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Dichroism of impurity crystals without inversion center induced by energy term linear in wavevector
0038-1098/83/060515-02503.00/0 Pergamon Press Ltd.
Solid State Communications, Vol. 45, No. 6, pp. 515-516, 1983. Printed in Great Britain.
DICHROISM OF IMPURITY CRYSTALS WITHOUT INVERSION CENTER INDUCED BY ENERGY TERM LINEAR IN WAVEVECTOR V.M. Edelstein Academy of Sciences of the USSR, Institute of Solid State Physics, Chernogolovka 142432, U.S.S.R.
(Received 4 October 1982 by V.M. Agranovich) The effect of a parity violation in hexagonal crystals without inversion center has been investigated. The transition ls-2s of an electron bound to a donor is shown to be dipole allowed. In a weak magnetic field the spinflip transitions take place only at a definite sign of a photon circulation. Their intensities is almost independent of the value of the magnetic field.
IN WURTZITE STRUCTURES such as CdS the crystal symmetry group for lack of an inversion center allow energy term which is linear in wavevector [ 1]
Vp = ~ ( p x s)c
(1)
where s is the spin-operator and e is the unit vector parallel to the hexagonal axis. This term in the Hamiltonian of the conduction electron originates from the spin-orbit coupling of the electron with the electrostatic field E [2] s(p x E) + h.c.
hcoe, which, in turn, exceeds the characteristic energy of the fine structure ( 2 s - 2 p splitting). Further on the z-axis of the right coordinate system is assumed to be directed along the crystal axis e. The hydrogen-like Hamiltonian for the unperturbed donor has the form Ho -
(2)
In CdS the conduction-band states are formed from the wavefunctions of the valent shell of Cd-atoms and due to hexagonal perturbation of tetrahedron configuration the resulting field of S2--ions surrounding Cd2+-ions has the non-zero value on the sites of Cd-ions. Using the conventional k - p method equation (1) may be deduced from equation (2). On the ground of the Hamiltonian (1) there has been plenty of past calculations of pure spin transitions of the donor electrons in a magnetic field-combined resonance [3]. At a weak magnetic field H the matrix elements of such transitions are very small, as being proportional to H. The primary significance of the Hamiltonian (1) consists in alternating its sign by the coordinate inversion. For this reason it mixes the states with the opposite parity and equal e-components of the total angular momentum. Hence, interaction (1) admixes s- and p-like states of the electron bound to the donor and makes possible the dipole spin-flip transition ls-2s. In this note we present the calculations of the intensity of these transitions in a weak magnetic field H oriented along e-axis so that the energy of donor ionization Ry -me4/2h2e2o considerably exceeds the cyclotron energy 515
p2
e2
2m
eor
+
lelh
2mc
lelh 1I-I+ - - g
2moc
H.
(3)
Here the irrelevant diamagnetic term is neglected. The perturbation is the sum of the dipole interaction with the light, proportional to =r, and the parity violating Hamiltonian Vp = ~-~
--
,
e,
A = 1Hxr.
(4)
The term (e/c)A in equation (4) will be always omitted excluding the cases when the matrix element of the momentum p equals zero. The amplitude of the ls~-2st transition according to the second order perturbation theory is given by the expression M
=
Ma+M b
Ma = ~, (2stl=r[nt)(nflVplls~)
(2s1'[ VI,[n+)(n $1= rl ls+)
Mb = En
E~s -- E~
(5)
Let us first consider Ma. The energy denominators in this sum are all of the order of Ry, hence the effect of the magnetic field may be neglected. The coordinate and spin parts of the eigenfunctions of rio are factorized, therefore it holds the relationship
im
(ntl (p x ~)ells~,) -
Els - - E n
h
(ntl (r x a ) e l l s $ )
(6)
516
DICHROISM OF IMPURITY CRYSTALS
from which we find that
Xm ( 2 s l x - - i y l 2 p ,
2h: imX (2st[ (=r)[(r x , ) c ] Ils$).
(7)
2h z
Ma -
Some caution must be taken in evaluating the sum representingMb, where there are small denominators corresponding to the intermediate 2p-states. In the reduced sum M~, where the 2p-states are excluded, one may neglect the influence of the magnetic field and use relationship (6) as well. Then,
Vol. 45, No. 6 lz = 1)(2p, lz = ll=rlls)
(13)
2g(m/mo)- 1
After standard calculations the sum of equations (9) and (13) looks as follows (a-~y~ M = --0.57~a
~" = 1 +
2
e°h2 ex--iey),
-1.
m
gmo
a -
me 2 ,
1
(14)
It may be shown that the matrix element of the M~ -
imX S" 2h2n~p
(8)
(2stl(r x o ) clh$)(n$ trlls$).
Adding and subtracting from equation (8) the contribution of the 2p-states, we obtain imX
M; -
2h 2 (2stl[(r x o ) cl(=r)lls$) imX
2h 2 2p
(9)
(2sTl(r x o ) e l 2p,l,)(2p$[ t r l l s $ ) .
The first term cancels the matrix element Ma, in the second one due to the equation (tp(r x o ) e l $ ) = - i
(It °
x i l)o
--X --iy
(10)
= -- i ( x -- iy)
only 2p-state with lz = 1 gives a nonzero result. The last point to consider is the calculation of the contribution of 2p-states to M b. Since in our approximation, neglecting the fine structure energy, (2st p[ 2p) = 0, only the vector-potential term in Hamiltonian (4) V~ -
Xle[H
--.±r± 4hc
the opposite sign and polarization dependence ex + icy. Thus, both these transitions occur at opposite signs of the circular polarization of the photon propagating along the e-axis. Since in the initially dipole allowed l s - 2 p transition the electron spin conserves, the energy of the absorbed photon because of the Zeeman shift differs from that of the l s - 2 s spin-flip transition. As yet no information is available on the experiments, explicitly determining the quantity X for conduction bands of any semiconductors. In the case of CdS X was indirectly evaluated [5] by studying the spectral width of the spin-flip Raman scattering of light by degenerate electrons, though the lineshape was analyzed using the simplest diffusive approximation. Observations on the transitions described above and measuring their intensity would allow one to ultimately determine the value of X. If, according to [5], one supposes that for CdS X _~ 1.6 × 10- ]o eV cm- 1 and adopts m/mo = 0.2, g = 1.8, then the intensity ratio Ils_2s/Ias_2p will be of the order of 3 × 10 -3 . Acknowledgements - The author is grateful to Professor
E.I. Rashba for helpful discussions. REFERENCES
(11) 1.
results in the 2 s - 2 p mixing. From the equation (l"[a±r±l$) =
l s t - 2 s $ transition differs from equation (14) only by
(i 0 Xoi t ) 1"
$
= x--iy
2. (12)
x+iy
it is seen that the nonzero contribution also results from the p-state with Iz = 1 only and is equal to
3. 4. 5.
E.I. Rashba, Fiz. Tverd. Tela 1,407 (1959); R.C. Gasella, Phys. Rev. Lett. 5,371 (1960). Silvan S. Schweber, A n Introduction to Relativistic Quantum Field Theory, Row, Peterson & Co., New York (1961). E.I. Rashba &V.I. Sheka, F T T 6, 141 (1964). E.I. Rashba, Usp. Fiz. Nauk 84, 557 (1964). R. Romestain, S. Geschwind & G.E. Devlin, Phys. Rev. Lett. 39, 1583 (1977).
Solid State Communications, Vol. 45, No. 6, pp. 515-516, 1983. Printed in Great Britain.
DICHROISM OF IMPURITY CRYSTALS WITHOUT INVERSION CENTER INDUCED BY ENERGY TERM LINEAR IN WAVEVECTOR V.M. Edelstein Academy of Sciences of the USSR, Institute of Solid State Physics, Chernogolovka 142432, U.S.S.R.
(Received 4 October 1982 by V.M. Agranovich) The effect of a parity violation in hexagonal crystals without inversion center has been investigated. The transition ls-2s of an electron bound to a donor is shown to be dipole allowed. In a weak magnetic field the spinflip transitions take place only at a definite sign of a photon circulation. Their intensities is almost independent of the value of the magnetic field.
IN WURTZITE STRUCTURES such as CdS the crystal symmetry group for lack of an inversion center allow energy term which is linear in wavevector [ 1]
Vp = ~ ( p x s)c
(1)
where s is the spin-operator and e is the unit vector parallel to the hexagonal axis. This term in the Hamiltonian of the conduction electron originates from the spin-orbit coupling of the electron with the electrostatic field E [2] s(p x E) + h.c.
hcoe, which, in turn, exceeds the characteristic energy of the fine structure ( 2 s - 2 p splitting). Further on the z-axis of the right coordinate system is assumed to be directed along the crystal axis e. The hydrogen-like Hamiltonian for the unperturbed donor has the form Ho -
(2)
In CdS the conduction-band states are formed from the wavefunctions of the valent shell of Cd-atoms and due to hexagonal perturbation of tetrahedron configuration the resulting field of S2--ions surrounding Cd2+-ions has the non-zero value on the sites of Cd-ions. Using the conventional k - p method equation (1) may be deduced from equation (2). On the ground of the Hamiltonian (1) there has been plenty of past calculations of pure spin transitions of the donor electrons in a magnetic field-combined resonance [3]. At a weak magnetic field H the matrix elements of such transitions are very small, as being proportional to H. The primary significance of the Hamiltonian (1) consists in alternating its sign by the coordinate inversion. For this reason it mixes the states with the opposite parity and equal e-components of the total angular momentum. Hence, interaction (1) admixes s- and p-like states of the electron bound to the donor and makes possible the dipole spin-flip transition ls-2s. In this note we present the calculations of the intensity of these transitions in a weak magnetic field H oriented along e-axis so that the energy of donor ionization Ry -me4/2h2e2o considerably exceeds the cyclotron energy 515
p2
e2
2m
eor
+
lelh
2mc
lelh 1I-I+ - - g
2moc
H.
(3)
Here the irrelevant diamagnetic term is neglected. The perturbation is the sum of the dipole interaction with the light, proportional to =r, and the parity violating Hamiltonian Vp = ~-~
--
,
e,
A = 1Hxr.
(4)
The term (e/c)A in equation (4) will be always omitted excluding the cases when the matrix element of the momentum p equals zero. The amplitude of the ls~-2st transition according to the second order perturbation theory is given by the expression M
=
Ma+M b
Ma = ~, (2stl=r[nt)(nflVplls~)
(2s1'[ VI,[n+)(n $1= rl ls+)
Mb = En
E~s -- E~
(5)
Let us first consider Ma. The energy denominators in this sum are all of the order of Ry, hence the effect of the magnetic field may be neglected. The coordinate and spin parts of the eigenfunctions of rio are factorized, therefore it holds the relationship
im
(ntl (p x ~)ells~,) -
Els - - E n
h
(ntl (r x a ) e l l s $ )
(6)
516
DICHROISM OF IMPURITY CRYSTALS
from which we find that
Xm ( 2 s l x - - i y l 2 p ,
2h: imX (2st[ (=r)[(r x , ) c ] Ils$).
(7)
2h z
Ma -
Some caution must be taken in evaluating the sum representingMb, where there are small denominators corresponding to the intermediate 2p-states. In the reduced sum M~, where the 2p-states are excluded, one may neglect the influence of the magnetic field and use relationship (6) as well. Then,
Vol. 45, No. 6 lz = 1)(2p, lz = ll=rlls)
(13)
2g(m/mo)- 1
After standard calculations the sum of equations (9) and (13) looks as follows (a-~y~ M = --0.57~a
~" = 1 +
2
e°h2 ex--iey),
-1.
m
gmo
a -
me 2 ,
1
(14)
It may be shown that the matrix element of the M~ -
imX S" 2h2n~p
(8)
(2stl(r x o ) clh$)(n$ trlls$).
Adding and subtracting from equation (8) the contribution of the 2p-states, we obtain imX
M; -
2h 2 (2stl[(r x o ) cl(=r)lls$) imX
2h 2 2p
(9)
(2sTl(r x o ) e l 2p,l,)(2p$[ t r l l s $ ) .
The first term cancels the matrix element Ma, in the second one due to the equation (tp(r x o ) e l $ ) = - i
(It °
x i l)o
--X --iy
(10)
= -- i ( x -- iy)
only 2p-state with lz = 1 gives a nonzero result. The last point to consider is the calculation of the contribution of 2p-states to M b. Since in our approximation, neglecting the fine structure energy, (2st p[ 2p) = 0, only the vector-potential term in Hamiltonian (4) V~ -
Xle[H
--.±r± 4hc
the opposite sign and polarization dependence ex + icy. Thus, both these transitions occur at opposite signs of the circular polarization of the photon propagating along the e-axis. Since in the initially dipole allowed l s - 2 p transition the electron spin conserves, the energy of the absorbed photon because of the Zeeman shift differs from that of the l s - 2 s spin-flip transition. As yet no information is available on the experiments, explicitly determining the quantity X for conduction bands of any semiconductors. In the case of CdS X was indirectly evaluated [5] by studying the spectral width of the spin-flip Raman scattering of light by degenerate electrons, though the lineshape was analyzed using the simplest diffusive approximation. Observations on the transitions described above and measuring their intensity would allow one to ultimately determine the value of X. If, according to [5], one supposes that for CdS X _~ 1.6 × 10- ]o eV cm- 1 and adopts m/mo = 0.2, g = 1.8, then the intensity ratio Ils_2s/Ias_2p will be of the order of 3 × 10 -3 . Acknowledgements - The author is grateful to Professor
E.I. Rashba for helpful discussions. REFERENCES
(11) 1.
results in the 2 s - 2 p mixing. From the equation (l"[a±r±l$) =
l s t - 2 s $ transition differs from equation (14) only by
(i 0 Xoi t ) 1"
$
= x--iy
2. (12)
x+iy
it is seen that the nonzero contribution also results from the p-state with Iz = 1 only and is equal to
3. 4. 5.
E.I. Rashba, Fiz. Tverd. Tela 1,407 (1959); R.C. Gasella, Phys. Rev. Lett. 5,371 (1960). Silvan S. Schweber, A n Introduction to Relativistic Quantum Field Theory, Row, Peterson & Co., New York (1961). E.I. Rashba &V.I. Sheka, F T T 6, 141 (1964). E.I. Rashba, Usp. Fiz. Nauk 84, 557 (1964). R. Romestain, S. Geschwind & G.E. Devlin, Phys. Rev. Lett. 39, 1583 (1977).