- Email: [email protected]
Electric-dipole and magnetic-dipole interference in the spin resonance of conduction electrons in Wurtzite semiconductors
Superlattices and Microstructures, Vol. 1, No.6, 1985
499
ELECTRIC-DIPOLE AND MAGNETIC-DIPOLE INTERFERENCE IN THE SPIN RESONANCE OF CONDUCTION ELECTRONS IN WURTZITE SEMICONDUCTORS Sudha Gopalan and S. Rodriguez Department of Physics, Purdue University, West Lafayette, IN 47907 (Received 29 September 1985) Recent studies of magnetotransmission in n-type InSb revealed an unusually strong conduction electron spin resonance which changed upon reversal of either the applied magnetic field Bo or the direction of propagation q of the incident radiation. This was attributed to an interference between electric-dipole and magnetic dipole interactions, the former being allowed by a spin-orbit interaction lacking inversion symmetry. We study here a similar phenomenon for conduction band electrons in semiconductors having the Wurtzite structure. 1.
Introduction
Recent measurements of magnetotransmission in n-type InSb, carried out in the parallel Voigt geometry (incident radiatiQn polarized parallel to a dc magnetic field Bo), revealed several interesting features. I ,2 The intensity of the electron-spin resonance was observed to be two orders of magnitude stronger than expEcted from magnetic-dipole transitions alone. This was attributed to a spin-orbit interaction which causes a mixing of electronic configurational and spin degrees of freedom, thereby allowing spin-flip transitions within the electric dipole approximation. The intensity of the spin resonance was also observed to show a strong angular dependence as the sample was rotated about the direction of propagation of the incident radiation. It was found thal reversal of either Bo or of the wave vector q of the incident radiation altered the spin resonance considerably. This result was understood from the microscopic point of view as an interference between the electric-dipole and, the less intense, magnetic-dipole interactions. A detailed theoretical study of this phenomenon and the selection rules for magneto-absorption in zinc-blende semiconductors were presented in Ref. 3 for various geometries. Excellent agreement with the experimental results was obtained, indicating the importance of inversion asymmetry in the study of spin resonance in semiconductors. It is natural to expect that this phenomenon should exist in other acentric semiconductors. In the present paper we develop a microscopic theory of spin resonance of conduction-band electrons in a wurtzite semiconductor. Awurtzite structure has the point symmetry C6V ' The interaction Hamiltonian due to spin-orbit coupling is of the form 4 ,5 0749-6036/85/060499+03 $02.0010
(1)
Here, c is a unit vector along the C6 axis and ~ is an asymmelry parameter characteristic of the crystal. 0 and p are, respectively, the spin and momentum operators. We notice that HI mixes eigenstates of components of angular momentum at right angles to c but conserves total (orbital and spin) angular momentum about that axis. Thus the perturbed states contain a mixture of up and down spins which in turn allow the electric dipole spin resonance transitions. The resulting transitions then provide a direct means of determining ~. This study was motivated by the experimental studies 6 ,7 of the electric dipole spin resonance in CdI_xMnxSe at far infrared frequencies. At low temperatures the spin resonance is mostly due to electrons bound to the donor states. However, as the temperature is increased, the conduction electron spin resonance dominates. 2. Microscopic Theory of Spin Resonance We describe an electron in the conduction band of a crystal with the wurtzite structure, within the framework of the effective mass approximation. We follow the approach described in Ref. 3. The Hamiltonian of the electrQn in the presence of a dc magnetic field Bo and including the spin orbit-coupling is given by H=Ho+H I 2 Fi 2 k =- + -1 gil +0
2m*
2
B
*tl
~ + + o + -2 C • (0 x K)
(2)
Here Fit = P + ~ 1. o witho B = v x 1.0 and c © 1985 Academic Press Inc. (London) Limited
Superlattices and Microstructures, Vol. 1, No.6, 1985
500
t
HI = e . (t x"k) describes the spin-orbit coupllng which is regarded as a small perturbation. We choose a co-ordinate system (~,n,t) whose t-axis is parallel to B o. The eigenvalues and eigenstates of Ho are described in detail in Ref. 3. We quote here some of the important results. The eigenstates of Ho are represented by In,kt,M,s) where tikI; and s = ±~ are the eigenva1ues of PI; and ~crt respectively. The integer n is the landau quantum number and Mis the eigenvalue of the I;-component of the angular momentum. The eigenvalue of Ho associated with In,kl;,M,s) is E (k) = fiIJl
ns I;
c
(n+~) +
fi2k 2 ( *) --f+ .9.2 !!!.... Fiwcs 2m m
• (3)
We note here that these landau energy levels are not affected by HI up to terms linear in The components of t can be expressed in terms of a+ = -i(n.c/2eBo)~(k~+ikT])
~.
(v'IHdv) Ev ' - Ev
o .
if v' t- v
(8)
=v
(9)
v'
The transformed Hamiltonian is K = e-
i5 He i5 = H + ~ A. t + i;fi o mc mc
eA + 2Cf[
A· [k,5]
c· (~7) cr x fI. + 2"1 gliB ~cr' *l) A
i gIlB[t,~]
•
B
(10)
t
We express Aand ~ in terms of = £E where E is the intensity of the electric component of the incident radiation and &is a unit vector parallel to the direction of polarization. That part of the interaction Hamiltonian which causes spin-flip transitions is given by
(5)
and its Herm~tian conjugate, a'khaving the properties a In,kt,M,s> = (n+l)2In+l,k ,M,s> and aln,kt,M,s> = n~ln-l, k~, M,s>. t In the presence of an external electromagnetic field 1 •
rt = - -c A, B = v x A)
(11)
where fi is a unit vector along the direction of propagation of the incident radiation, 2
characterized by the vector potential obtain,
Table 1. Intensity I (v'IHIlv) 1 of spin flip transitions described by ~s = ± 1, ~n = 0 in various geometries. The constant II m* used below is II =~m)'
A we
H = Ho + HI + H'
Geometry
2 2
n k + 2"1 gliB ~cr' * = -*l)
A
0
(~
+ '" c· cr A
7)
A
Parallel Voigt geometry tll~o' fille, Bo~c
x K .
Co
eli TfI. • +K + 2C1t Ae c· (+ + 1< cr x 7) fI. + 2"1 gliB +cr •
mc
(v'15Iv) = i
+
Here we is the cyclotron resonance frequency defineo by _ eBo (4) W - 1< c mc
2m
If we designate the eigenvectors of Ho , In,kl;,M,s), by Iv), Eq. (7) is equivalent to
* l)
(6)
We now eliminate HI to first order so that we can describe the interaction Hamiltonian as causing transitions between the unperturbed landau states. This is achieved by a unitary transformation e 15 where 5 is a time-dependent Hermitian operator. The operator 5 is selected so as to satisfy the condition (7)
tllBo ' fi ~ e, Bo~ e Perpendicular Voigt geometry EJ.Bo ' n~ e, Bollc Faraday geometry Bo"c"n, right circular polarization
f2[1 + A2] f2[1 - A]2
501
Superlattices and Microstructures, Vol. 1, No.6, 1985
E= ehg IC E
4mc
'
(12)
Acknowledgments--This work was supported by the National Science Foundation Grant No. DMR84-03325.
(13)
References
and, A=
4mc ti,2 wg lC
A
•
Here w is the angular frequency of the. incident radiation and c the dielectric constant of the crystal at that frequency. The matrix elements for spin-flip transitions calculated using Eq. (11) are shown in Table 1 for various geometries. We note that only in the parallel Voigt geometry and for n ~ C and ~o ~ c, the electric-dipole and the magnetic-dipole matrix elements are in phase. Hence reversing the magnetic field should change the intensity of the spin resonance line. Using an estimate of A = 6.05 x 10- 10 eV cm obtained from the results 6 for the donor bound electrons we get A = 8.51 for ti.w = 10.4 meV. The interference effect in this geometry can therefore be as much as 22X of the total intensity.
1. M. Dobrowolska, Y.-F. Chen, J.K. Furdyna, and S. Rodriguez, Phys. Rev. lett. 51, 134 (1983). 2. Y.-F. Chen. M. Dobrowolska. J.K. Furdyna. and S. Rodriguez. Phys. Rev. 832.890 (1985). 3. S. Gopalan. J.K. Furdyna. and ~ Rodriguez. Phys. Rev. 832, 903 (1985). 4. R. C. Casella~Phys. Rev. lett. i, 371 (1960). 5. R. Romeslain. S. Geschwind, and G.E. Devlin. Phys. Rev. lett. 39, 1583 (1977). 6. M. Dobrowolska. A:-Witowski, J.K. Furdyna. T. Ichiguchi. H.D. Drew, and P.A. Wolff. Phys. Rev. 829, 6652 (1984). 7. M. Dobrowolski, H.D. Drew, J.K. Furdyna. T. Ichiguchi. A. Witowski. and P.A. Wolff, Phys. Rev. lett. 49. 845 (1982).
499
ELECTRIC-DIPOLE AND MAGNETIC-DIPOLE INTERFERENCE IN THE SPIN RESONANCE OF CONDUCTION ELECTRONS IN WURTZITE SEMICONDUCTORS Sudha Gopalan and S. Rodriguez Department of Physics, Purdue University, West Lafayette, IN 47907 (Received 29 September 1985) Recent studies of magnetotransmission in n-type InSb revealed an unusually strong conduction electron spin resonance which changed upon reversal of either the applied magnetic field Bo or the direction of propagation q of the incident radiation. This was attributed to an interference between electric-dipole and magnetic dipole interactions, the former being allowed by a spin-orbit interaction lacking inversion symmetry. We study here a similar phenomenon for conduction band electrons in semiconductors having the Wurtzite structure. 1.
Introduction
Recent measurements of magnetotransmission in n-type InSb, carried out in the parallel Voigt geometry (incident radiatiQn polarized parallel to a dc magnetic field Bo), revealed several interesting features. I ,2 The intensity of the electron-spin resonance was observed to be two orders of magnitude stronger than expEcted from magnetic-dipole transitions alone. This was attributed to a spin-orbit interaction which causes a mixing of electronic configurational and spin degrees of freedom, thereby allowing spin-flip transitions within the electric dipole approximation. The intensity of the spin resonance was also observed to show a strong angular dependence as the sample was rotated about the direction of propagation of the incident radiation. It was found thal reversal of either Bo or of the wave vector q of the incident radiation altered the spin resonance considerably. This result was understood from the microscopic point of view as an interference between the electric-dipole and, the less intense, magnetic-dipole interactions. A detailed theoretical study of this phenomenon and the selection rules for magneto-absorption in zinc-blende semiconductors were presented in Ref. 3 for various geometries. Excellent agreement with the experimental results was obtained, indicating the importance of inversion asymmetry in the study of spin resonance in semiconductors. It is natural to expect that this phenomenon should exist in other acentric semiconductors. In the present paper we develop a microscopic theory of spin resonance of conduction-band electrons in a wurtzite semiconductor. Awurtzite structure has the point symmetry C6V ' The interaction Hamiltonian due to spin-orbit coupling is of the form 4 ,5 0749-6036/85/060499+03 $02.0010
(1)
Here, c is a unit vector along the C6 axis and ~ is an asymmelry parameter characteristic of the crystal. 0 and p are, respectively, the spin and momentum operators. We notice that HI mixes eigenstates of components of angular momentum at right angles to c but conserves total (orbital and spin) angular momentum about that axis. Thus the perturbed states contain a mixture of up and down spins which in turn allow the electric dipole spin resonance transitions. The resulting transitions then provide a direct means of determining ~. This study was motivated by the experimental studies 6 ,7 of the electric dipole spin resonance in CdI_xMnxSe at far infrared frequencies. At low temperatures the spin resonance is mostly due to electrons bound to the donor states. However, as the temperature is increased, the conduction electron spin resonance dominates. 2. Microscopic Theory of Spin Resonance We describe an electron in the conduction band of a crystal with the wurtzite structure, within the framework of the effective mass approximation. We follow the approach described in Ref. 3. The Hamiltonian of the electrQn in the presence of a dc magnetic field Bo and including the spin orbit-coupling is given by H=Ho+H I 2 Fi 2 k =- + -1 gil +0
2m*
2
B
*tl
~ + + o + -2 C • (0 x K)
(2)
Here Fit = P + ~ 1. o witho B = v x 1.0 and c © 1985 Academic Press Inc. (London) Limited
Superlattices and Microstructures, Vol. 1, No.6, 1985
500
t
HI = e . (t x"k) describes the spin-orbit coupllng which is regarded as a small perturbation. We choose a co-ordinate system (~,n,t) whose t-axis is parallel to B o. The eigenvalues and eigenstates of Ho are described in detail in Ref. 3. We quote here some of the important results. The eigenstates of Ho are represented by In,kt,M,s) where tikI; and s = ±~ are the eigenva1ues of PI; and ~crt respectively. The integer n is the landau quantum number and Mis the eigenvalue of the I;-component of the angular momentum. The eigenvalue of Ho associated with In,kl;,M,s) is E (k) = fiIJl
ns I;
c
(n+~) +
fi2k 2 ( *) --f+ .9.2 !!!.... Fiwcs 2m m
• (3)
We note here that these landau energy levels are not affected by HI up to terms linear in The components of t can be expressed in terms of a+ = -i(n.c/2eBo)~(k~+ikT])
~.
(v'IHdv) Ev ' - Ev
o .
if v' t- v
(8)
=v
(9)
v'
The transformed Hamiltonian is K = e-
i5 He i5 = H + ~ A. t + i;fi o mc mc
eA + 2Cf[
A· [k,5]
c· (~7) cr x fI. + 2"1 gliB ~cr' *l) A
i gIlB[t,~]
•
B
(10)
t
We express Aand ~ in terms of = £E where E is the intensity of the electric component of the incident radiation and &is a unit vector parallel to the direction of polarization. That part of the interaction Hamiltonian which causes spin-flip transitions is given by
(5)
and its Herm~tian conjugate, a'khaving the properties a In,kt,M,s> = (n+l)2In+l,k ,M,s> and aln,kt,M,s> = n~ln-l, k~, M,s>. t In the presence of an external electromagnetic field 1 •
rt = - -c A, B = v x A)
(11)
where fi is a unit vector along the direction of propagation of the incident radiation, 2
characterized by the vector potential obtain,
Table 1. Intensity I (v'IHIlv) 1 of spin flip transitions described by ~s = ± 1, ~n = 0 in various geometries. The constant II m* used below is II =~m)'
A we
H = Ho + HI + H'
Geometry
2 2
n k + 2"1 gliB ~cr' * = -*l)
A
0
(~
+ '" c· cr A
7)
A
Parallel Voigt geometry tll~o' fille, Bo~c
x K .
Co
eli TfI. • +K + 2C1t Ae c· (+ + 1< cr x 7) fI. + 2"1 gliB +cr •
mc
(v'15Iv) = i
+
Here we is the cyclotron resonance frequency defineo by _ eBo (4) W - 1< c mc
2m
If we designate the eigenvectors of Ho , In,kl;,M,s), by Iv), Eq. (7) is equivalent to
* l)
(6)
We now eliminate HI to first order so that we can describe the interaction Hamiltonian as causing transitions between the unperturbed landau states. This is achieved by a unitary transformation e 15 where 5 is a time-dependent Hermitian operator. The operator 5 is selected so as to satisfy the condition (7)
tllBo ' fi ~ e, Bo~ e Perpendicular Voigt geometry EJ.Bo ' n~ e, Bollc Faraday geometry Bo"c"n, right circular polarization
f2[1 + A2] f2[1 - A]2
501
Superlattices and Microstructures, Vol. 1, No.6, 1985
E= ehg IC E
4mc
'
(12)
Acknowledgments--This work was supported by the National Science Foundation Grant No. DMR84-03325.
(13)
References
and, A=
4mc ti,2 wg lC
A
•
Here w is the angular frequency of the. incident radiation and c the dielectric constant of the crystal at that frequency. The matrix elements for spin-flip transitions calculated using Eq. (11) are shown in Table 1 for various geometries. We note that only in the parallel Voigt geometry and for n ~ C and ~o ~ c, the electric-dipole and the magnetic-dipole matrix elements are in phase. Hence reversing the magnetic field should change the intensity of the spin resonance line. Using an estimate of A = 6.05 x 10- 10 eV cm obtained from the results 6 for the donor bound electrons we get A = 8.51 for ti.w = 10.4 meV. The interference effect in this geometry can therefore be as much as 22X of the total intensity.
1. M. Dobrowolska, Y.-F. Chen, J.K. Furdyna, and S. Rodriguez, Phys. Rev. lett. 51, 134 (1983). 2. Y.-F. Chen. M. Dobrowolska. J.K. Furdyna. and S. Rodriguez. Phys. Rev. 832.890 (1985). 3. S. Gopalan. J.K. Furdyna. and ~ Rodriguez. Phys. Rev. 832, 903 (1985). 4. R. C. Casella~Phys. Rev. lett. i, 371 (1960). 5. R. Romeslain. S. Geschwind, and G.E. Devlin. Phys. Rev. lett. 39, 1583 (1977). 6. M. Dobrowolska. A:-Witowski, J.K. Furdyna. T. Ichiguchi. H.D. Drew, and P.A. Wolff. Phys. Rev. 829, 6652 (1984). 7. M. Dobrowolski, H.D. Drew, J.K. Furdyna. T. Ichiguchi. A. Witowski. and P.A. Wolff, Phys. Rev. lett. 49. 845 (1982).