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Theory of negative magnetoresistance in three-dimensional systems
Solid State Communications, Vol. 34, pp. 431-432. Pergamon Press Ltd. 1980. PrInted in Great Britain. THEORYOF NEGATWE MAGNETORESISTANCE IN THREE-DIMENSIONAL SYSTEMS A. Kawabata Department of ~iysics, Gakushwn University, Mejiro, Toshima-ku, Tokyo, 171 Japan (Received 8 Januwy 1980 by W Sasaki) Theory of negative magnetoresistance in two-dimensional systems due to delocalizatioiiof electrons by magnetic field by Hikami, Laildn, and Nagoaka is extended to the case of three-dimension. The increase in conductivity by magnetic field is independent of the direction of2r~’2, the current relative to r that of the magnetic field, and is proportional to H 1 being energy relaxation time of electrons,when the magnetic field His small, If r0 is large enough at low temperature and H is not too small, it is independent of the parameter characterizing the system, and is of the form 0.918’~JH mho cm~(H in kOe). RECENTLY it was pointed out by Abrahams ef al. [1] that in two-dimensional/electron systems with random potential all the wave functions are localized and that the conductivity vanishes~at absolute zero temperature. Thistolocalization is lifted by magnetic field,shown and itby gives rise negative magnetoresistance, as was Hikami et a!. [2] on the basis of the method developed by Gor’kov ~t a!. [3]. Recent experimental data by
the energy relaxation time of an electron due to inelastic scattering [5] (we assume that there is no local. ized spin nor spin—flip scattering due to spin—orbit interaction). The cut 2/A2, off parameter N0is are where A =q0, VpT the given meanby q0 1/A and N0 1 free path of an electron, VF being Fermi velocity. Below we will see that the value of magnetoconductivity does not depend on q 0 and N0, as long as q01,N0 ~ 1. In fact equation (3) is valid under the conditions
Kawaguchi and ICawaji [4] are at large consistent with the theoretical predictions. The anomolous term which gives negative divergent contnbution to the conductivity at zero temperature and zero magnetic field gives a considerable negative contribution, although not divergent,mechanism also in three-dimensional case, suppression of this by magnetic field willand result in negative magnetoresistance. It is easy to extend the calculation in [2] to three-
2
f1/mvpr,
X/l~1,
~r,
where ~ = eH/mc is cyclotron frequency. We perform the integration over 2 q,~in equation (3), and we obtain e a~(Jl,T) = it hi —
N
1
0
dimensional systems. The conductivity o(H, T)under magnetic field H and at temperature Tis given by o(H, T) = 0~ + o 1(H, T), (1) where a0is the normal part 2r/m, (2) = ne n, ~r• r being electron density, effective mass, and the relaxation time of an electron due to Impurity (elastic) scattering, respectively. The anomolous part is Independent of the direction of the current, relative to the magnetic field, and Is given t!Y e2D a~(FI,T)
=
—
~
Nit 0
(4)
q01
~J(N+1/2 + 8)tan 2s,/(N + 1/2 + 5)’ (5)
X
2/4TeD. In thelimit ofH-+O(l-~o°)we can where 8 =1 write the right-hand side of equation (5) in the form of an integral, and we can rewrite it as N,+ ~ ~ — — I, 1 . ~ -i q~l 3 an , irhl .‘ ~i~x+8) 2~~x+8
-‘s—--
0
(6) It is easy to see that the right-hand side is independent ofI as it should be. Hence the magnetoconductivity i~o(H,T) = o~(J1,T) — o~(0,T) is given by
7.0
2
I dq 1 Ii _j
~~w’HT’ “
‘
I
N0
x’c
,~,14Dr~(N+ 1/2)+Dq~+
/
— —
(3)
—
3hl .~ ir
[ ~ sJ(xdx+ 5) tan” 2s/(x +5) 1N1,+i
1 I
N_o\sJ(N+ 1/2 + 5)~
1/re’
where i~= c~i/eH,D is diffusion coefficient, and T~i5
-i
q0 2~J(N+1/2 + 8) (7)
431
432
NEGATNE MAGNETORESISTANCE IN THREE-DIMENSIONAL SYSTEMS
Vol. 34, No.6
Since q01,N0 )‘ 1 by assumption, we take the limit q01,
ized spin and spin flip scattering due to spin-orbit inter-
N0 -~ co, and we obtain 2hl &,(H, T) = 2ir
action suppress negative magnetoresistance. As for localized spin, it provides magnetoresistance [7]. anothermechanism of negative Experimentally, negative magnetoresistance has been observed on various doped semiconductors [8—15]. Present theory shows that it is a general phenomenon to be observed in any kind of materials which exhibit
f
2[~,/(N+ i + 5) 1
—
~/(N + 8) — ~/(N + 1/2 + 8)j~
(8)
If we put Xe = Vpr can of bethe written in the metallic conductivity. the upperoflimit ofisi~to(H, T) 2/4XX~with 0, the8 use relation D =form vj~r/3.AS for a given value of H isSince independent 0~, as seen in 8we= will 3l see below r 0 can be much larger than r~ i.e. equation [91,it is easier to observe I the phenomena in A6 )‘ A, at low temperature, and 5 can be larger or doped semiconductors than in metals. smaller than unity under the conditions (4). In the case S ~ 1 (low temperature or strong magnetic field) equation(8) reduces to Acknowledgements — The author is grateful to Prof. S. Kawaji for valuable discussions and for readingthe 1 (H in kOe) manuscript before useful discussions with Dr.publication. S. Hikami. He Thisappreciates~the work is supported ~a(H,T) = = 0.918%JHmho cm (9) by Grant-in-Aid from the Ministry of Education, Science where and Culture. C,
=
=
~ I2MN+ 1)—sIN]—
1
+ 1/2))
0.605.
(10)
It is to be noted that in this case ~a(H, 7’) does not depend on the parameters characterizing the system. When 8 ) 1, from equation (8) we obtain e2 Ac(H, T) ~ 261,211! ~
1
0(N + 1/2 + ~)S/2 2 r dx e 26ir2h1 (x +8)~~’
4. (11)
and hence Ao(H, T)
REFERENCES 1. E. Abrahams, P.W. Anderson, D.C. Licciardello & T.V. Rarnakrishnan,Fhys. Rev. Lett. 42,673 (1979). 2. S. Hikami. A.I. larkij4 & Y. Nagaoka, Frog. Theor. P11)75. 63, 707(1980). 3. LP. Gor’kov, A.!. Larkin & D.E. Khmel’nitzkii, ZhETFLett. 30, 248 (1979).
5. (12)
6.
This value can be larger than normal transverse magnetoconductivity [6],— a when 2, r~~ r 0(~,~r)of energy relaxation of The possible mechanisms an electron are electron—phonon interaction (r~~ T3) and electron—electron interaction (r 6 ~ r~2), and, for instance, in a typical case of highly doped 8secforsemiconducboth mechtors r6 is estimated to be 10~—l0 anisms at T = 1 K. Since a typical value of r is 10~3sec,
7. 8.
=
uo(r/r)3t2(w~r)2/l2sJ3.
the condition r 6 r can easily be satisfied. For nand = 1018 7cmsec’ cm~ and m =M0.2 obtain Vp sjHcm I0 (Bin kOe), the A 10~cm. l =we 0.811 x iCi~ condition 1> A can be satisfied up to magnetic field of ~‘
several kOe. It is not difficult to extend4 the theory to the case 1< A, and it will be discussed somewhere else. So far we have assumed the absence of spin dependent scattering. As is shown in [2], scattering by local-
Y. Kawaguchi & S. Kawaji,J. Fhys. Soc. Japan 48,699 (1980); State Y. Kawaguchi, Y. Kitahara & S. Kawaji,Solid Commun.~26, 701 (1978); Y.Kawaguchi,Y.Kitahara&S.Kawaji,Surf ScL 73, 520 (1978).
P.W. Anderson, E. Abrahams & T.V. Ramakrishnan,
Phys. Rev. Lett. 43, 718 (1979). C. Kittel, Quantum Theory of Solids Chapter 18.
Wiley, New York (1963). Y. Toyozawa,J. Phys. Soc. Japan 17,986 (1962). W. Sasald,hvc. mt. Conf I4iys. Semicond. Kyoto, p. 543. (1. Pliys. Soc. Japan 21, Suppl. (1966)). 9. Semicond. M. Balkanski & A. Geismar,hvc. Conf Phys. Kyoto, p. 554. (J. Phys.mt. Soc. Japan 21, Suppl. (1966)):
10. Soc. C. Yamanouchi, K. Mizuguchi Japan 22, 859 (1967). & W. Sasaki, /. Phys. ~ Y. Ootsuka, S. Kobayashi, S. Ikehara, W. (1979). Sasaki & J. Kondo,Solid State Cominun. 30, 169 12. 13.
14. 15.
H. Roth, W.D. Straub,W. Bernard & J.E. Mulliern Jr.,Thys. Rev. Lett. 11,328 (1963). O.V. Emelianenko & D.N. Nasledov,IZh. Techn. Fir. 28, 1177(1958) (Soy. Fhys.—Tech. Phys. 3,
1094 (1959)). J.F. Woods & C.Y. Chen,Phys. Rev. 135, A1462 (19M) Y. Katayama & S. Tanaka, Phys. Rev. 153,873 (1967).
the energy relaxation time of an electron due to inelastic scattering [5] (we assume that there is no local. ized spin nor spin—flip scattering due to spin—orbit interaction). The cut 2/A2, off parameter N0is are where A =q0, VpT the given meanby q0 1/A and N0 1 free path of an electron, VF being Fermi velocity. Below we will see that the value of magnetoconductivity does not depend on q 0 and N0, as long as q01,N0 ~ 1. In fact equation (3) is valid under the conditions
Kawaguchi and ICawaji [4] are at large consistent with the theoretical predictions. The anomolous term which gives negative divergent contnbution to the conductivity at zero temperature and zero magnetic field gives a considerable negative contribution, although not divergent,mechanism also in three-dimensional case, suppression of this by magnetic field willand result in negative magnetoresistance. It is easy to extend the calculation in [2] to three-
2
f1/mvpr,
X/l~1,
~r,
where ~ = eH/mc is cyclotron frequency. We perform the integration over 2 q,~in equation (3), and we obtain e a~(Jl,T) = it hi —
N
1
0
dimensional systems. The conductivity o(H, T)under magnetic field H and at temperature Tis given by o(H, T) = 0~ + o 1(H, T), (1) where a0is the normal part 2r/m, (2) = ne n, ~r• r being electron density, effective mass, and the relaxation time of an electron due to Impurity (elastic) scattering, respectively. The anomolous part is Independent of the direction of the current, relative to the magnetic field, and Is given t!Y e2D a~(FI,T)
=
—
~
Nit 0
(4)
q01
~J(N+1/2 + 8)tan 2s,/(N + 1/2 + 5)’ (5)
X
2/4TeD. In thelimit ofH-+O(l-~o°)we can where 8 =1 write the right-hand side of equation (5) in the form of an integral, and we can rewrite it as N,+ ~ ~ — — I, 1 . ~ -i q~l 3 an , irhl .‘ ~i~x+8) 2~~x+8
-‘s—--
0
(6) It is easy to see that the right-hand side is independent ofI as it should be. Hence the magnetoconductivity i~o(H,T) = o~(J1,T) — o~(0,T) is given by
7.0
2
I dq 1 Ii _j
~~w’HT’ “
‘
I
N0
x’c
,~,14Dr~(N+ 1/2)+Dq~+
/
— —
(3)
—
3hl .~ ir
[ ~ sJ(xdx+ 5) tan” 2s/(x +5) 1N1,+i
1 I
N_o\sJ(N+ 1/2 + 5)~
1/re’
where i~= c~i/eH,D is diffusion coefficient, and T~i5
-i
q0 2~J(N+1/2 + 8) (7)
431
432
NEGATNE MAGNETORESISTANCE IN THREE-DIMENSIONAL SYSTEMS
Vol. 34, No.6
Since q01,N0 )‘ 1 by assumption, we take the limit q01,
ized spin and spin flip scattering due to spin-orbit inter-
N0 -~ co, and we obtain 2hl &,(H, T) = 2ir
action suppress negative magnetoresistance. As for localized spin, it provides magnetoresistance [7]. anothermechanism of negative Experimentally, negative magnetoresistance has been observed on various doped semiconductors [8—15]. Present theory shows that it is a general phenomenon to be observed in any kind of materials which exhibit
f
2[~,/(N+ i + 5) 1
—
~/(N + 8) — ~/(N + 1/2 + 8)j~
(8)
If we put Xe = Vpr can of bethe written in the metallic conductivity. the upperoflimit ofisi~to(H, T) 2/4XX~with 0, the8 use relation D =form vj~r/3.AS for a given value of H isSince independent 0~, as seen in 8we= will 3l see below r 0 can be much larger than r~ i.e. equation [91,it is easier to observe I the phenomena in A6 )‘ A, at low temperature, and 5 can be larger or doped semiconductors than in metals. smaller than unity under the conditions (4). In the case S ~ 1 (low temperature or strong magnetic field) equation(8) reduces to Acknowledgements — The author is grateful to Prof. S. Kawaji for valuable discussions and for readingthe 1 (H in kOe) manuscript before useful discussions with Dr.publication. S. Hikami. He Thisappreciates~the work is supported ~a(H,T) = = 0.918%JHmho cm (9) by Grant-in-Aid from the Ministry of Education, Science where and Culture. C,
=
=
~ I2MN+ 1)—sIN]—
1
+ 1/2))
0.605.
(10)
It is to be noted that in this case ~a(H, 7’) does not depend on the parameters characterizing the system. When 8 ) 1, from equation (8) we obtain e2 Ac(H, T) ~ 261,211! ~
1
0(N + 1/2 + ~)S/2 2 r dx e 26ir2h1 (x +8)~~’
4. (11)
and hence Ao(H, T)
REFERENCES 1. E. Abrahams, P.W. Anderson, D.C. Licciardello & T.V. Rarnakrishnan,Fhys. Rev. Lett. 42,673 (1979). 2. S. Hikami. A.I. larkij4 & Y. Nagaoka, Frog. Theor. P11)75. 63, 707(1980). 3. LP. Gor’kov, A.!. Larkin & D.E. Khmel’nitzkii, ZhETFLett. 30, 248 (1979).
5. (12)
6.
This value can be larger than normal transverse magnetoconductivity [6],— a when 2, r~~ r 0(~,~r)of energy relaxation of The possible mechanisms an electron are electron—phonon interaction (r~~ T3) and electron—electron interaction (r 6 ~ r~2), and, for instance, in a typical case of highly doped 8secforsemiconducboth mechtors r6 is estimated to be 10~—l0 anisms at T = 1 K. Since a typical value of r is 10~3sec,
7. 8.
=
uo(r/r)3t2(w~r)2/l2sJ3.
the condition r 6 r can easily be satisfied. For nand = 1018 7cmsec’ cm~ and m =M0.2 obtain Vp sjHcm I0 (Bin kOe), the A 10~cm. l =we 0.811 x iCi~ condition 1> A can be satisfied up to magnetic field of ~‘
several kOe. It is not difficult to extend4 the theory to the case 1< A, and it will be discussed somewhere else. So far we have assumed the absence of spin dependent scattering. As is shown in [2], scattering by local-
Y. Kawaguchi & S. Kawaji,J. Fhys. Soc. Japan 48,699 (1980); State Y. Kawaguchi, Y. Kitahara & S. Kawaji,Solid Commun.~26, 701 (1978); Y.Kawaguchi,Y.Kitahara&S.Kawaji,Surf ScL 73, 520 (1978).
P.W. Anderson, E. Abrahams & T.V. Ramakrishnan,
Phys. Rev. Lett. 43, 718 (1979). C. Kittel, Quantum Theory of Solids Chapter 18.
Wiley, New York (1963). Y. Toyozawa,J. Phys. Soc. Japan 17,986 (1962). W. Sasald,hvc. mt. Conf I4iys. Semicond. Kyoto, p. 543. (1. Pliys. Soc. Japan 21, Suppl. (1966)). 9. Semicond. M. Balkanski & A. Geismar,hvc. Conf Phys. Kyoto, p. 554. (J. Phys.mt. Soc. Japan 21, Suppl. (1966)):
10. Soc. C. Yamanouchi, K. Mizuguchi Japan 22, 859 (1967). & W. Sasaki, /. Phys. ~ Y. Ootsuka, S. Kobayashi, S. Ikehara, W. (1979). Sasaki & J. Kondo,Solid State Cominun. 30, 169 12. 13.
14. 15.
H. Roth, W.D. Straub,W. Bernard & J.E. Mulliern Jr.,Thys. Rev. Lett. 11,328 (1963). O.V. Emelianenko & D.N. Nasledov,IZh. Techn. Fir. 28, 1177(1958) (Soy. Fhys.—Tech. Phys. 3,
1094 (1959)). J.F. Woods & C.Y. Chen,Phys. Rev. 135, A1462 (19M) Y. Katayama & S. Tanaka, Phys. Rev. 153,873 (1967).