- Email: [email protected]
Relaxation of the electron spin in quantum dots via one- and two-phonon processes
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 316 (2007) e937–e939 www.elsevier.com/locate/jmmm
Relaxation of the electron spin in quantum dots via one- and two-phonon processes C. Calero, E.M. Chudnovsky, D.A. Garanin Department of Physics and Astronomy, Lehman College, City University of New York, 250 Bedford Park Boulevard West, Bronx, NY 10468-1589, USA Available online 23 March 2007
Abstract We have studied direct and Raman processes of the decay of electron spin states in a quantum dot via radiation of phonons corresponding to elastic twists. Universal dependence of the spin relaxation rate on the strength and direction of the magnetic field has been obtained in terms of the electron gyromagnetic tensor and macroscopic elastic constants of the solid. Published by Elsevier B.V. PACS: 72.25.Rb; 73.21.La Keywords: Quantum dot; Spin–phonon relaxation; Gyromagnetic tensor
Electron spin relaxation in solids is related to such important applied problems as electron spin resonance and quantum computing. Many interactions contribute to the relaxation time of the electron spin in a semiconductor quantum dot. In principle, all of them can be eliminated (such as impurities, nuclear spins, etc.) but the interaction with phonons cannot. Thus, spin–phonon interaction provides the most fundamental upper bound on the lifetime of electron spin states. The existing methods of computing electron spin–phonon rates in semiconductors rely upon phenomenological models of spin–orbit interaction, see, e.g., Refs. [1–10]. These models contain unknown constants that must be obtained from experiment. Meantime, the spin–orbit coupling determines the difference of the electron g-tensor from the unit tensor. The question then arises whether the effect of the spin–orbit coupling on spin–phonon relaxation can be expressed via the difference between the electron gyromagnetic tensor gab and the vacuum tensor g0 dab . Since gab can be measured independently, this would enable one to compare the computed relaxation rates with experiment without any fitting parameters. In this paper we show that this can be done under certain reasonable simplifying assumptions. We Corresponding author.
E-mail address: [email protected] (C. Calero). 0304-8853/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.jmmm.2007.03.174
obtain spin–phonon relaxation rates due to direct onephonon processes and due to two-phonon Raman processes. Zeeman interaction of the electron with an external magnetic field, H, is given by the Hamiltonian H^ Z ¼ mB gab Sa H b ,
(1)
where mB is the Bohr magneton and S ¼ s=2 is the dimensionless electron spin, with sa being Pauli matrices. One can choose the axes of the coordinate system along the principal axes of the tensor gab . Then, gab ¼ ga dab . The perturbation of Eq. (1) by phonons has been studied in the past [3,4,6] by writing all terms of the expansion of gab on the strain tensor, uab , permitted by symmetry. This gives spin–phonon interaction of the form Aabgr uab S g H r with unknown coefficients Aabgr . To avoid this uncertainty we limit our consideration to local rotations generated by transverse phonons. The argument for doing this is threefold. Firstly, the rate of the transition accompanied by the emission or absorption of a phonon is inversely proportional to the fifth power of the sound velocity [11]. Since the velocity of the transverse sound is always smaller than the velocity of the longitudinal sound [12], the transverse phonons must dominate the transitions. Secondly, we notice that interaction of the electron spin with a local elastic twist generated by a transverse phonon does not
ARTICLE IN PRESS e938
C. Calero et al. / Journal of Magnetism and Magnetic Materials 316 (2007) e937–e939
contain any unknown constants. Consequently, it gives parameter-free lower bound on the spin relaxation. Thirdly, for a dot that is sufficiently rigid to permit only tiny local rotations as a whole under an arbitrary elastic deformation, the emission or absorption of a quantum of the elastic twist is the only spin–phonon relaxation channel. The angle of the local rotation of the crystal lattice in the presence of the deformation, uðrÞ, is given by [12] df ¼ 12 r u.
(2)
As stated above, the gyromagnetic tensor gab is determined by the local environment of the quantum dot. In the presence of long wave deformations of the lattice, the whole environment is rotated so that the gyromagnetic tensor becomes gab ¼ Raa0 Rbb0 ga0 b0 ,
(3)
where Rab is the 3 3 rotation matrix given by dfðrÞ. One can thus write the total Hamiltonian in the form ^ ¼H ^ 0 þH ^ ph , H Z
(4)
where (5)
^ ph describes free harmonic phonons. In the above and H formulae, df must be understood as an operator. Indeed, from the canonical quantization of phonons and Eq. (2) one obtains sffiffiffiffiffiffiffiffiffi _ X ½ik ekl eikr df ¼ (6) ðakl þ aykl Þ, pffiffiffiffiffiffiffiffi 8rV kl okl where r is the mass density, V is the volume of the crystal, ekl are unit polarization vectors, l ¼ t1 ; t2 ; l denotes polarization, and okl ¼ vl k is the phonon frequency. 1. Direct processes In order to account for spin transitions accompanied by the emission or absorption of one phonon one needs to consider terms up to first order in phonon amplitudes. Therefore, with the help of the expansion of the rotation matrix to the first order in df, Rab ¼ dab abg dfg , we obtain the full Hamiltonian: ^ 0¼H ^ Z þH ^ ph , H
(7)
where ^ s2ph ¼ mB abg ðga gb ÞH b dfg Sa . H
(8)
Spin–phonon transitions occur between the eigenstates of ^ 0 , which are direct products of spin and phonon states H jC i ¼ jc i jf i.
To obtain the relaxation rate one can use the Fermi golden rule. The matrix element corresponding to the decay of the spin jCþ i ! jC i can be evaluated with the help of Eqs. (8) and (6), we obtain X ^ s2ph jCþ i ¼ p_ffiffiffiffi V kl hnk0 þ 1jakl þ aykl jnk0 i, hC jH V kl eikr V kl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ½k ekl , 8r_okl
(9)
^ Z with energies E and Here jc i are the eigenstates of H ^ ph with energies E ph . jf i are the eigenstates of H Spin–phonon processes conserve energy, E þ þ E phþ ¼
ð11Þ
where the components of vector K are given by K g mB abg ðga gb ÞH b hc jS a jcþ i.
(12)
Note that only the transverse phonons are considered in the summations. For the direct process, the decay rate, GD , is, then _o0 GD ¼ G0 coth , (13) 2kB T where
1 ^ 0 ¼ mB ga0 b0 ðR1 H Z a0 a S a ÞðRb0 b H b Þ,
^ s2ph ; ^ ¼H ^ 0þH H
^ sph of Eq. (7), the states jf i differ E þ E ph . For H by one emitted or absorbed phonon with wave vector k. We will use the following designations: jfþ i jnk i; jf i jnk þ 1i. (10)
_o0 E þ E ¼ mB
X
!1=2 g2g H 2g
(14)
g
is the distance between the two spin levels, and 1X G0 ¼ jV kl j2 2pdðokl o0 Þ. (15) V kl R P Using Eq. (11) and replacing k by ðV =ð2pÞ3 Þ d3 k, one obtains G0 ¼
1 jKj2 o30 , 12p_ rv5t
(16)
where vt is the velocity of the transverse sound. A straightforward calculation of the spin matrix elements yields m2 X jKj2 ¼ B ðg gb Þ2 8 ab¼x;y;z a " # ðga þ gb Þ2 H 2a H 2b 2 2 Ha þ Hb P . ð17Þ 2 g ðgg H g Þ Then, the decay rate can be written in the final form _ mB H 5 F T ðnÞ, (18) GD ¼ 3pr _vt where n H=H and 2 !3=2 !1=2 3 X X m H 5 g2g n2g coth4 B g2g n2g F T ðnÞ ¼ 2k T B g g " # ðga þ gb Þ2 n2a n2b 1 X 2 2 2 ðg gb Þ na þ nb P . 2 32 ab a g ðgg ng Þ ð19Þ
ARTICLE IN PRESS C. Calero et al. / Journal of Magnetism and Magnetic Materials 316 (2007) e937–e939
If the field is directed along the z-axis, Eq. (19) simplifies to g3 gm H F T ðez Þ ¼ z ½ðgz gx Þ2 þ ðgz gy Þ2 coth z B . (20) 2kB T 32 When components of gab are of order unity and kB T\mB H, then GD ðkB T=_ÞðmB H=E t Þ4 , where we have introduced E t ð_3 rv5t Þ1=4 102 K. 2. Raman processes Spin–lattice relaxation by Raman scattering is a twophonon mechanism consisting of a spin transition accompanied by the absorption of a phonon and the emission of another phonon of different frequency. In spite of being a second-order process, its contribution can be very important, since the phase space of the phonons triggering the transition is not limited to the distance between the levels as in the direct case. To describe such processes, we need to consider terms up to second order in phonon amplitudes in the Hamiltonian. To this end we expand the 3 3 rotation matrix to second order in df Rab ¼ dab abg dfg þ 12½dfa dfb dab ðdfÞ2 ,
(21)
^ s2ph ¼ H ^ ð1Þ þ H ^ ð2Þ , H s2ph s2ph
(24)
To obtain the relaxation rate of the transition jCþ i ! jC i one needs to evaluate the matrix element of the ^ ð2Þ process, which is the sum of the matrix element with H s2ph ð1Þ ^ and that with H s2ph in the second order: M R ¼ ð1þ1Þ M ð2Þ , where R þ MR ^ ð2Þ M ð2Þ R ¼ hC jHs2ph jCþ i,
(25)
and ð1Þ
M ð1þ1Þ ¼ R
ð1Þ
^ ^ X hC jH s2ph jCx ihCx jHs2ph jCþ i x¼
E þ þ _okE x ð1Þ
þ
ð1Þ
^ ^ X hC jH s2ph jCx ihCx jHs2ph jCþ i x¼
E þ E x _oq
.
~ ab ¼ M ab þ M ba where M ph2R ph2R ph2R with M ab ph2R ¼ hnq þ 1jdfa jnq ihnk 1jdfb jnk i ¼
_2 ½k eklk a ½q eqlq b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnq þ 1Þnk . 8rV _oklk _oqlq
ð28Þ
The Raman rate of the transition jCþ i ! jC i can be obtained by using the Fermi golden rule
ð23Þ
Again, we will study the spin–phonon transitions between ^ 0 , jC i ¼ jc i jf i. Here, the eigenstates of H ^ Z with energies E again,jc i are the eigenstates of H ^ ph with energies E ph .We and jf i the eigenstates of H consider Raman processes, in which a phonon with wave vector k is absorbed and a phonon with a wave vector q is emitted. We will use the following designations for the phonon states: jf i jnk 1; nq þ 1i.
ð27Þ
The ratio of Raman and direct rates is of order GR =GD 102 ðkB T=mB HÞ2 ðkB T=E t Þ4 . Consequently, Raman processes dominate over direct processes at high temperature and low field.
^ ð2Þ ¼ 1 mB ðga þ gb ÞS a H b dfa dfb H s2ph 2
jfþ i jnk ; nq i;
mB H ~ xz ½ð2gy gx gz ÞM ph2R 4 yz ~ ið2gx gy gz ÞM ph2R ,
M R ¼ M ð2Þ R ¼
(22)
^ ð1Þ is given by Eq. (8) and where H s2ph
mB ga ðS dfÞa ðH dfÞa .
The intermediate phonon states are jnk 1; nq i in the first term and jnk ; nq þ 1i in the second term. Raman processes may dominate over direct processes for a small energy difference between the spin states, _o0 5kB T. Thus, we will consider terms to the lowest order in H. For simplicity we will study the case where the field is directed along the z-axis. Then the matrix element becomes
GR ¼ ½ð2gy gx gz Þ2 þ ð2gx gy gz Þ2 p3 kB T mB H 2 kB T 6 . Et Et 3024 _
and obtain the full Hamiltonian from Eqs. (4) and (5) ^ ¼H ^ 0þH ^ s2ph ; H
e939
ð26Þ
ð29Þ
Acknowledgment This work has been supported by the NSF Grant no. EIA-0310517.
References [1] G. Dresselhaus, Phys. Rev. 100 (1955) 580. [2] E.I. Rashba, Sov. Phys.—Solid State 2 (1960) 1109; Yu.A. Bychkov, E.I. Rashba, JETP Lett. 39 (1984) 78. [3] H. Hasegawa, Phys. Rev. 118 (1960) 1523. [4] L. Roth, Phys. Rev. 118 (1960) 1534. [5] A.V. Khaetskii, Yu.V. Nazarov, Phys. Rev. 61 (2000) 12639; A.V. Khaetskii, Yu.V. Nazarov, Phys. Rev. 64 (2001) 125316. [6] B.A. Glavin, K.W. Kim, Phys. Rev. B68 (2003) 045308. [7] E.I. Rashba, Phys. Rev. B68 (2003) 241315. [8] E.I. Rashba, A.I. Efros, Phys. Rev. Lett. 91 (2003) 126405. [9] V.N. Golovach, A. Khaetskii, D. Loss, Phys. Rev. Lett. 93 (2004) 016601. [10] C. Tahan, R. Joynt, Phys. Rev. 71 (2005) 075315. [11] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Clarendon Press, Oxford, 1970. [12] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Pergamon, New York, 1970.
Journal of Magnetism and Magnetic Materials 316 (2007) e937–e939 www.elsevier.com/locate/jmmm
Relaxation of the electron spin in quantum dots via one- and two-phonon processes C. Calero, E.M. Chudnovsky, D.A. Garanin Department of Physics and Astronomy, Lehman College, City University of New York, 250 Bedford Park Boulevard West, Bronx, NY 10468-1589, USA Available online 23 March 2007
Abstract We have studied direct and Raman processes of the decay of electron spin states in a quantum dot via radiation of phonons corresponding to elastic twists. Universal dependence of the spin relaxation rate on the strength and direction of the magnetic field has been obtained in terms of the electron gyromagnetic tensor and macroscopic elastic constants of the solid. Published by Elsevier B.V. PACS: 72.25.Rb; 73.21.La Keywords: Quantum dot; Spin–phonon relaxation; Gyromagnetic tensor
Electron spin relaxation in solids is related to such important applied problems as electron spin resonance and quantum computing. Many interactions contribute to the relaxation time of the electron spin in a semiconductor quantum dot. In principle, all of them can be eliminated (such as impurities, nuclear spins, etc.) but the interaction with phonons cannot. Thus, spin–phonon interaction provides the most fundamental upper bound on the lifetime of electron spin states. The existing methods of computing electron spin–phonon rates in semiconductors rely upon phenomenological models of spin–orbit interaction, see, e.g., Refs. [1–10]. These models contain unknown constants that must be obtained from experiment. Meantime, the spin–orbit coupling determines the difference of the electron g-tensor from the unit tensor. The question then arises whether the effect of the spin–orbit coupling on spin–phonon relaxation can be expressed via the difference between the electron gyromagnetic tensor gab and the vacuum tensor g0 dab . Since gab can be measured independently, this would enable one to compare the computed relaxation rates with experiment without any fitting parameters. In this paper we show that this can be done under certain reasonable simplifying assumptions. We Corresponding author.
E-mail address: [email protected] (C. Calero). 0304-8853/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.jmmm.2007.03.174
obtain spin–phonon relaxation rates due to direct onephonon processes and due to two-phonon Raman processes. Zeeman interaction of the electron with an external magnetic field, H, is given by the Hamiltonian H^ Z ¼ mB gab Sa H b ,
(1)
where mB is the Bohr magneton and S ¼ s=2 is the dimensionless electron spin, with sa being Pauli matrices. One can choose the axes of the coordinate system along the principal axes of the tensor gab . Then, gab ¼ ga dab . The perturbation of Eq. (1) by phonons has been studied in the past [3,4,6] by writing all terms of the expansion of gab on the strain tensor, uab , permitted by symmetry. This gives spin–phonon interaction of the form Aabgr uab S g H r with unknown coefficients Aabgr . To avoid this uncertainty we limit our consideration to local rotations generated by transverse phonons. The argument for doing this is threefold. Firstly, the rate of the transition accompanied by the emission or absorption of a phonon is inversely proportional to the fifth power of the sound velocity [11]. Since the velocity of the transverse sound is always smaller than the velocity of the longitudinal sound [12], the transverse phonons must dominate the transitions. Secondly, we notice that interaction of the electron spin with a local elastic twist generated by a transverse phonon does not
ARTICLE IN PRESS e938
C. Calero et al. / Journal of Magnetism and Magnetic Materials 316 (2007) e937–e939
contain any unknown constants. Consequently, it gives parameter-free lower bound on the spin relaxation. Thirdly, for a dot that is sufficiently rigid to permit only tiny local rotations as a whole under an arbitrary elastic deformation, the emission or absorption of a quantum of the elastic twist is the only spin–phonon relaxation channel. The angle of the local rotation of the crystal lattice in the presence of the deformation, uðrÞ, is given by [12] df ¼ 12 r u.
(2)
As stated above, the gyromagnetic tensor gab is determined by the local environment of the quantum dot. In the presence of long wave deformations of the lattice, the whole environment is rotated so that the gyromagnetic tensor becomes gab ¼ Raa0 Rbb0 ga0 b0 ,
(3)
where Rab is the 3 3 rotation matrix given by dfðrÞ. One can thus write the total Hamiltonian in the form ^ ¼H ^ 0 þH ^ ph , H Z
(4)
where (5)
^ ph describes free harmonic phonons. In the above and H formulae, df must be understood as an operator. Indeed, from the canonical quantization of phonons and Eq. (2) one obtains sffiffiffiffiffiffiffiffiffi _ X ½ik ekl eikr df ¼ (6) ðakl þ aykl Þ, pffiffiffiffiffiffiffiffi 8rV kl okl where r is the mass density, V is the volume of the crystal, ekl are unit polarization vectors, l ¼ t1 ; t2 ; l denotes polarization, and okl ¼ vl k is the phonon frequency. 1. Direct processes In order to account for spin transitions accompanied by the emission or absorption of one phonon one needs to consider terms up to first order in phonon amplitudes. Therefore, with the help of the expansion of the rotation matrix to the first order in df, Rab ¼ dab abg dfg , we obtain the full Hamiltonian: ^ 0¼H ^ Z þH ^ ph , H
(7)
where ^ s2ph ¼ mB abg ðga gb ÞH b dfg Sa . H
(8)
Spin–phonon transitions occur between the eigenstates of ^ 0 , which are direct products of spin and phonon states H jC i ¼ jc i jf i.
To obtain the relaxation rate one can use the Fermi golden rule. The matrix element corresponding to the decay of the spin jCþ i ! jC i can be evaluated with the help of Eqs. (8) and (6), we obtain X ^ s2ph jCþ i ¼ p_ffiffiffiffi V kl hnk0 þ 1jakl þ aykl jnk0 i, hC jH V kl eikr V kl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ½k ekl , 8r_okl
(9)
^ Z with energies E and Here jc i are the eigenstates of H ^ ph with energies E ph . jf i are the eigenstates of H Spin–phonon processes conserve energy, E þ þ E phþ ¼
ð11Þ
where the components of vector K are given by K g mB abg ðga gb ÞH b hc jS a jcþ i.
(12)
Note that only the transverse phonons are considered in the summations. For the direct process, the decay rate, GD , is, then _o0 GD ¼ G0 coth , (13) 2kB T where
1 ^ 0 ¼ mB ga0 b0 ðR1 H Z a0 a S a ÞðRb0 b H b Þ,
^ s2ph ; ^ ¼H ^ 0þH H
^ sph of Eq. (7), the states jf i differ E þ E ph . For H by one emitted or absorbed phonon with wave vector k. We will use the following designations: jfþ i jnk i; jf i jnk þ 1i. (10)
_o0 E þ E ¼ mB
X
!1=2 g2g H 2g
(14)
g
is the distance between the two spin levels, and 1X G0 ¼ jV kl j2 2pdðokl o0 Þ. (15) V kl R P Using Eq. (11) and replacing k by ðV =ð2pÞ3 Þ d3 k, one obtains G0 ¼
1 jKj2 o30 , 12p_ rv5t
(16)
where vt is the velocity of the transverse sound. A straightforward calculation of the spin matrix elements yields m2 X jKj2 ¼ B ðg gb Þ2 8 ab¼x;y;z a " # ðga þ gb Þ2 H 2a H 2b 2 2 Ha þ Hb P . ð17Þ 2 g ðgg H g Þ Then, the decay rate can be written in the final form _ mB H 5 F T ðnÞ, (18) GD ¼ 3pr _vt where n H=H and 2 !3=2 !1=2 3 X X m H 5 g2g n2g coth4 B g2g n2g F T ðnÞ ¼ 2k T B g g " # ðga þ gb Þ2 n2a n2b 1 X 2 2 2 ðg gb Þ na þ nb P . 2 32 ab a g ðgg ng Þ ð19Þ
ARTICLE IN PRESS C. Calero et al. / Journal of Magnetism and Magnetic Materials 316 (2007) e937–e939
If the field is directed along the z-axis, Eq. (19) simplifies to g3 gm H F T ðez Þ ¼ z ½ðgz gx Þ2 þ ðgz gy Þ2 coth z B . (20) 2kB T 32 When components of gab are of order unity and kB T\mB H, then GD ðkB T=_ÞðmB H=E t Þ4 , where we have introduced E t ð_3 rv5t Þ1=4 102 K. 2. Raman processes Spin–lattice relaxation by Raman scattering is a twophonon mechanism consisting of a spin transition accompanied by the absorption of a phonon and the emission of another phonon of different frequency. In spite of being a second-order process, its contribution can be very important, since the phase space of the phonons triggering the transition is not limited to the distance between the levels as in the direct case. To describe such processes, we need to consider terms up to second order in phonon amplitudes in the Hamiltonian. To this end we expand the 3 3 rotation matrix to second order in df Rab ¼ dab abg dfg þ 12½dfa dfb dab ðdfÞ2 ,
(21)
^ s2ph ¼ H ^ ð1Þ þ H ^ ð2Þ , H s2ph s2ph
(24)
To obtain the relaxation rate of the transition jCþ i ! jC i one needs to evaluate the matrix element of the ^ ð2Þ process, which is the sum of the matrix element with H s2ph ð1Þ ^ and that with H s2ph in the second order: M R ¼ ð1þ1Þ M ð2Þ , where R þ MR ^ ð2Þ M ð2Þ R ¼ hC jHs2ph jCþ i,
(25)
and ð1Þ
M ð1þ1Þ ¼ R
ð1Þ
^ ^ X hC jH s2ph jCx ihCx jHs2ph jCþ i x¼
E þ þ _okE x ð1Þ
þ
ð1Þ
^ ^ X hC jH s2ph jCx ihCx jHs2ph jCþ i x¼
E þ E x _oq
.
~ ab ¼ M ab þ M ba where M ph2R ph2R ph2R with M ab ph2R ¼ hnq þ 1jdfa jnq ihnk 1jdfb jnk i ¼
_2 ½k eklk a ½q eqlq b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnq þ 1Þnk . 8rV _oklk _oqlq
ð28Þ
The Raman rate of the transition jCþ i ! jC i can be obtained by using the Fermi golden rule
ð23Þ
Again, we will study the spin–phonon transitions between ^ 0 , jC i ¼ jc i jf i. Here, the eigenstates of H ^ Z with energies E again,jc i are the eigenstates of H ^ ph with energies E ph .We and jf i the eigenstates of H consider Raman processes, in which a phonon with wave vector k is absorbed and a phonon with a wave vector q is emitted. We will use the following designations for the phonon states: jf i jnk 1; nq þ 1i.
ð27Þ
The ratio of Raman and direct rates is of order GR =GD 102 ðkB T=mB HÞ2 ðkB T=E t Þ4 . Consequently, Raman processes dominate over direct processes at high temperature and low field.
^ ð2Þ ¼ 1 mB ðga þ gb ÞS a H b dfa dfb H s2ph 2
jfþ i jnk ; nq i;
mB H ~ xz ½ð2gy gx gz ÞM ph2R 4 yz ~ ið2gx gy gz ÞM ph2R ,
M R ¼ M ð2Þ R ¼
(22)
^ ð1Þ is given by Eq. (8) and where H s2ph
mB ga ðS dfÞa ðH dfÞa .
The intermediate phonon states are jnk 1; nq i in the first term and jnk ; nq þ 1i in the second term. Raman processes may dominate over direct processes for a small energy difference between the spin states, _o0 5kB T. Thus, we will consider terms to the lowest order in H. For simplicity we will study the case where the field is directed along the z-axis. Then the matrix element becomes
GR ¼ ½ð2gy gx gz Þ2 þ ð2gx gy gz Þ2 p3 kB T mB H 2 kB T 6 . Et Et 3024 _
and obtain the full Hamiltonian from Eqs. (4) and (5) ^ ¼H ^ 0þH ^ s2ph ; H
e939
ð26Þ
ð29Þ
Acknowledgment This work has been supported by the NSF Grant no. EIA-0310517.
References [1] G. Dresselhaus, Phys. Rev. 100 (1955) 580. [2] E.I. Rashba, Sov. Phys.—Solid State 2 (1960) 1109; Yu.A. Bychkov, E.I. Rashba, JETP Lett. 39 (1984) 78. [3] H. Hasegawa, Phys. Rev. 118 (1960) 1523. [4] L. Roth, Phys. Rev. 118 (1960) 1534. [5] A.V. Khaetskii, Yu.V. Nazarov, Phys. Rev. 61 (2000) 12639; A.V. Khaetskii, Yu.V. Nazarov, Phys. Rev. 64 (2001) 125316. [6] B.A. Glavin, K.W. Kim, Phys. Rev. B68 (2003) 045308. [7] E.I. Rashba, Phys. Rev. B68 (2003) 241315. [8] E.I. Rashba, A.I. Efros, Phys. Rev. Lett. 91 (2003) 126405. [9] V.N. Golovach, A. Khaetskii, D. Loss, Phys. Rev. Lett. 93 (2004) 016601. [10] C. Tahan, R. Joynt, Phys. Rev. 71 (2005) 075315. [11] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Clarendon Press, Oxford, 1970. [12] L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Pergamon, New York, 1970.