Transport in semiconductor superlattices

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Transport in semiconductor superlattices
Original motivation for semiconductor
superlattices

• L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61
  (1970)
• Observation of negative differential
  conductivity (NDC) due to Bloch oscillations
  in superlattices
• ---------
• ) F. Bloch, Z. Physik 52, 555 (1928)

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Previous lecture minibands in a superlattice
superlattice
LA
LB
d
anisotropic 3-dimensional minibands in a
superlattice (Tight-binding appr.)
ec(k) Ec Ec,m (Dc,m/2)coskzd
(?2/2mc)(kx2 ky2)
tailoring of the band structure
band off-set DEc depends on
Al-content x
miniband spacing Ec,m depends on well width
LA
miniband width Dc,m depends on barrier
width LB
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What are Bloch Oscillations
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What are Bloch Oscillations
? Periodic motion in space
Bloch frequency nB
Bloch oscillations are a nice concept, but cannot
be observed in (natural (!)) semiconductors,
...
... but in man-made semiconductor superlattices
they can!
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Comparison bulk semiconductor vs. superlattice

• band width reduced by about 2 o.o.m.? strongly
  reduced scattering rates (LO phonons)

• BZ reduced by about 2 o.o.m.? strongly increased
  Bloch frequency

• ? condition for Bloch oscillations nBtm gt
  1 can be fulfilled in superlattices

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Superlattice at zero field
tunneling between wells gt formation of a
miniband
miniband dispersion e(kz) (D/2) 1-cos(kzd)
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Superlattice under applied bias Bloch
oscillations
quantum mechanical description eigenstates of
an electron in a periodic structure electric
field potential eFz Wannier-Stark ladder with
spacingDW eFd
Semiclassical description
electron performs Bloch oscillationswith
frequency nB eFd/h in momentum space and in
real space
hnB
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Original motivation for semiconductor
superlattices

• L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61
  (1970)
• Observartion of negative differential
  conductivity (NDC) due to Bloch oscillations
  in superlattices (Esaki model!)

---------------) F. Bloch, Z. Physik 52, 555
(1928)
current density j j enltvdrgt
average drift velocity ltvdrgt (1/t)?vgr(t)
exp(-t/t) dt
for tight-binding band vgr(k) (1/?) (d/dk)
e(k) with e(k) (D/2)1 - coskd
vgr(k) (Dd/2 ?) sinkd
with k(t) (eF/?)t and wB eFd/? vgr(t)
vgrmax sinwBt, with vgrmax Dd/2?
Average drift velocity ltvdrgt (1/t)vgrmax
?sin(wBt) exp(-t/t) dt vgrmax wBt 1
(wBt)2-1
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drift velocity versus electric field F
vgrmax/2
Average drift velocity ltvdrgt (1/t)vgrmax
?sin(wBt) exp(-t/t) dt vgrmax wBt 1
(wBt)2-1
ltvdrgt (et/m) F ? t F, with m ?2/(Dd2),
for wB ltlt 1/t Drude regime
ltvdrgt D /(eFt) ? t -1 F-1, with L D /(eFt),
for wB gtgt 1/t Hopping regime
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drift velocity versus electric field F
Similar results for
Semiclassical treatment (0 lt eFd lt D) M.C.
simulations, e.g.
Quantum mechanical picture (?/t lt eFd eFd gt D no
problem!) hopping WS ladder
S. Rott, N. Linder, and G. H. Döhler, Phys. Rev.
B 65 , 195301 (2002)
A. Wacker, A.-P. Jauho, S. Rott, A. Markus, P.
Binder, and G.H. Döhler, Phys. Rev. Lett. 83 ,
836-839 (1999)
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Experimental verification of NDC regime, Bloch
oscillations and WS ladder
(1) Measurements of I vs. U in (weakly) doped
superlattices) (Grahn et al.)
(2) ps time-of-flight measurements in (weakly)
doped superlattices) field dependence (Grahn et
al.)
(3) THz emission from coherently excited
electrons with fs-pulse excitation (Roskos et
al.)

(4) Wannier-Stark ladder from spectrally
resolved absorption or photo-current measurements
(Sai-Halasz et al. and ourselves at Erlangen
University)
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(3) Coherent submillimeter-wave emission from
Bloch oscillations in a semiconductor
superlattice
H. Roskos et al., PRL 70, 3319 (1993)
t0
tTB/4
tTB/2
t3TB/3
tTB
Lowest conduc-tion mini-bandeFz
fs-pulse hn Egeffat t0

All the carriers oscillate coherently and emit
coherent THz radiation with nB eFd/h due to
periodic acceleration and deceleration
uppermost va-lence mini-bandeFz
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(4) Optical study (C.W.) of Wannier-Stark ladder
- spectrally resolved absorption or
photo-current measurements
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(4) Optical study (C.W.) of Wannier-Stark ladder
- spectrally resolved absorption or
photo-current measurements
Theory
Experiment
K.H. Schmidt, N. Linder, G.H. Döhler, H.T. Grahn,
K. Ploog, H. Schneider,Phys. Rev. Lett. 72 ,
2769-2772 (1994)
N. Linder, K.H. Schmidt, W. Geißelbrecht, H.T.
Grahn, K. Ploog, H. Schneider, G.H. Döhler, Phys.
Rev. B 52 , 17352 - 17365 (1995)
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See you tomorrow!subject optical properties of
nanostructures