Lecture 3: optical phenomena in QWs and Superlattices

1
Lecture 3 optical phenomena in QWs and
Superlattices

2
Recapitulation optical transitions in
semiconductors
Conduction band
Probability for an optical transition between
individual conduction and a valence band states
in a bulk semiconductor
wv,k?c,k ? ltc,kpv,kgt2 d(ec(k)
ev(k) ?w) d(k k ? q)
energy conservation
momentum conservation(q 2p/l ? 5x104 cm-1,
negligible compared with ty-pical k-values
?2p/a ?108cm-1)
interband dipole matrixelement
Absorption coefficient at the photon frequency w
(integral over all possible microscopic
absorption processes!)
a(w) ? (1/2p)3???wv,k?c,k d3k? ?
Nvc(3)(ecv(k)) ltc,kpv,kgt2d(ecv(k)-?w)
combined den-sity of states
a(w) ? (1/2p)3???wv,k?c,k d3k? ? (?w-Eg)1/2
ltc,kpv,kgt2
calculation of matrixelement (for negligible q)
ltc,kpv,kgt ???exp(-ikr) uc,k(r)(?/i)(d/dr)
exp(-ikr) uv,k(r) d3r
ltc,kpv,kgt ???uc,k(r)(?/i)(d/dr) uv,k(r)
d3r, as ltuc,kuv,kgt 0
Valence band
ltc,kpv,kgt ? ???uc,0(r)(?/i)(d/dr) uv,0(r)
d3r pc,v
pcv interband dipole matrixelement
pcv2/2m0 ? 25 eV for all s.c.s
3
Optical transitions in QWs
LA
Probability for an optical transition between
individual conduction and a valence band states
in a QW
wv,mk?c,n,k? ltc,nkpv,m,kgt2 d(ec,n(k)
ev,m(k) ?w) d(k k ? q)
energy conservation
momentum conservation(q 2p/l ? 5x104 cm-1,
negligible compared with ty-pical k-values
?2p/a ?108cm-1)
interband dipole matrixelement
Absorption coefficient at the photon frequency w
(integral over all possible microscopic
absorption processes)
a(w)???(1/2p)2??wv,m,k?c,n,k d2k ? Nvc(2)(e)
ltc,n,kpv,m,kgt2d(e-?w)
m n
2-d combined density of states
Nvc(2)(e) (mcv/p?2) const.(!)
with mcv-1 mc-1 mv-1
a(w) ? ???(?w - en,m) ltc,n,kpv,m,kgt2
m n
calculation of matrixelement (next slide)
4
Optical transitions in QWs
LA
a(w) ? ???(?w - en,m) ltc,n,kpv,m,kgt2
m n
calculation of matrixelement
ltc,n,kpv,m,kgt ???exp(-ikr)xn(z)uc,k(r)(?/
i)(d/dr)exp(ikr)xm(z)uv,k(r)d3r
ltc,n,kpv,m,kgt ?xn(z)uc,k(r)(?/i)(d/dr)
xm(z)uv,k(r)dz
ltc,n,kpv,m,kgt pc,v?xn(z) xm(z) dz ? pc,v
dn,m (quasi-selection rule!)
Details of band structure
conduction, heavy, light and spin split-off
valence bands (index i)
While uc,0(r) has s-character, uv,i,0(r) has
p-character, i.e.
orbit angular momentum l 1, spin ½ ? j
3/2, 1/2
? j 3/2 jz ?3/2 heavy holes,
? j 3/2 jz ?1/2 light holes,
(? j 1/2 jz ?1/2 spin splitt-off holes)
5
Optical transitions in QWs
LA
ltc,0pv,i,0gt ???uc,k(r)(?/i)(d/dr) uv,i,k(r)
d3r, as ltuc,kuv,kgt 0
in QW symmetry is broken ? orientation of j in
z-direction
heavy holes (v,h) 3/2, 3/2gt lz1gtsz
1/2gt
3/2,-3/2gt lz -1gtsz -1/2gt
light holes 3/2, 1/2gt c0(lz 0gtsz
1/2gt) c1(lz1gtsz -1/2gt)
3/2,-1/2gt c-0(lz 0gtsz -1/2gt)
c-1(lz -1gtsz 1/2gt)
with (c?0/c?1)2 31
For heavy holes
ltc,0pxv,h,0gt ltc,0pyv,h,0gt (1/2)½ pc,v
ltc,0pzv,h,0gt 0
For light holes
ltc,0pxv,h,0gt ltc,0pyv,h,0gt (1/6)½ pc,v
ltc,0pzv,h,0gt (2/3)½ pc,v
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First observation of quantized states in QW
absorption
T 4.2 K
Dingle, Wiegmann and Henry, PRL 33, 827 (1974)
7
Well width dependence of Exciton enegies
incorrect band- offsets!
Dingle, Wiegmann and Henry, PRL 33, 827 (1974)
8
absorption in high-quality QWs
Miller, Kleinman, Tsang and Gossard, PRB 24, 1134
(1981)
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Polarization dependent absorption in QWs
LA
For heavy holes
ltc,0pxv,h,0gt ltc,0pyv,h,0gt (1/2)½ pc,v
LA
ltc,0pzv,h,0gt 0
For light holes
ltc,0pxv,h,0gt ltc,0pyv,h,0gt (1/6)½ pc,v
x
ltc,0pzv,h,0gt (2/3)½ pc,v
qz
Normal incidence
?
hh/lh absorption 3/1
z
y
ex
Wave guide absorption
TE - mode same as for normal incidence
hh/lh absorption 3/1
qz
TM - mode
ez
lh-absorption only strong!
?
ey
10
generation of spin-polarized electrons by
absorption in QWs
LA
Electrons 1/2, 1/2gt sgt 1/2gt
1/2,-1/2gt sgt -1/2gt
heavy holes 3/2, 3/2gt lz1gtsz 1/2gt
3/2,-3/2gt lz -1gtsz -1/2gt
-1/2
1/2
light holes 3/2, 1/2gt c0(lz 0gtsz 1/2gt)
c1(lz1gtsz -1/2gt)
3/2,-1/2gt c-0(lz 0gtsz -1/2gt)
c-1(lz -1gtsz 1/2gt)
s -
s -
s
s
circular polarized photon (with q (0, 0, qz))
s 1, 1gt
s - 1,-1gt
-3/2
3/2
-1/2
1/2
s - polarized light at hnhh generates pure -1/2
polarized electrons
s - - polarized light at hnhh generates pure 1/2
polarized electrons
11
heavy/light hole mixing in QWs
10 nm GaAs-QW in Al0.25Ga0.75As
Sanders and Bajaj, Phys. Rev. B 35, 2308 (1987)