Energy Dependence of Luminescence Photon Circular
Polarisation: Measurement of Strain in III-V Semiconductors
W Wang, K Allaart and D Lenstra
Vrije Universiteit Amsterdam, Department of Physics and Astronomy, De Boelelaan
1081, 1081 HV, Amsterdam, The Netherlands, and COBRA Research Institute,
Eindhoven, The Netherlands
[email protected]
Abstract. The energy dependence of luminescence photon circular polarisation, after injection
of spin-polarised electrons, is characteristic for strain in a semiconductor. We calculate the
energy dependence of the correlation between the electron spin polarisation Ps and the circular
polarisation of the emitted light Pcir within the Luttinger-Kohn model with uniaxial strain. The
correlation factor is found to vary with luminescence photon energy, between roughly -0.9 and
0.8, depending on the valence band splitting energy due to the strain. This provides a method
for the detection of strain in a semiconductor.
1. Introduction
The optical emission from a semiconductor with spin polarized electron injection will have substantial
circularly polarized components. In an unstrained semiconductor the polarisation correlation is simply
Pcir  0.5Ps , independent of the photon energy [1-3]. This relation follows directly from the
selection rules for the optical transitions and the equal weight of all magnetic substates in case of
isotropic material. When the semiconductor is strained, tensile or compressively, the intrinsic
rotational symmetry is broken. Thereby the polarisation correlation will become complicated, varying
with the emission photon energy, between roughly -0.9 and 0.8, depending on the strength and type of
the strain. We show that this provides a method for measurements of strain in a semiconductor, which
can be an alternative method, in addition to the spectral analysis of the linear polarisation degree [4].
In the non-magnetic semiconductors, the luminescence circular polarisation variation measurements
could show more accurate results.
The spin polarisation Ps of the conduction band electrons is defined as [1-3]
Ps  N   N   N   N  
(1)
with N  the density of electrons with spin parallel and antiparallel to the magnetization direction. The
circular polarisation Pcir of the light is defined as
Pcir  I   I   I   I  
(2)
with I  the intensities of left and right circular polarized components of the emitted light, defined
with respect to the detection direction. We shall restrict ourselves here to the case that this direction
coincides with that of the system quantization direction. The emitted light intensities for photon
energy  are determined by the total spontaneous emission rate [5-6]
I 
2e 2 nr  1
m02 c 3 2 2 3


0



d | k | d | k |2 sin  f e ( Ekc ) f h ( Ekv ) L( Ekcv ) | M D (k ) |
0
(3)
where nr is the refractive index. The integration is over the electronic wave vector k of the carriers
involved in the transition;  is the polar angle of k with respect to the magnetization axis. A
Lorentzian transition line shape [7]
L( Ekcv ) 
( /  s ) 2
 [( Ekcv   ) 2  ( /  s ) 2 ]
(4)
is assumed in which Ekcv is the energy difference between the initial band energy E kc and final band
energy E kv with wave vector k ;  s is the intraband relaxation time. The Fermi distributions f e and
f h of electrons and holes respectively are those for the intraband quasi equilibrium, which is reached
very fast as compared to the spontaneous emission life times. The label D on the dipole transition
matrix element M D (k ) indicates that left or right handedness must be defined with respect to the
detector direction. All factors in the integrand of Eq. (3) are functions of both magnitude of k and
polar angle  .
To derive a relation between the electron spin polarisation Ps and circular polarisation Pcir , one
must know the energy band structure of a semiconductor. That gives the injected electron distributions
f e , f h , the optical transition energy Ekcv , the line shape function L( Ekcv ) , and also the band
wavefunctions that determine M D (k ) , and therewith the optical transition probabilities. All these
factors are influenced directly by the strain in semiconductors, and conversely. Therefore the
polarisation correlation of Pcir and Ps will provide a signature of the strain in a semiconductor.
2. Analysis and Results
The conduction band states are denoted as | S  for spin up and | S  for spin down. The HH and
LH band wavefunctions can be expressed in terms of the basis states J, mJ
at k  0 with
mJ  3 / 2,1 / 2,1 / 2,3 / 2 the projection of J on the fixed z -axis. By k  p theory, one is able to
find the wavefunctions of the states in the vicinity of k  0 as a superposition of these basis states.
The Luttinger-Kohn Hamiltonian matrix [8] for uniaxial strain with the z -axis as symmetry axis,
has twofold degenerate eigenenergies [5]
1/ 2

1 2 2
3 2 4 



2
2 2
2 2 2
(5)
 1 | k | 2( '  2 (k z  k  ))  3 3 k z k    k    ,
2
4


 


in which  '   m0 /  2 , with  the band splitting energy at k  0 due to the strain, k z | k | cos  ,
k | k | sin  , parameters  1 ,  2 ,  3 and  are material dependent. The energy bands as functions of
the electron wave vector, for different angles  of it, are plotted in figure 1. Figure 1a is for tensile
strain with   10 meV and figure 1b for compressive strain with   10 meV. The curves are for
  0,  / 6,  / 3,  / 2 of the k vector; the order is indicated by the arrows. The upper energy will be
referred to as E h and as the heavy-hole band, the lower as E l and as the light-hole band; a
2
E
2m0
nomenclature that does not uniquely correspond to large or small effective mass, which is now
obviously an anisotropic tensor in k space.
The two eigenvectors corresponding to E h and those for E l can be written in a compact matrix
multiplication form as superpositions of the m J basis states as
 h1 
  b Rh 0  c    3 / 2 
 2



 
 h 
1   c 0 Rh b   1 / 2 
(6)

 1  
 Rl b  c 
   1/ 2 
0
N
l
i




 
2 
c  b Rl    3 / 2 
 0
 l 
with Rh  H h    Eh , Rh   H h    Eh , and N i | Ri | 2  | c | 2  | b | 2 , i  h, l . Here we
follow the notation of reference [8]: b  3h0 (k x  ik y )k z , c  3 / 2h0 [( k x2  k y2 )  2ik x k y ] , and
H h  h0 / 2[ 1k 2   2 (2k z2  k 2 )] , H l  h0 / 2[ 1k 2   2 (2k z2  k2 )] with h0   2 / m0 .
(b)
(a)
Figure 1. Angle
 dependent energy band dispersion of E l and Eh , where  is the angle between the k -
vector of the electron and the strain symmetry axis. The dashed lines are the LH band and the solid lines the HH
band. The curves are for   0,  / 6,  / 3,  / 2 ; the order is denoted by the arrows. (a) tensile strain with
  10 meV, (b) compressive strain with   10 meV.
(a)
Figure 2. Dependence of the components
| f mi J |2 on the strain  for the HH band (a) and LH band (b). The
solid line gives the circular polarisation of the emitted light
factor
M D (k ) only.
(b)
 Pcir for each of the bands if one considers the
The optical transition matrix element between the conduction bands and valence bands is
c r v 
f mi J c r mJ ,

(7)
mJ
where the expansion coefficient f mi J of the valence band wavefunction on the basis states are given by
a row of the matrix in Eq. (6); r is the dipole transition operator for left or right circular polarized
light.
Strain influences all factors in the integrand of Eq. (3), but a dominant factor for the correlation
between electron spin and photon circular polarisation is the transition matrix element M D (k ) of
Eq.(3), namely its discrimination of emitted light σ+ and σ− . In figure 2 we show the magnitude of the
components, | f mhJ | 2 and | f ml J | 2 , of the HH and LH bands as a function of band splitting energy 
due to a strain, integrated over all directions of k , at k  0.01a0 , with a0  2 / a ; a the lattice
constant. In the isotropic case,   0 , we have the integrated | f mi J | 2  1 / 2 for mJ  1 / 2,3 / 2 in
both HH and LH bands. When  increases, i.e. compressive strain, m J  3 / 2 components begin to
dominate the HH band wavefunctions, which contribute most to the transition rates. The transition
from a | S  conduction band state to the m J  3 / 2 component of the valence band states gives
Pcir  1 (seen by the observer). Similarly Pcir = +1 for a transition from | S  to mJ  3 / 2 . So
if one considers the effect of the matrix element M D (k ) only, one has
Pcir   Ps for large
compressive strain. For tensile strain the mJ  1 / 2 components dominate in the HH band. The
correlation between Pcir and Ps is then just opposite, but the size of the matrix element is a factor
3
smaller due to partial spin flip. This explains why for negative  the value Pcir  Ps is approached
more slowly than Pcir   Ps for positive  . It also explains the relation Pcir  0.5Ps in case all m J
components have equal weight, as is the case without strain.
Figure 3. The strain dependent correlation between the electron polarisation Ps and the circular polarisation of
the emitted light Pcir , tensile strain case.
These simple features, shown in figure 2, are still visible in the transition rates (3) which include
the strain effects in all factors, such as the energy band dispersion Eq. (5). The calculated polarisation
correlations of Pcir and Ps as a function of the strain with realistic parameters for GaAs are shown in
figure 3 and 4 for tensile and compressive cases respectively. For tensile strain, with band splitting
energy  decreasing from 0 to -10 meV, the ratio of Pcir and Ps increases from -0.5 to 0.5. The ratio
approaches 0.8 at   20 meV. For compressive strain, the ratio between Pcir and Ps decreases
from -0.5 to -0.9, when the strain induced band splitting energy  increases from 0 to 10 meV.
Further increase of  makes the ratio approach -1. Comparing figure 3 and 4, one finds significantly
different polarisation correlation in various strained cases, both regarding the values and the trends as
a function of photon energy. This enables a way to investigate the strain in semiconductors by
measuring the correlation between the injected electron spin polarisation and the luminescence circular
polarisation.
Figure 4. The strain dependent correlation between the electron polarisation Ps and the circular polarisation of
the emitted light Pcir , compressive strain case.
3. Conclusion
We have investigated the strain dependent correlation between the electron spin polarisation Ps
and the circular polarisation of the emitted light Pcir . This correlation factor varies with luminescence
photon energy, between roughly -0.9 and 0.8, depending on the band splitting energy due to the strain.
This provides a method for detection of strain in a semiconductor.
The authors acknowledge financial support by the Freeband Communication Impulse of the
technology programme of the Netherlands' Ministry of Economic Affairs.
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