should not exhibit (in contrast to predictions of [11]) any magnetic field effect on the activation energy (which agrees with the classical results [3]). Then, we will clarify the magnetic field dependence of the ε2 conductivity for different limiting cases allowing to discriminate a signature of the ε2 conductivity in the magnetoresistance measurements. We will show in particular that this dependence can be considered as a simple increase of the ε2 only at strong enough fields while for the weak fields (in contrast to the earlier considerations [5]) the magnetic field contribution appears to contain additional temperature dependence (∝ 1/T ). We will start with the free energy for electrons within the impurity band including a possibility of double
occupation of the sites which has a form (compare with [5]): X 1 {εi (Ni,+ + Ni,− ) − µ0 gH (Ni,+ − Ni,−)
2
i
+(2εi + U )ni − µ(Ni,+ + Ni,− + 2ni ) +T [Ni,+ ln Ni,+ + Ni,− ln Ni,− +ni ln ni + (1 − Ni,+ − Ni,− − ni )]} (1)
Here Ni,+ and Ni,− are the occupation numbers of the site i corresponding to single electron occupation with a spin direction along and opposite to the magnetic field, correspondingly, ni is the occupation number corresponding to double occupation of the site, U is the Hubbard energy, µ is the chemical potential, εi is the site energy, µ0 is the Bohr magneton while g is the Lande factor. We will assume that the occupation of the upper Hubbard band is small enough and in any case N+ >> n which allows to neglect n in course of estimates of N+,− . In this case minimizing the free energy of equation (1) one obtains exp(µ0 gH /2T ) , exp[(εi − µ)/T ] + 2 cosh(µ0 gH /2T ) µ0 gH N− = N+ exp(− ) T n = {exp[(2εi + U − 2µ)/T ] N+ =
+2 cosh(µ0 gH /2T ) exp[(εi + U − µ)/T ]}−1 (2)
Note that neglecting a possibility of double occupation and taking into account that in the absence of the spin-orbital effects the relation between N+ and N− is not sensitive to the site energy one could rewrite equation (1) introducing the “site occupation number” N = N+ + N− and the “spin occupation number” Ns = N− /N (the latter describes the occupation of the higher state of the Zeeman doublet). As it is easily seen, the spin occupation numbers are not sensitive to the site while a presence of the spin degrees of
Vol. 108, No. 6
MAGNETORESISTANCE RELATED TO ON-SITE SPIN CORRELATIONS
freedom leads only to a shift of the “orbital” chemical potential by a quantity −(µ0 gH /2) + T ln(1 − Ns ). Correspondingly, the equation for the chemical potential has a form: X
(N+ + N− )
i
=
X i
1 exp[(εi − µ)/T ][2 cosh(µ0 gH /2T )]−1 + 1
=N
(3)
where N = ND − NA is the total number of electrons within the impurity band, ND and NA are the total numbers of donors and accepters. Note that here we have assumed that the position of µ˜ is pinned within the impurity band which is possible if n << NA . This fact makes difference with the situation of the impurity level considered in [5]. Thus one can estimate the chemical potential as µ=µ ˜ − T ln(2 cosh
µ0 gH
2T
)
(4)
where µ˜ = µ(H = 0). We would like to note that for a purely ε3 conductivity (n = 0) the picture discussed above contradicts with the paper [11]; in this paper the different scenario for the occupation numbers was considered assuming a presence of “impurity subbands” for “spin-up” and “spin-down” electrons and leading to the giant negative magnetoresistance. To our opinion, the assumption in question does not hold for the single-occupied sites because the two spin states can not coexist for a given “orbital” number and thus the corresponding states can not be considered as statistically independent. In contrast, according to our analysis the conductivity is dominated by the “spin-up” electrons (N+ ) and the only effect of the magnetic field is a shift of all of the site energies by a quantity −T ln 2 cosh(µ0 gH /2T ). Thus the activation energy ε3 does not depend on H. Now let us consider the ε2 conductivity which is controlled by an occupation of the second Hubbard band at the percolation level; the latter we will denote as ε p . Thus we will make use of equation (2) putting here εi = ε p :
357
while for H ≠ 0 there is a positive magnetoresistance. Note that this magnetoresistance is similar to one considered in [8] for the variable range hopping conductivity. An important difference is that for the nearest neighbor hopping the effect corresponds to renormalization of the activation energy rather than to renormalization of the density of states [8]. For the case µ0 gH >> 2T one easily sees that σ ∝ exp(−
µ0 gH
T
−
ε2
T
(7)
)
that is in this case the magnetic field renormalizes the ε2 value leading to a giant positive magnetoresistance.
We would like to note that the activation term for the ε3 conductivity—as shown above—is not affected by
the magnetic field and thus a presence of giant positive magnetoresistance is a signature of the upper Hubbard band conductivity. While linear at moderate fields, the magnetoresistance is expected to saturate at high enough fields when the field-independent ε3 channel (for which the activation term is field-independent) starts to dominate over the suppressed ε2 channel. The behavior in question is seen from the following estimate (taking for clarity the ε3 case) σ = C1 exp(−
ε2 + µ0 gH
ε3
) + C2 exp(− ) = T T ε3 C1 µ0 gH ε3 − ε2 C2 exp(− ) 1 + exp(− + ) (8) T C2 T T
When the exponential in the brackets is much less than unity, the temperature dependence is described as ε3 C1 µ0 gH ε3 − ε2 ∝ exp − + exp(− + )
T
C2
T
T
Thus the corresponding saturation field is estimated as µ0 gHsat ∼ ε3 − ε2
For the weak field limit µ0 gH << 2T one has σ ∝ exp −(
ε2
T
+
(µ0 gH )2
4T 2
)
(9)
So the activation energy term can be estimated as
1 (µ0 gH )2 4 T σ ∝n= −1 An important factor is the apparently “universal” co2ε p + U − 2µ˜ + 2T ln 2 cosh(µ0 gH /2T ) exp +1 efficient µ0 g characterizing an enhancement of ε2 by T the magnetic field. Nevertheless we would like to note (5) that the “universal” magnetic field dependence holds Here we have assumed that ε p − µ˜ >> µ0 gH /2. Thus only for the field region corresponding to linear field dependence. we in particular obtain that for H = 0 We have compared the predictions given above with ε2 = 2ε p + U − 2µ ˜ (6) some existing experimental data for the magnetic field δε2 =
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MAGNETORESISTANCE RELATED TO ON-SITE SPIN CORRELATIONS
behavior of the activation energy for different materials (or magnetoresistance curves for different temperatures in the region of ε2 conductivity) [7,12–16,18]. All the materials were on the insulating side of the metal– insulator transition at H=0 except GaAs [16] where the metal–insulator transition was stimulated by the magnetic field. The corresponding experimental curves ε2 (H ) clearly demonstrate linear plots at large magnetic fields. Basing on these plots we have estimated the values of gµ0 , the estimates are given in Table where we compare these values with known values of g for the corresponding materials. One sees that for n-type materials [13, 14, 16, 18]. there is an excellent agreement. Note that we have also considered a possible role of the anizotropy of g- factor for many-valley semiconductors related to the valleys anizotropy which gives rise to a difference between g- factors for the magnetic field directions parallel and normal to the valley axis (gk < g⊥ . Although the difference in question can be large, a presence of several valleys with different orientations typically leads to the averaging and the anisotropy of the effective g factor typically appears to be not too strong. Nevertheless the experimental data (see e.g. [14] demonstrate a dependence of the activation energy on the magnetic field direction. Thus we have estimated the effective g-factors with respect to the situation of [14] taking into account that the main contribution is given by the valleys ensuring the lowest possible g (and thus of the activation energy). The results are given in Table and also demonstrate a good agreement with experiment. We would like also to note that the averaging mentioned above is least pronounced for a situation when the magnetic field is parallel to axis of one of the valleys, in this case the activation energy is dominated by a corresponding valley with g = gk . The situation is not as clear for p-type materials [7,15] because the estimates of g for shallow accepters are not as simple as for shallow donors due to the degeneracy of the valence band. In this concern we would like to note that the analysis of the magnetoresistance for ε2 hopping conductivity corresponding to the intermediate range of doping gives a possibility for direct estimates of g-factors for situations when these factors can not be extracted by other methods. Now let us discuss a possible role of magnetic field effect on the overlapping integrals. As it is well known, the main effect is the temperature independent positive magnetoresistance. However the overlapping integrals also affect the width of the impurity band and thus the value of ε3 for impurity concentration not far
Vol. 108, No. 6
from insulator–metal transition (see e.g. [3]). Another possibility for the overlapping integrals to affect the activation energy is related to correlated hopping [7]. Thus we should in principle also compare the experimentally observed behavior of the activation energy with the behavior expected for these mechanisms. Here we would like to note that for the overlapping integrals effect one does not expect any universal behavior of the sort predicted for the spin ε2 magnetoresistance. In particular, the wave shrinkage effects are obviously sensitive to the Bohr radii. In contrast, the experimental data (see Table ) exhibit universal magnetic field behavior (depending only on the g-factor value) for different materials, different impurities types and concentrations and do not show any dependence on the Bohr radii (like P and Sb in Ge). Moreover, the agreement between the values of g extracted from experimental magnetoresistance curves with the help of the universal ε2 slope discussed above with known values can hardly be considered as an accidental one. So these features allow us to exclude the orbital effects from the considerations. To conclude, we have shown that an increase of the activation energy in the nearest neighbor hopping conductivity is a signature of a contribution of the upper Hubbard band (ε2 channel). While being quadratic in terms of H in weak magnetic fields µ0 gH < 2kB T , the enhancement is linear in strong magnetic fields: ∆ε2 = µ0 gH; as it is seen, in the latter case the behavior is a universal one allowing to discriminate the mechanism in question. Note that while an appearence of µ0 gH addition to the activation energy is clearly of the same nature as for the mechanism considered for varuable range hopping [8–10], the universal ε2 (H ) slope depending exclusively on the value of g (in contrast to the VRH situation where the experimental behavior is not as universal) gives an advantage which even can ensure an independent method to estimate g-factors. The results are in good agreement with experimental data for n-type materials where g values are well known; this fact is to our opinion a strong evidence of the ε2 character of hopping for the data in question. Acknowledgements—We are grateful to Polyanskaya T.A. for initiating discussion. This paper was supported by Russian Science Foundation (grant N 9702-18280a). REFERENCES 1. Mott, N.F. and Davis, G.A., Electronic Processes in Non- Crystalline Materials Oxford, 1979. 2. Gershenson, E.M., Mel’nikov, A.P. and Rabinovich, R.I., in: Electron–electron interactions in
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MAGNETORESISTANCE RELATED TO ON-SITE SPIN CORRELATIONS
disordered systems, eds A.L. Efros and M. Pollak, North-Holland, Amsterdam, p.483, 1985. 3. Efros, A.L. and Shklovskii, B.I., Electronic properties of doped semiconductors, Springer-Verlag, 1984. 4. Andreev, A.G., Zabrodskii, A.G., Zvyagin, I.P. and Egorov, S.V., Sov. Phys. Semicond., 31, 1997, 1174. 5. Yamanouchi, C., J. Phys. Soc. Jap., 18, 1963, 1775. 6. Mikoshiba, N., Rev. Mod. Phys., 40, 1968, 833. 7. Knotek, M.L., Phys. Rev. B., 16, 1977, 2629. 8. Kurobe, A. and Kamimura, H., J. Phys. Soc. Jap., 51, 1982, 1904. 9. Meir, Y., Europhys. Lett., 33, 1996, 171. 10. Matveev, K.A., et al., Phys. Rev. B, 52, 1995, 5289. 11. Shchamkhalova, B.S. and Tkach, Y.Y., Solid State Comm., 99, 1996, 261.
359
12. Matveev, G.A., Sokolov, V.I. and Tsidilkovskii, I.M., Sov. Phys. Semicond., 8, 1974, 1781. 13. Mansfield, R., Abboudy, S. and Fozooni, P., Phil. Mag. B, 57, 1988, 777. 14. Chroboczek, J.A., Prohovsky, E.W. and Sladek, R.J., Phys. Rev., 169, 1968, 593. 15. Gershenson, E.M., et al. Sov. Phys. Semicond., 8, 1974, 1057. 16. Matveev, G.A., Sov. Phys. Semicond., 15, 1981, 2333. 17. Damon, D.H. and Geritsen, A.N., Phys. Rev., 127, 1962, 405. 18. Halbo, L. and Sladek, R.J., Phys. Rev., 173, 1968, 794.
0038–1098/98 $ - see front matter
Pergamon
PII: S0038–1098(98)00358-5
MAGNETORESISTANCE RELATED TO ON-SITE SPIN CORRELATIONS IN THE NEAREST NEIGHBOR HOPPING CONDUCTIVITY N.V. Agrinskaya and V.I. Kozub ∗ A.F.Ioffe Physical–Technical Institute, 194021, St. Petersburg, Russia (Received 28 July 1998; accepted 4 August 1998 by A.L. Efros)
We consider the magnetoresistance in the nearest neighbor hopping conductivity with an account of both Hubbard and spin correlations. It is shown that no spin magnetoresistance exists for the conductivity within the lower Hubbard band. In contrast, the ε2 conductivity involving the upper Hubbard band states exhibits an increase of activation energy which in high magnetic fields has a universal form µ0 gH. The results obtained are in good agreement with existing experimental data for temperature dependent magnetoresistance in different doped matec 1998 Published by Elsevier Science Ltd. All rights reserved rials. Keywords: A. Semiconductors, C. Impurities in semiconductors, D. Electronic transport
The hopping conductivity over states within the second Hubbard band (involving double-occupied impurity sites) has been considered already in early years of studies of the conductivity over localized states [1, 2]; it is controlled by a specific activation energy ε2 and thus is often told about as the ε2 conductivity. A presence of this channel of the hopping conductivity was never disputed; its importance is expected to be the most for intermediate impurity concentrations when the Hubbard energy (U = e2 /κa, a being the Bohr radius, κ being the dielectric constant) is of the same order as a width of the impurity band (ε3 ∝ (e2 N 1/3 /κ), N being the impurity concentration). Note however that the very temperature dependence does not allow to discriminate clearly between the different exponential laws especially when the exponents are not strongly different. This fact and obvious difficulties of analytical calculations of the ε2 value lead to the situation when during last years the nearest neighbor hopping was typically considered as related to the “standard” hopping within the impurity band (ε3 conductivity) [3]. At the same time one expects that the on-site Hubbard correlation can give rise to specific physical behavior which can allow to discriminate between the ∗
E-mail: [email protected]
channels in question. In particular, we would like to mention the paper [4] where a specific role of the ε2 conductivity in thermoelectric effect was emphasized. In what follows we will discuss another aspect of the ε2 conductivity related to on-site spin correlations which lead to positive spin magnetoresistance for the ε2 conductivity. Until now a lot of papers reported a magnetic field induced enhancement of the activation energy for the hopping conductivity (see e.g. references given in Table ); the corresponding positive magnetoresistance was an addition to positive hopping magnetoresistance of the temperature-independent factor (related to wave functions shrinkage). As early as at the beginning of sixties Yamanouchi [5] had ascribed this magnetoresistance to spin effects in the ε2 conductivity. However the author had considered a simplified model dealing with impurity level rather than with impurity band. Then, the paper had not considered analytically different limiting cases and was mainly concentrated on the numerical analysis for the region of relatively weak fields and high temperatures. The analysis had given an apparent quadratic field dependence of the ε2 value: ε2 = ε2 (H = 0) + γH 2 which was extensively cited. However the citations were not typically accompanied by any discussion of the physical picture. Then, the magnetic field induced enhancement of
355
356
MAGNETORESISTANCE RELATED TO ON-SITE SPIN CORRELATIONS
Vol. 108, No. 6
Table 1. Mater.
g-factor
fields, (T)
gµ0 × 1020 (exper.)
K
ND , (cm−3)
Ref.
n-Ge:Sb n-Ge:P; H k [001] ¯ ] n-Ge:P; H k [110 n-InP n-GaAs n-GaAs p-Ge:Ga p-InSb
1.6∗ 1.7∗∗ 1.25∗∗ 1.3 0.45 0.45 — —
5–18 10–14.6 10–14.6 4–7 20–35 8–14 5–10 2.5–4
1.6 1.8 1.1 1.3 0.4 0.6 0.5 3
0.1 0.08 0.08 0.35 0.7 — 0.4 0.65
1.2 ×1017 2 ×1016 2 ×1016 8 ×1015 5 ×1016 5 ×1015 3–5 ×1016 1.5 ×1016
[12] [14] [14] [13] [16] [18] [7] [15]
* g-factor for shallow donors in Ge was taken from [17]. ** effective g-factors estimated on the base of the values of g⊥ and gk given in [14].
the activation energy was sometimes ascribed to an admixture of excited impurity state due to the quadratic Zeeman effect [6]. Other authors, however, ascribed the field dependence of the activation energy to orbital effects (including in particular correlated hopping [7]), related to a magnetic field effect on overlapping integrals [3]. Some later a role of the double occupied centers leading to giant positive magnetoresistance was considered for the case of variable range hopping by Kurobe and Kamimura [8]; in the situation considered the double occupied states were expected to exist at the Fermi level. In what follows the corresponding behavior was studied extensively (see e.g. [9,10]) However possible effects on the nearest neighbor hopping were not discussed. Then, we should also mention the recent paper by Shchamkhalova and Tkach [11] suggesting a spin mechanism leading to giant negative magnetoresistance for the ε3 channel. Thus one notes that the situation with magnetic field effect on the activation energies in the nearest neighbor hopping is still not a conventional one. In what follows we will give a short analysis of the problem. We will show that the ε3 conductivity should not exhibit (in contrast to predictions of [11]) any magnetic field effect on the activation energy (which agrees with the classical results [3]). Then, we will clarify the magnetic field dependence of the ε2 conductivity for different limiting cases allowing to discriminate a signature of the ε2 conductivity in the magnetoresistance measurements. We will show in particular that this dependence can be considered as a simple increase of the ε2 only at strong enough fields while for the weak fields (in contrast to the earlier considerations [5]) the magnetic field contribution appears to contain additional temperature dependence (∝ 1/T ). We will start with the free energy for electrons within the impurity band including a possibility of double
occupation of the sites which has a form (compare with [5]): X 1 {εi (Ni,+ + Ni,− ) − µ0 gH (Ni,+ − Ni,−)
2
i
+(2εi + U )ni − µ(Ni,+ + Ni,− + 2ni ) +T [Ni,+ ln Ni,+ + Ni,− ln Ni,− +ni ln ni + (1 − Ni,+ − Ni,− − ni )]} (1)
Here Ni,+ and Ni,− are the occupation numbers of the site i corresponding to single electron occupation with a spin direction along and opposite to the magnetic field, correspondingly, ni is the occupation number corresponding to double occupation of the site, U is the Hubbard energy, µ is the chemical potential, εi is the site energy, µ0 is the Bohr magneton while g is the Lande factor. We will assume that the occupation of the upper Hubbard band is small enough and in any case N+ >> n which allows to neglect n in course of estimates of N+,− . In this case minimizing the free energy of equation (1) one obtains exp(µ0 gH /2T ) , exp[(εi − µ)/T ] + 2 cosh(µ0 gH /2T ) µ0 gH N− = N+ exp(− ) T n = {exp[(2εi + U − 2µ)/T ] N+ =
+2 cosh(µ0 gH /2T ) exp[(εi + U − µ)/T ]}−1 (2)
Note that neglecting a possibility of double occupation and taking into account that in the absence of the spin-orbital effects the relation between N+ and N− is not sensitive to the site energy one could rewrite equation (1) introducing the “site occupation number” N = N+ + N− and the “spin occupation number” Ns = N− /N (the latter describes the occupation of the higher state of the Zeeman doublet). As it is easily seen, the spin occupation numbers are not sensitive to the site while a presence of the spin degrees of
Vol. 108, No. 6
MAGNETORESISTANCE RELATED TO ON-SITE SPIN CORRELATIONS
freedom leads only to a shift of the “orbital” chemical potential by a quantity −(µ0 gH /2) + T ln(1 − Ns ). Correspondingly, the equation for the chemical potential has a form: X
(N+ + N− )
i
=
X i
1 exp[(εi − µ)/T ][2 cosh(µ0 gH /2T )]−1 + 1
=N
(3)
where N = ND − NA is the total number of electrons within the impurity band, ND and NA are the total numbers of donors and accepters. Note that here we have assumed that the position of µ˜ is pinned within the impurity band which is possible if n << NA . This fact makes difference with the situation of the impurity level considered in [5]. Thus one can estimate the chemical potential as µ=µ ˜ − T ln(2 cosh
µ0 gH
2T
)
(4)
where µ˜ = µ(H = 0). We would like to note that for a purely ε3 conductivity (n = 0) the picture discussed above contradicts with the paper [11]; in this paper the different scenario for the occupation numbers was considered assuming a presence of “impurity subbands” for “spin-up” and “spin-down” electrons and leading to the giant negative magnetoresistance. To our opinion, the assumption in question does not hold for the single-occupied sites because the two spin states can not coexist for a given “orbital” number and thus the corresponding states can not be considered as statistically independent. In contrast, according to our analysis the conductivity is dominated by the “spin-up” electrons (N+ ) and the only effect of the magnetic field is a shift of all of the site energies by a quantity −T ln 2 cosh(µ0 gH /2T ). Thus the activation energy ε3 does not depend on H. Now let us consider the ε2 conductivity which is controlled by an occupation of the second Hubbard band at the percolation level; the latter we will denote as ε p . Thus we will make use of equation (2) putting here εi = ε p :
357
while for H ≠ 0 there is a positive magnetoresistance. Note that this magnetoresistance is similar to one considered in [8] for the variable range hopping conductivity. An important difference is that for the nearest neighbor hopping the effect corresponds to renormalization of the activation energy rather than to renormalization of the density of states [8]. For the case µ0 gH >> 2T one easily sees that σ ∝ exp(−
µ0 gH
T
−
ε2
T
(7)
)
that is in this case the magnetic field renormalizes the ε2 value leading to a giant positive magnetoresistance.
We would like to note that the activation term for the ε3 conductivity—as shown above—is not affected by
the magnetic field and thus a presence of giant positive magnetoresistance is a signature of the upper Hubbard band conductivity. While linear at moderate fields, the magnetoresistance is expected to saturate at high enough fields when the field-independent ε3 channel (for which the activation term is field-independent) starts to dominate over the suppressed ε2 channel. The behavior in question is seen from the following estimate (taking for clarity the ε3 case) σ = C1 exp(−
ε2 + µ0 gH
ε3
) + C2 exp(− ) = T T ε3 C1 µ0 gH ε3 − ε2 C2 exp(− ) 1 + exp(− + ) (8) T C2 T T
When the exponential in the brackets is much less than unity, the temperature dependence is described as ε3 C1 µ0 gH ε3 − ε2 ∝ exp − + exp(− + )
T
C2
T
T
Thus the corresponding saturation field is estimated as µ0 gHsat ∼ ε3 − ε2
For the weak field limit µ0 gH << 2T one has σ ∝ exp −(
ε2
T
+
(µ0 gH )2
4T 2
)
(9)
So the activation energy term can be estimated as
1 (µ0 gH )2 4 T σ ∝n= −1 An important factor is the apparently “universal” co2ε p + U − 2µ˜ + 2T ln 2 cosh(µ0 gH /2T ) exp +1 efficient µ0 g characterizing an enhancement of ε2 by T the magnetic field. Nevertheless we would like to note (5) that the “universal” magnetic field dependence holds Here we have assumed that ε p − µ˜ >> µ0 gH /2. Thus only for the field region corresponding to linear field dependence. we in particular obtain that for H = 0 We have compared the predictions given above with ε2 = 2ε p + U − 2µ ˜ (6) some existing experimental data for the magnetic field δε2 =
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MAGNETORESISTANCE RELATED TO ON-SITE SPIN CORRELATIONS
behavior of the activation energy for different materials (or magnetoresistance curves for different temperatures in the region of ε2 conductivity) [7,12–16,18]. All the materials were on the insulating side of the metal– insulator transition at H=0 except GaAs [16] where the metal–insulator transition was stimulated by the magnetic field. The corresponding experimental curves ε2 (H ) clearly demonstrate linear plots at large magnetic fields. Basing on these plots we have estimated the values of gµ0 , the estimates are given in Table where we compare these values with known values of g for the corresponding materials. One sees that for n-type materials [13, 14, 16, 18]. there is an excellent agreement. Note that we have also considered a possible role of the anizotropy of g- factor for many-valley semiconductors related to the valleys anizotropy which gives rise to a difference between g- factors for the magnetic field directions parallel and normal to the valley axis (gk < g⊥ . Although the difference in question can be large, a presence of several valleys with different orientations typically leads to the averaging and the anisotropy of the effective g factor typically appears to be not too strong. Nevertheless the experimental data (see e.g. [14] demonstrate a dependence of the activation energy on the magnetic field direction. Thus we have estimated the effective g-factors with respect to the situation of [14] taking into account that the main contribution is given by the valleys ensuring the lowest possible g (and thus of the activation energy). The results are given in Table and also demonstrate a good agreement with experiment. We would like also to note that the averaging mentioned above is least pronounced for a situation when the magnetic field is parallel to axis of one of the valleys, in this case the activation energy is dominated by a corresponding valley with g = gk . The situation is not as clear for p-type materials [7,15] because the estimates of g for shallow accepters are not as simple as for shallow donors due to the degeneracy of the valence band. In this concern we would like to note that the analysis of the magnetoresistance for ε2 hopping conductivity corresponding to the intermediate range of doping gives a possibility for direct estimates of g-factors for situations when these factors can not be extracted by other methods. Now let us discuss a possible role of magnetic field effect on the overlapping integrals. As it is well known, the main effect is the temperature independent positive magnetoresistance. However the overlapping integrals also affect the width of the impurity band and thus the value of ε3 for impurity concentration not far
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from insulator–metal transition (see e.g. [3]). Another possibility for the overlapping integrals to affect the activation energy is related to correlated hopping [7]. Thus we should in principle also compare the experimentally observed behavior of the activation energy with the behavior expected for these mechanisms. Here we would like to note that for the overlapping integrals effect one does not expect any universal behavior of the sort predicted for the spin ε2 magnetoresistance. In particular, the wave shrinkage effects are obviously sensitive to the Bohr radii. In contrast, the experimental data (see Table ) exhibit universal magnetic field behavior (depending only on the g-factor value) for different materials, different impurities types and concentrations and do not show any dependence on the Bohr radii (like P and Sb in Ge). Moreover, the agreement between the values of g extracted from experimental magnetoresistance curves with the help of the universal ε2 slope discussed above with known values can hardly be considered as an accidental one. So these features allow us to exclude the orbital effects from the considerations. To conclude, we have shown that an increase of the activation energy in the nearest neighbor hopping conductivity is a signature of a contribution of the upper Hubbard band (ε2 channel). While being quadratic in terms of H in weak magnetic fields µ0 gH < 2kB T , the enhancement is linear in strong magnetic fields: ∆ε2 = µ0 gH; as it is seen, in the latter case the behavior is a universal one allowing to discriminate the mechanism in question. Note that while an appearence of µ0 gH addition to the activation energy is clearly of the same nature as for the mechanism considered for varuable range hopping [8–10], the universal ε2 (H ) slope depending exclusively on the value of g (in contrast to the VRH situation where the experimental behavior is not as universal) gives an advantage which even can ensure an independent method to estimate g-factors. The results are in good agreement with experimental data for n-type materials where g values are well known; this fact is to our opinion a strong evidence of the ε2 character of hopping for the data in question. Acknowledgements—We are grateful to Polyanskaya T.A. for initiating discussion. This paper was supported by Russian Science Foundation (grant N 9702-18280a). REFERENCES 1. Mott, N.F. and Davis, G.A., Electronic Processes in Non- Crystalline Materials Oxford, 1979. 2. Gershenson, E.M., Mel’nikov, A.P. and Rabinovich, R.I., in: Electron–electron interactions in
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