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Weak localization and negative magnetoresistance in wurtzite-type crystals
Solid State Communications, Vol. 100, No. 2, pp. 95-99, 1996 Copyright @I996 Published by Elwier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/96 $12.00 + .OO
Pergamon
PII:SOO38-1098(96)00462-O
WEAK LOCALIZATION AND NEGATIVE MAGNETORESISTANCE IN WURTZITE-TYPE CRYSTALS F.G. Pikus” and G.E. Piku@ “University of California, Santa Barbara, CA 93106, U.S.A. bA.F. Ioffe Physicotechnical Institute 194021St Petersburg, Russia (Received 30 May 1996; accepted 19 June 1996 by A.H. MacDonald)
We have developed a theory of the negative magnetoresistance due to the weak localization in uniaxial wurtzite-type crystals, in which the spin splitting of the conduction band is linear in the wave vector, unlike the cubic A3Bs crystals Unlike earlier theories, we take into account the correlation between the electron motion in spin and co-ordinate spaces. It is shown that as a result of this correlation the magnetoresistance depends on the orientation of the magnetic field with respect to the main axis of the crystal even if the effective mass is isotropic. The new theory allows to accurately determine the value of the spin splitting constant in uniaxial crystals. Copyright @ 1996Published by Elsevier Science Ltd Keywords: A. semiconductors, D. electronic transport, D. quantum localization.
The et%& of the negative magnetoresistance observed in highly doped semiconductors and metals in weak magnetic fields is known to be caused by the weak localization. The weak localization, which is the result of the constructive interference of two electron waves traveling along a closed path in opposite directions and scattering by the same impurities, leads to an additional contribution to the resistance. In a magnetic field the two waves acquire a phase difference (P/*0, where @ is the magnetic flux through the electron trajectory and 90 = Rc/2e is the flux quantum. As a result, the interference conditions am violated and the additional resistance decreases, which manifests itself as a negative magnetoresistance, i.e. increase of the conductivity [l]. If one takes into account the electron spin, the singlet state of the two waves with the total momentum J = 0 gives a negative contribution into the resistance (antilocalization), while the triplet states with J = 1 give a positive contribution [2]. In certain situations the triplet contribution is suppressed by the spin relaxation, while the singlet contribution is not. Then a weak magnetic field first reduces the negative contribution of the singlet state, which results in a
positive magnetoresistance. The negative magnetoresistance takes over in larger fields, which suppress the triplet states stronger than the spin relaxation does. According to [2,3] this change of the sign of magnetoresistance in bulk crystals can be observed for Elliott-Yafet and Dyakonov-Peml me&anisms of the spin relaxation. On the other hand, the spin relaxation dominated by the scattering on paramagnetic impurities suppresses the singlet contribution, and the magnetoresistance is negative in the whole range of magnetic fields where the weak localization is important. Therefore, the type of the magnetoresistance is determined by the dominant spin relaxation mechanism. In the non-centrosymmetric semiconductors (with the exception of narrow-gap and gapless semiconductors) this is the Dyakonov-Pexel mechanism, which is caused by the spin splitting of the conduction band [4,5]. In the uniaxial wurtzite-type crystals, unlike the cubic crystals A3Bs, the spin splitting of the conduction band is linear in the wave vector k, as described by the following Hamiltonian [6]:
95
3f = alakl,.
(1)
NEGATIVE MAGNETORESISTANCE IN WURTZITE-TYPE CRYSTALS
96
where z is the main axis C6 of the. crystal It was shown [7-91 that in this case the theory of the weak localization must take into account the correlation between the electron motion in co-ordinate and spin spaces, similarly to the 20 structures The weak localization contribution to the conductivity can be expressed through the Cooperon C as follows [l-3]:
Hem D = :$I, is the diffusion coefficient, ~1 is the transport time, TO is the momentum lifetime, YiS the electron velocity, qiax = 1/ DTI, v. = (2m)3/2&/(27r)2 is the density of states at the Fermi level E,U,and m is the effective mass. The summation in Eq. (2) is carried over spin indices or,B= + l/2. * . For simplicity we assume an isotropic electron spectrum. Then the-matrix equation for the Cooperon can be written similarly to Refs [7-91:
hck,k’ = To
X
1 - i(@To
- i(0 + P)RTo -
&q)*T;
-[(a -+fdfi).ro]*- 2(V,q)(U + p)fiT,2
-z 1 C&k/
(4).
Here, according to Eq. (I), f&(g)
=
nsinSsincp,
ny(g) = -hz sin 9 cos 9, G = akF,
&,k’-g
do,
(4)
has the following harmonics as its eigenfunctions: (5) Cz = c&,Y:(s’, Q.+). These formulas are which is written in the coordinate system with z’ ]I k’, and 9’ and Cp’are the polar and axial angles of k in this system. The following property is well-known for the spherical harmonics:
=
Y”,(o~) Jon'* W(6) cosn0sin 6d0
=y”,(ok)(&$).
Here ml2 W(6)(1- cosne)sinede,
1 -=
Tll
(n # O),
I0 n/2
J% JVk,,12
Ck,k’(4) = 1V~.KI2+27TVoTo
Jww’,
Vol. 100, No. 2
1 -= W(0) sin 8d9. (7) To I o (3) it follows from (4-6) that the eigenfunctions (5) have the corresponding eigenvalues A,,,,,,= 1 - TO/T,, (n # 0), and As = 1. Therefore, the only large component of the Cooperon is the scalar CQOI 0. Since the right-hand side of Eq. (4) contains linear in g terms, it is also necessary to retain the harmonics Cl,,. These can be found from (4) if we substitute CC,&’= Cl,, in the right-hand side and Ck,K = CQ+ CI,~ in the left-hand side, and keep only terms linear in g. Then, using Eqs (6,7) we can express Ct through Co:
g vg = -, m
and TV is the phase-breaking time. In Eq. (3) the Pauli matrices o act on the first pair of spin indices, while the matrices p - on the second pair. The angles 9 and Q, are that of the vector I in the coordinate system with x ]I c6, and the integration is over solid angle 4n: do, = sin 9 d9 dp. Vk,g is the scattering matrix element, which includes the density of the scatterers Then, for the scattering probability we have
C:.ky
the scattering probability depends only on the scattering angle 8, then the equation * Throughout this paper we take h = 1 except in the tiaal expressions
cl,my!@k)
1
In = -_I ; .(
-
1
[b’kqho
+ (0
+ P)nTo]
(-h
> (8)
We now substitute C = C-J+ Ct to the right-hand side of Eq. (4), express Ci through CJ using Eq. (8), and integrate both sides over d Ok and d OK to cancel out all components &,, except CQ.We then arrive to the following equation for the matrix s = 27rvo~&: LJfs=
W(k, k’) = 27rvs 1v&r I2 . If
=
1,
(9)
where H = D4’+ (a + p)*fi*T1
x[(a,+Ph?y -
+ (~?DT#'~~
(a,+
@3x]+
$.
w
NEGATIVE MAGNETORESISTANCE IN WURTZITE-TYPE CRYSTALS
Vol. 100, No. 2
The Green’s function for the equation (10) can be written as *
97
6 -_ 4eBD Rc ’
(16)
to introduce new operators D
where ‘P&(q) and E,(q) and the eigenfunctions and eigenvalues of the operator H:
(I=
0 -
l/2
(17)
6
The Hamiltonian (13, 14) can be then written as HWq)
(12)
= E,Y’(q).
The basis of eigenfunctions Y’ can be converted by the appropriate choice of their linear combinations to a basis of singlet and triplet states. The Hamiltonian (10) is then split into two matrices, since it has no nonzero matrix element between singlet and triplet functions. For the singlet state with J = 0 and antisymmetric wave function Y” the Hamiltonian is simple:
x3(q) = Dq2+ -$
= Dq: +
61aatl+ k,
3& (q2) = Dqz + 6{aat} + $
(18) + 2 (2 - Ji)
+2i(DT,)1’2Q (J+at - J-a),
n2.rl
(19)
In the basis of the eigenfunctions of the operator {aaf} = $(aat + ata) these operators have the following non-zero matrix elements
(13)
while for the triplet state with J = 1 and symmetric wave functions Yk (m = 0, kl) it is convenient to introduce the momentum operator J = (a+ P) 12 and write the Hamiltonian in this form: 3-f,(q) =Dq2 + 2 (2 - JI’) \
3-&q,)
(n-llaIn)=(nlatIn-l)=Ji;,
1 (nl (aat In) =n+ -. 2
(20)
According to Eqs. (2), (15) the weak localization correction to the conductivity can be written as
R’T,
I
+
2i(DT1)‘12Q(J+q_ - J-q+) +
1
4*SqA,
(14)
(21)
G,
where
where q+ = qx + iq,
and J2 = $
(Jx * Jy) .
S(qJ =
Since the eigenfunctions Y, are orthonormal, it follows
from Eq. (11) that
nz (22) (-&+m&kJn n=O
EF(qz) are the eigenvalues of 3&-,and Jft and nmax= 116~1.For B IIz these eigenvalues can be calculated
analytically, similarly to Ref. [A. Since we are only (15) interested in the change of the conductivity with the magnetic field, we subtract Ac(B = 0) from Eq. (21) Due to the anisotropy of the spin splitting (1) the to obtain the magnetoconductivity &r(B) = Aa (B) Hamiltonian 3& in a magnetic field depends on the *a(O). In 60, following Ref. [lo], we can extend field direction. In a magnetic field B 11z we can use both the integration over qz and summation over n to infulity: the commutation relation S(q) =
-& +$, &*
&Y(B) =
where * Note that the formula (11) is valid only when the operator 3fis hermitian. It is therefore, inapplicable, for example, when the Zecman splitting in the magnetic field is taken into account.
-- e2 [+(x)dx-2f3 4n2Rl Tr 0
-f@'+
(5 2%)+
+ T)
jj(%)], (23)
98 F(x) =
NEGATIVE MAGNETORESISTANCE 1, 00
IN WURTZITE-TYPE
24 + 1 + b al(ao +
I
Vol. 100, No. 2
CRYSTALS
1 ’ a I “‘I
I ”
t ’ 1”
* 11
b) - 2b
3ai + 2a,b - I - 2(2n + 1)b +in=, (a, + b)@ - 1) - 2b[(2n + I)a, - 11 ( 1 -- 2 @l-l a,,_r +b
’
\
I’
_-____** /’ Lt.,
and 1 = (cR/eB)“’ is a magnetic length. For B I z we can choose axis x ]I B we can introduce a and ct similarly to Eq. (17): D(q; + q;) = @a+],
2D*‘2qy = i~3”~(a+-a).
Then the Hamiltonian (14) becomes
(25)
-2000
.I
(.
/’
-1ooa
1.
*
1
*,
\
\
‘. . . .
I,, 1000
3 ._____-_ .
*
Ii
2000
El, &as Fig. 1. Theoretical magnetoconductivity h(B) for B I z (curve 1) and B I( z (curve 2) for HP = 3Gs and HSO = 445Gs. The curve 3 shows the magnetoconductivity as obtained from the theory of Ref. [3] Eq. (28) for the same values of the parameters.
This formula differs from the one derived in Ref. [3] and used in, for example, Ref. [l l] in the value of -a) J, -2(2D~,)~/~fh&2$)Hso: insteadof Hso = 1/6T,,, with I/T,, = l/r,,,= +i(26T,)1’2R(U+ = 2n271 [4,5], the Refs. [3,11] used ~~~~~~= 1/2Tszz In this case the eigenvalues of the Hamiltonian can n2r1. be computed only numerically, restricting the matrix In Fig. 1 the magnetoconductivity c%(B) as obsize to some large, but linite, value. It is convenientto hined using Eqs. (19 - 24), (26), and (27) is shown for two orientation of the magnetic field B II z and express the sum of the reciprocal eigenvalues E?(qx) B I z for the same values of HP, = 3Gs and Hw = through the minors IDdqJ I of the diagonal elements L?fl,,and the determinant ID( of Jfr as follows: 445Gs. The dashed line shows 6a from Eq. (28) for the same values of the parameters. One can see that 6a signiticantly depends on the field direction and differs strongly from the result of Eq. (28), which does ;m&&=Q”;$)l -*;%J) nmx (27) not take into account the correlation of the electron motion in spin and co-ordinate spaces. f c J-&‘(qJ. The results of magnetoconductivity measurements i for CdSe were published in Ref. [ 111.The samples used The 6rst sum converges rapidly for i - 00. The sumhad electron density n = 8.6 x 10” cmm3 and kF1 = mation in the second sum can also be extended to infinity if we subtract its value at B = 0 similarly to 2.62, which corresponds to D = i !kFl = 7.84 s for m = 0.13 mu, mc being the free electron mass. In Fig. 2 Eq. (23). If we omit the linear in R and q (or a, at) terms we reproduce the experimental results of Ref. [I I] for in Eqs (19) and (26), the first terms in Eqs. (23) and 6a at T = 0.06 K and compare it with the results of our theory. In Ref. [l I] the direction of the magnetic (27) also vanish. In this approximation the magnetofield is not specified; it is noted that at T = 1.5 K conductivity 6a does not depend on the orientation h(B) does not depend on the field orientation (at this of the magnetic field and is given by this formula: temperature HP 2 HSO and the dependence 60(B) does not have a minimum at small B, and the role 2f; 6aU3) = of the linear terms in Eqs (19), (26) becomes nonessential). The solid line in Fig. 2 shows the fit by the HP, +2%) B -h (%)I. (28) theoretical magnetoconductivity (Eqs. (21 - 24)). for +f3
3f,(qx) = Dq;+s(aa+f
+-$
& [ (2+ + >
(
NEGATIVE
Vol. 100, No. 2
J,
E g 0
0.1
MAGNETORESISTANCE
0
i
‘\O .\ \
o-
I \
;;’
\\
\\
I
G
0
\
_
I’ ,’ 1’
\
y-o.,-
IN WURTZITE-TYPE
CRYSTALS
99
the first order in R/T~EF = 2/kFl. For the sample used in Ref. [l l] 2/kFl = 0.7, which means that the theory should not give very accurate results. In conclusion, we have developed a theory of the weak localization and the negative magnetoresistance in wurtzite-type semiconductors. It is demonstrated that if the spin splitting of the conduction band is linear in the wave vector it is necessary to take into account the correlation between the electron motion in co-ordinate and spin spaces. This correlation and the anisotropy of the spin splitting result in the mangetoresistance being dependent on the magnetic field direction even for an isotropic electron effective mass. Acknowledgements-F.
- 2000
0
-1000 B.
1000
2000
Gauss
Fig. 2. Comparision of the measured in Ref. [ 1I] (circles) and theoretical magnetoconductivity. Solid line show the result of our theory for &, = 3 Gs and Hso = 445 Gs; dashed line is the best fit fit obtained y;lrn;$ theory of Ref. [3] (HP = 1.7 Gs and Hso =
G. I? acknowledges support by the NSF Grant DMR 93-08011, the Center for Quantized Electronic Structures (QUEST) of UCSB and by the Quantum Institute of UCSB. G. E. I? acknowledges support by RFFI Grant 96-02- 17849 and by the Volkswagen Foundation. REFERENCES
1. Altshuler, B., Khmelnitskii, D., Larkin, A. and Lee, I?, Phys. Rev. B, 22, 1980,5142. 2. Hikami, S., Larkin, A. and Nagaoka, Y., Progr. B \I z. The fitting was done by weighted explicit orthogonal distance regression using the software package ODRPACK [12]. The weights were selected to increase the importance of the low-field (B I 1kG.s)part of the magnetoconductivity curve. The parameters of the best fit are HP = 3 Gs and HSO = 445 Gs (these are the same values we used in Fig. 1). The dashed line shows the best fit obtained from Eq. (28), the parameters of this fit are HP = 1.7 Gs and Hm = 141 Gs. One can see that the agreement with experiment is not as good as that of the solid curve. The above value of HSO = 445 Gs corresponds to a = 3.9 x 10T2 eV. A. The standard deviation of CYas computed by the fitting procedure is 3%. In Ref. [l l] authors have obtained a = 2.5 x 10s2 eV* &they have used the notation h = 20( for the spin splitting coefficient). Taking into account the error in the definition of rsXXp ointed above this result should be divided by a. In Ref. [13] the constant h was measured for Cdl_,Mn,Se using electric-dipole spin resonance and found to be a = h/2 = (3.25 f 1.2) x lop2 eV. &extrapolated to x = 0). However, for more accurate comparison with our theory we must know the precise orientation of the magnetic field. We should also note that the weak localization theory includes only the corrections to the conductivity which are of
Theor. Phys., 63,1980,707. 3. Altshuler, B.L., Aronov, A.G., Larkin, A.I. and Khmelnitskii, D.E., JETP, 54, 1981, 411; JETP, 81, 1981,788. 4. Dyakonov, M.I. and Perel, V.I., JETP, 33, 1971,
1053. 5. Pikus, G.E. and Titkov, A., Spin Relaxation under Optical orientation, in Optical Orientation (edited by F. Mayer and B. Zakharchenya), North Holland, Amsterdam, 1984. 6. Rashba, E.I. and Sheka, V.I., Fiz. Tverd. Tela, Sb. II, 1959, 162; Rashba, E.I. and Sheka, V.I., Sov. Phys.-Solid State, 3, 1961, 1257. 7. Iordanskii, S.V., Lyanda-Geller, Yu.B. and Pikus, G.E., JETP Lett., 60,1994,206; Pisma v JETP 60, 1994,199. 8. Pikus, F.G. and Pikus, G.E., Phys. Rev. B, 51,1995, 16928. 9. Knap, W. et. al, Phys. Rev. B, 53, 1996, 3912. 10. Kawabata, A., Solid State Commun., 34,1980,431. 11. Sawicki, M. et. al, Phys. Rev. Lett., 56, 1986,508. 12. Boggs, PT. et. al, NIST preprint NISTIR 92-4834. 13. Dobrowolska, M. et. al, Phys. Rev. B 29, 1984, 6628.
Pergamon
PII:SOO38-1098(96)00462-O
WEAK LOCALIZATION AND NEGATIVE MAGNETORESISTANCE IN WURTZITE-TYPE CRYSTALS F.G. Pikus” and G.E. Piku@ “University of California, Santa Barbara, CA 93106, U.S.A. bA.F. Ioffe Physicotechnical Institute 194021St Petersburg, Russia (Received 30 May 1996; accepted 19 June 1996 by A.H. MacDonald)
We have developed a theory of the negative magnetoresistance due to the weak localization in uniaxial wurtzite-type crystals, in which the spin splitting of the conduction band is linear in the wave vector, unlike the cubic A3Bs crystals Unlike earlier theories, we take into account the correlation between the electron motion in spin and co-ordinate spaces. It is shown that as a result of this correlation the magnetoresistance depends on the orientation of the magnetic field with respect to the main axis of the crystal even if the effective mass is isotropic. The new theory allows to accurately determine the value of the spin splitting constant in uniaxial crystals. Copyright @ 1996Published by Elsevier Science Ltd Keywords: A. semiconductors, D. electronic transport, D. quantum localization.
The et%& of the negative magnetoresistance observed in highly doped semiconductors and metals in weak magnetic fields is known to be caused by the weak localization. The weak localization, which is the result of the constructive interference of two electron waves traveling along a closed path in opposite directions and scattering by the same impurities, leads to an additional contribution to the resistance. In a magnetic field the two waves acquire a phase difference (P/*0, where @ is the magnetic flux through the electron trajectory and 90 = Rc/2e is the flux quantum. As a result, the interference conditions am violated and the additional resistance decreases, which manifests itself as a negative magnetoresistance, i.e. increase of the conductivity [l]. If one takes into account the electron spin, the singlet state of the two waves with the total momentum J = 0 gives a negative contribution into the resistance (antilocalization), while the triplet states with J = 1 give a positive contribution [2]. In certain situations the triplet contribution is suppressed by the spin relaxation, while the singlet contribution is not. Then a weak magnetic field first reduces the negative contribution of the singlet state, which results in a
positive magnetoresistance. The negative magnetoresistance takes over in larger fields, which suppress the triplet states stronger than the spin relaxation does. According to [2,3] this change of the sign of magnetoresistance in bulk crystals can be observed for Elliott-Yafet and Dyakonov-Peml me&anisms of the spin relaxation. On the other hand, the spin relaxation dominated by the scattering on paramagnetic impurities suppresses the singlet contribution, and the magnetoresistance is negative in the whole range of magnetic fields where the weak localization is important. Therefore, the type of the magnetoresistance is determined by the dominant spin relaxation mechanism. In the non-centrosymmetric semiconductors (with the exception of narrow-gap and gapless semiconductors) this is the Dyakonov-Pexel mechanism, which is caused by the spin splitting of the conduction band [4,5]. In the uniaxial wurtzite-type crystals, unlike the cubic crystals A3Bs, the spin splitting of the conduction band is linear in the wave vector k, as described by the following Hamiltonian [6]:
95
3f = alakl,.
(1)
NEGATIVE MAGNETORESISTANCE IN WURTZITE-TYPE CRYSTALS
96
where z is the main axis C6 of the. crystal It was shown [7-91 that in this case the theory of the weak localization must take into account the correlation between the electron motion in co-ordinate and spin spaces, similarly to the 20 structures The weak localization contribution to the conductivity can be expressed through the Cooperon C as follows [l-3]:
Hem D = :$I, is the diffusion coefficient, ~1 is the transport time, TO is the momentum lifetime, YiS the electron velocity, qiax = 1/ DTI, v. = (2m)3/2&/(27r)2 is the density of states at the Fermi level E,U,and m is the effective mass. The summation in Eq. (2) is carried over spin indices or,B= + l/2. * . For simplicity we assume an isotropic electron spectrum. Then the-matrix equation for the Cooperon can be written similarly to Refs [7-91:
hck,k’ = To
X
1 - i(@To
- i(0 + P)RTo -
&q)*T;
-[(a -+fdfi).ro]*- 2(V,q)(U + p)fiT,2
-z 1 C&k/
(4).
Here, according to Eq. (I), f&(g)
=
nsinSsincp,
ny(g) = -hz sin 9 cos 9, G = akF,
&,k’-g
do,
(4)
has the following harmonics as its eigenfunctions: (5) Cz = c&,Y:(s’, Q.+). These formulas are which is written in the coordinate system with z’ ]I k’, and 9’ and Cp’are the polar and axial angles of k in this system. The following property is well-known for the spherical harmonics:
=
Y”,(o~) Jon'* W(6) cosn0sin 6d0
=y”,(ok)(&$).
Here ml2 W(6)(1- cosne)sinede,
1 -=
Tll
(n # O),
I0 n/2
J% JVk,,12
Ck,k’(4) = 1V~.KI2+27TVoTo
Jww’,
Vol. 100, No. 2
1 -= W(0) sin 8d9. (7) To I o (3) it follows from (4-6) that the eigenfunctions (5) have the corresponding eigenvalues A,,,,,,= 1 - TO/T,, (n # 0), and As = 1. Therefore, the only large component of the Cooperon is the scalar CQOI 0. Since the right-hand side of Eq. (4) contains linear in g terms, it is also necessary to retain the harmonics Cl,,. These can be found from (4) if we substitute CC,&’= Cl,, in the right-hand side and Ck,K = CQ+ CI,~ in the left-hand side, and keep only terms linear in g. Then, using Eqs (6,7) we can express Ct through Co:
g vg = -, m
and TV is the phase-breaking time. In Eq. (3) the Pauli matrices o act on the first pair of spin indices, while the matrices p - on the second pair. The angles 9 and Q, are that of the vector I in the coordinate system with x ]I c6, and the integration is over solid angle 4n: do, = sin 9 d9 dp. Vk,g is the scattering matrix element, which includes the density of the scatterers Then, for the scattering probability we have
C:.ky
the scattering probability depends only on the scattering angle 8, then the equation * Throughout this paper we take h = 1 except in the tiaal expressions
cl,my!@k)
1
In = -_I ; .(
-
1
[b’kqho
+ (0
+ P)nTo]
(-h
> (8)
We now substitute C = C-J+ Ct to the right-hand side of Eq. (4), express Ci through CJ using Eq. (8), and integrate both sides over d Ok and d OK to cancel out all components &,, except CQ.We then arrive to the following equation for the matrix s = 27rvo~&: LJfs=
W(k, k’) = 27rvs 1v&r I2 . If
=
1,
(9)
where H = D4’+ (a + p)*fi*T1
x[(a,+Ph?y -
+ (~?DT#'~~
(a,+
@3x]+
$.
w
NEGATIVE MAGNETORESISTANCE IN WURTZITE-TYPE CRYSTALS
Vol. 100, No. 2
The Green’s function for the equation (10) can be written as *
97
6 -_ 4eBD Rc ’
(16)
to introduce new operators D
where ‘P&(q) and E,(q) and the eigenfunctions and eigenvalues of the operator H:
(I=
0 -
l/2
(17)
6
The Hamiltonian (13, 14) can be then written as HWq)
(12)
= E,Y’(q).
The basis of eigenfunctions Y’ can be converted by the appropriate choice of their linear combinations to a basis of singlet and triplet states. The Hamiltonian (10) is then split into two matrices, since it has no nonzero matrix element between singlet and triplet functions. For the singlet state with J = 0 and antisymmetric wave function Y” the Hamiltonian is simple:
x3(q) = Dq2+ -$
= Dq: +
61aatl+ k,
3& (q2) = Dqz + 6{aat} + $
(18) + 2 (2 - Ji)
+2i(DT,)1’2Q (J+at - J-a),
n2.rl
(19)
In the basis of the eigenfunctions of the operator {aaf} = $(aat + ata) these operators have the following non-zero matrix elements
(13)
while for the triplet state with J = 1 and symmetric wave functions Yk (m = 0, kl) it is convenient to introduce the momentum operator J = (a+ P) 12 and write the Hamiltonian in this form: 3-f,(q) =Dq2 + 2 (2 - JI’) \
3-&q,)
(n-llaIn)=(nlatIn-l)=Ji;,
1 (nl (aat In) =n+ -. 2
(20)
According to Eqs. (2), (15) the weak localization correction to the conductivity can be written as
R’T,
I
+
2i(DT1)‘12Q(J+q_ - J-q+) +
1
4*SqA,
(14)
(21)
G,
where
where q+ = qx + iq,
and J2 = $
(Jx * Jy) .
S(qJ =
Since the eigenfunctions Y, are orthonormal, it follows
from Eq. (11) that
nz (22) (-&+m&kJn n=O
EF(qz) are the eigenvalues of 3&-,and Jft and nmax= 116~1.For B IIz these eigenvalues can be calculated
analytically, similarly to Ref. [A. Since we are only (15) interested in the change of the conductivity with the magnetic field, we subtract Ac(B = 0) from Eq. (21) Due to the anisotropy of the spin splitting (1) the to obtain the magnetoconductivity &r(B) = Aa (B) Hamiltonian 3& in a magnetic field depends on the *a(O). In 60, following Ref. [lo], we can extend field direction. In a magnetic field B 11z we can use both the integration over qz and summation over n to infulity: the commutation relation S(q) =
-& +$, &*
&Y(B) =
where * Note that the formula (11) is valid only when the operator 3fis hermitian. It is therefore, inapplicable, for example, when the Zecman splitting in the magnetic field is taken into account.
-- e2 [+(x)dx-2f3 4n2Rl Tr 0
-f@'+
(5 2%)+
+ T)
jj(%)], (23)
98 F(x) =
NEGATIVE MAGNETORESISTANCE 1, 00
IN WURTZITE-TYPE
24 + 1 + b al(ao +
I
Vol. 100, No. 2
CRYSTALS
1 ’ a I “‘I
I ”
t ’ 1”
* 11
b) - 2b
3ai + 2a,b - I - 2(2n + 1)b +in=, (a, + b)@ - 1) - 2b[(2n + I)a, - 11 ( 1 -- 2 @l-l a,,_r +b
’
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and 1 = (cR/eB)“’ is a magnetic length. For B I z we can choose axis x ]I B we can introduce a and ct similarly to Eq. (17): D(q; + q;) = @a+],
2D*‘2qy = i~3”~(a+-a).
Then the Hamiltonian (14) becomes
(25)
-2000
.I
(.
/’
-1ooa
1.
*
1
*,
\
\
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I,, 1000
3 ._____-_ .
*
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2000
El, &as Fig. 1. Theoretical magnetoconductivity h(B) for B I z (curve 1) and B I( z (curve 2) for HP = 3Gs and HSO = 445Gs. The curve 3 shows the magnetoconductivity as obtained from the theory of Ref. [3] Eq. (28) for the same values of the parameters.
This formula differs from the one derived in Ref. [3] and used in, for example, Ref. [l l] in the value of -a) J, -2(2D~,)~/~fh&2$)Hso: insteadof Hso = 1/6T,,, with I/T,, = l/r,,,= +i(26T,)1’2R(U+ = 2n271 [4,5], the Refs. [3,11] used ~~~~~~= 1/2Tszz In this case the eigenvalues of the Hamiltonian can n2r1. be computed only numerically, restricting the matrix In Fig. 1 the magnetoconductivity c%(B) as obsize to some large, but linite, value. It is convenientto hined using Eqs. (19 - 24), (26), and (27) is shown for two orientation of the magnetic field B II z and express the sum of the reciprocal eigenvalues E?(qx) B I z for the same values of HP, = 3Gs and Hw = through the minors IDdqJ I of the diagonal elements L?fl,,and the determinant ID( of Jfr as follows: 445Gs. The dashed line shows 6a from Eq. (28) for the same values of the parameters. One can see that 6a signiticantly depends on the field direction and differs strongly from the result of Eq. (28), which does ;m&&=Q”;$)l -*;%J) nmx (27) not take into account the correlation of the electron motion in spin and co-ordinate spaces. f c J-&‘(qJ. The results of magnetoconductivity measurements i for CdSe were published in Ref. [ 111.The samples used The 6rst sum converges rapidly for i - 00. The sumhad electron density n = 8.6 x 10” cmm3 and kF1 = mation in the second sum can also be extended to infinity if we subtract its value at B = 0 similarly to 2.62, which corresponds to D = i !kFl = 7.84 s for m = 0.13 mu, mc being the free electron mass. In Fig. 2 Eq. (23). If we omit the linear in R and q (or a, at) terms we reproduce the experimental results of Ref. [I I] for in Eqs (19) and (26), the first terms in Eqs. (23) and 6a at T = 0.06 K and compare it with the results of our theory. In Ref. [l I] the direction of the magnetic (27) also vanish. In this approximation the magnetofield is not specified; it is noted that at T = 1.5 K conductivity 6a does not depend on the orientation h(B) does not depend on the field orientation (at this of the magnetic field and is given by this formula: temperature HP 2 HSO and the dependence 60(B) does not have a minimum at small B, and the role 2f; 6aU3) = of the linear terms in Eqs (19), (26) becomes nonessential). The solid line in Fig. 2 shows the fit by the HP, +2%) B -h (%)I. (28) theoretical magnetoconductivity (Eqs. (21 - 24)). for +f3
3f,(qx) = Dq;+s(aa+f
+-$
& [ (2+ + >
(
NEGATIVE
Vol. 100, No. 2
J,
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0.1
MAGNETORESISTANCE
0
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the first order in R/T~EF = 2/kFl. For the sample used in Ref. [l l] 2/kFl = 0.7, which means that the theory should not give very accurate results. In conclusion, we have developed a theory of the weak localization and the negative magnetoresistance in wurtzite-type semiconductors. It is demonstrated that if the spin splitting of the conduction band is linear in the wave vector it is necessary to take into account the correlation between the electron motion in co-ordinate and spin spaces. This correlation and the anisotropy of the spin splitting result in the mangetoresistance being dependent on the magnetic field direction even for an isotropic electron effective mass. Acknowledgements-F.
- 2000
0
-1000 B.
1000
2000
Gauss
Fig. 2. Comparision of the measured in Ref. [ 1I] (circles) and theoretical magnetoconductivity. Solid line show the result of our theory for &, = 3 Gs and Hso = 445 Gs; dashed line is the best fit fit obtained y;lrn;$ theory of Ref. [3] (HP = 1.7 Gs and Hso =
G. I? acknowledges support by the NSF Grant DMR 93-08011, the Center for Quantized Electronic Structures (QUEST) of UCSB and by the Quantum Institute of UCSB. G. E. I? acknowledges support by RFFI Grant 96-02- 17849 and by the Volkswagen Foundation. REFERENCES
1. Altshuler, B., Khmelnitskii, D., Larkin, A. and Lee, I?, Phys. Rev. B, 22, 1980,5142. 2. Hikami, S., Larkin, A. and Nagaoka, Y., Progr. B \I z. The fitting was done by weighted explicit orthogonal distance regression using the software package ODRPACK [12]. The weights were selected to increase the importance of the low-field (B I 1kG.s)part of the magnetoconductivity curve. The parameters of the best fit are HP = 3 Gs and HSO = 445 Gs (these are the same values we used in Fig. 1). The dashed line shows the best fit obtained from Eq. (28), the parameters of this fit are HP = 1.7 Gs and Hm = 141 Gs. One can see that the agreement with experiment is not as good as that of the solid curve. The above value of HSO = 445 Gs corresponds to a = 3.9 x 10T2 eV. A. The standard deviation of CYas computed by the fitting procedure is 3%. In Ref. [l l] authors have obtained a = 2.5 x 10s2 eV* &they have used the notation h = 20( for the spin splitting coefficient). Taking into account the error in the definition of rsXXp ointed above this result should be divided by a. In Ref. [13] the constant h was measured for Cdl_,Mn,Se using electric-dipole spin resonance and found to be a = h/2 = (3.25 f 1.2) x lop2 eV. &extrapolated to x = 0). However, for more accurate comparison with our theory we must know the precise orientation of the magnetic field. We should also note that the weak localization theory includes only the corrections to the conductivity which are of
Theor. Phys., 63,1980,707. 3. Altshuler, B.L., Aronov, A.G., Larkin, A.I. and Khmelnitskii, D.E., JETP, 54, 1981, 411; JETP, 81, 1981,788. 4. Dyakonov, M.I. and Perel, V.I., JETP, 33, 1971,
1053. 5. Pikus, G.E. and Titkov, A., Spin Relaxation under Optical orientation, in Optical Orientation (edited by F. Mayer and B. Zakharchenya), North Holland, Amsterdam, 1984. 6. Rashba, E.I. and Sheka, V.I., Fiz. Tverd. Tela, Sb. II, 1959, 162; Rashba, E.I. and Sheka, V.I., Sov. Phys.-Solid State, 3, 1961, 1257. 7. Iordanskii, S.V., Lyanda-Geller, Yu.B. and Pikus, G.E., JETP Lett., 60,1994,206; Pisma v JETP 60, 1994,199. 8. Pikus, F.G. and Pikus, G.E., Phys. Rev. B, 51,1995, 16928. 9. Knap, W. et. al, Phys. Rev. B, 53, 1996, 3912. 10. Kawabata, A., Solid State Commun., 34,1980,431. 11. Sawicki, M. et. al, Phys. Rev. Lett., 56, 1986,508. 12. Boggs, PT. et. al, NIST preprint NISTIR 92-4834. 13. Dobrowolska, M. et. al, Phys. Rev. B 29, 1984, 6628.