NanoVision 2020

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PHYS 2235 Physics of NanoMaterials S. J.
Xu Department of Physics (Lecture 5)
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Transport in Two-Dimensional Electron Systems

• Contents of Lecture 5
• Scattering mechanisms
• The modulated-doped GaAs/AlGaAs heterojunction
• Miniband transport in superlattice
• Wannier-Stark localization

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Scattering Mechanisms

• The MOBILITY of electrons in two-dimensional
  systems is limited by SCATTERING from a number of
  DIFFERENT sources COULOMB SCATTERING from
  IONIZED IMPURITIES is an ELASTIC process that is
  known to be dominant at LOW temperatures in
  bulk materials In two-dimensional systems
  ADDITIONAL Coulomb scatterers can arise from
  the presence of SURFACE STATES and FIXED
  CHARGES at the interface

• SHOWN IS THE TEMPERATURE DEPENDENCE OF
  THEMOBILITY IN BULK GaAs
• SCATTERING FROM IONIZED IMPURITIES IS THE
  DOMINANTMECHANISM AT LOW TEMPERATURES
• NOTE HOW SCATTERING FROM UNIONIZED IMPURITIES
  GIVESA TEMPERATURE INDEPENDENT CONTRIBUTION TO
  THE MOBILITY AT LOW TEMPERATURES
• AT HIGHER TEMPERATURES COULOMB SCATTERING IS
  LESSIMPORTANT AND SCATTERING FROM PHONONS IS
  INSTEADTHE DOMINANT MECHANISM

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• INELASTIC scattering of electrons is caused by
  the quantized VIBRATIONAL modes of the crystal
  lattice which are known as PHONONS In the
  semiconductors of interest here the presence of a
  DIATOMIC basis leads to TWO distinct types of
  lattice vibrations ACOUSTIC phonons are
  generated when the atoms in the basis vibrate
  with the SAME phase while OPTICAL phonons
  result when the basis atoms vibrate OUT of phase
  with each other Phonon scattering is
  dominant at HIGH temperatures but weakens below
  77 K

AN ACOUSTIC PHONON THE BASIS ATOMS MOVE IN PHASE
AN OPTICAL PHONON THE BASIS ATOMS MOVE OUT OF
PHASE
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• Phonons may be quantified in terms of their
  ANGULAR FREQUENCY and WAVENUMBER which in turn
  are related to each other by their DISPERSION
  RELATION In three-dimensional
  semiconductors with a diatomic basis the optical
  and acoustic modes each show THREE branches
  due to the three possible modes of vibration in
  the crystal There are normally TWO
  transverse branches and ONE longitudinal branch
  although for special symmetry directions in
  the crystal the transverse branches may be
  DEGENERATE

LO LONGITUDINAL OPTICAL PHONONTO TRANSVERSE
OPTICAL PHONONLA LONGITUDINAL ACOUSTIC
PHONONTA TRANSVERSE ACOUSTIC PHONON
PHONON DISPERSION CURVES FOR Si AND GaAs NOTE
THE DEGENERACY OF THE TRANSVERSE MODES
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• The dispersion curves reveal important
  information about phonon scattering in
  semiconductors The HIGH-ENERGY optical
  phonons have a low probability of excitation at
  low temperatures due to the SMALL thermal
  energy available At low temperatures
  ACOUSTIC phonon scattering dominates and is
  predicted to yield an INVERSE variation of
  mobility with temperature that is seen in CLEAN
  systems in which the obscuring effect of
  impurity scattering is suppressed

• THE HISTORICAL EVOLUTION OF THE
  TEMPERATUREDEPENDENCE OF THE MOBILITY IN
  GaAs/AlGaAsHETEROSTRUCTURES
• FOR 2 lt T lt 30 K THE BETTER QUALITY SAMPLES
  SHOWTHE 1/T VARIATION EXPECTED FOR ACOUSTIC
  PHONONSCATTERING
• ABOVE 50 K OPTICAL PHONON SCATTERING
  DOMINATESAND A STRONGER TEMPERATURE VARIATION IS
  FOUND THATIS COMMON TO ALL SAMPLES

L. Pfeiffer et al. Appl. Phys. Lett. 55 1888
(1989)
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The Modulation-Doped GaAs/AlGaAs Heterojunction

• An important innovation that allows very HIGH
  mobility to be achieved in heterojunctions is the
  introduced of an UNDOPED SPACER LAYER between the
  doped region and the channel layer This
  MODULATION DOPING removes the ionized donors from
  the GaAs so that their influence on
  scattering is strongly REDUCED The low
  temperature mobility is LIMITED by scattering
  from these REMOTE donors and from other
  unintentional IMPURITIES incorporated during the
  growth process

• THIS FIGURE SHOWS THE DEPENDENCE OF THE
  TWO-DIMENSIONAL-ELECTRON-GAS MOBILITY ON THE
  SPACER-LAYER THICKNESS(Wsp) IN THE GaAs/AlGaAs
  HETEROJUNCTION
• THE LOW-TEMPERATURE MOBILITY INCREASES WITH
  SPACER LAYERTHICKNESS DUE TO THE REDUCTION IN
  SCATTERING FROM IONIZEDDONORS IN THE AlGaAs
  LAYER
• AT THE HIGHEST TEMPERATURES HOWEVER THE MOBILITY
  IS LIMITEDBY OPTICAL-PHONON SCATTERING AND IS
  THEREFORE INDEPENDENTOF THE SPACER-LAYER
  THICKNESS
• AT INTERMEDIATE TEMPERATURES THE MOBILITY SHOWS
  THESIGNATURE OF ACOUSTIC PHONON SCATTERING

K. Hirakawa and H. SakakiPhys. Rev. B 33, 8291
(1986)
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• The mobility of the two-dimensional electron gas
  in a heterostructure can be FURTHER improved by
  growing a GaAs/AlGaAs SUPERLATTICE prior to the
  growth of the heterostructure itself The
  superlattice has the effect of SMOOTHING out
  irregularities in the crystal structure by
  PREVENTING the propagation of disorder or
  impurities as the crystal is grown

• THE SMOOTHING ACTION OF A SUPERLATTICE BUFFER
  LAYER
• THE ORIGINAL SURFACE OF THE GaAs SUBSTRATE IS
  VERY ROUGHON THE ATOMIC SCALE AND WOULD LEAD TO
  THE INTRODUCTIONOF SIGNIFICANT DISORDER IF
  GROWTH WERE PERFORMED DIRECTLYON TOP OF THIS
• BY GROWING JUST A FEW PERIODS OF A SUPERLATTICE
  STRUCTUREHOWEVER THE IRREGULARITIES IN THE
  SURFACE ARE EFFECTIVELYSMOOTHED OUT LEAVING A
  BETTER-DEFINED SURFACE FOR THEINITIATION OF
  HETEROJUNCTION GROWTH
• WHILE THE DETAILS OF THIS PROCESS ARE ONLY
  PARTLY UNDERSTOODIT IS THOUGHT THAT THE Al IN
  THE AlAs LAYERS IS PLAYS A CRUCIAL ROLE IN
  GETTERING IMPURITIES

M. J. Kelly, Low Dimensional SemiconductorsOxford
University Press, Oxford (1995)
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Perpendicular Transport
Standard Drude Picture of Electronic Transport in
Solids
The electron motion in solids under action of a
static field is drift transport carriers move
ballistically until they change their momentum by
a scattering process. The drift velocity of the
carriers is determined by a balance between the
momentum and energy gain from the field during
ballistic motion and by the momentum and energy
changes due to elastic and inelastic scattering
processes. The overall current due to the moving
carriers is then characterized by Ohms law
j s F.
The conductivity is in the Drude picture given by
where t is the momentum relaxation time, n the
carrier density and m the carrier mass.
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Bloch Oscillations of Electronic Transport in
Superlattice
In 1928, F. Bloch predicted that electron notion
in a periodic potential under the influence of a
static force is oscillatory motion!
An electron in a periodic potential subject to an
external electric field F changes its k-vector
according to
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If E(k)E0E1sin(kd), then
Bloch oscillations in the semi-classical picture.
An electron at k 0 starts to move with a
velocity in k-space once the field is turned on.
Until the edge of the first Brillouin zone, it
gains energy. When it leaves the first Brillouin
zone, the energy starts to decrease and the
velocity (dashed line) becomes negative. When it
reaches the centre of the second Brillouin zone,
it has returned to its original spatial position.
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The time period of the oscillatory motion is
given by
However, the period has to be shorter than the
collision time, which is currently impossible in
real bulk solids because the atomic lattice
constant is so small. For example, TB10-11 s
for F10 kV/cm and d3.5 Å. But the Bloch
oscillation should become possible in a
superlattice whose lattice period d is 10 to 50
times larger.
Taking collisions into account, Esaki and Tsu
calculated the drift velocity of electron in an
infinite superlattice using a classic method. The
velocity increment in a time interval dt is
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The average drift velocity imposed by collisions
occurring with a frequency t-1 is
Assuming EE02E1cos(kd), one finds
where ?eFtd/ph and 1/mSL(1/ h2)(?2E/?k2) . The
?d versus F curve has a maximum for p?1 and
exhibits an negative differential velocity (NDV)
beyond this value. The condition to be fulfilled
on t to achieve NDV is about 6 times easier than
that required to achieve Bloch oscillations.
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Observation of Esaki-Tsu NDV in Superlattice
Measured current-voltage characteristics of
undoped GaAs/AlAs superlattices with different
miniband width.
A. Sibille, et al, Phys. Rev. Lett. 64, 52 (1990).
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Observation of Wannier-Stark Localization in
Superlattice
An alternative description of Bloch oscillation
is so-called WannierStark localization. The
energy spectrum of the lattice subjected to an
electric field is no longer a continuous band. It
consists of the Wannier-Stark ladders with energy
Schematic diagram of optical transitions in a
biased superlattice characterized by spatially
direct and indirect transitions, leading to a fan
chart when the field is swept (inset).
K. Leo, Semicond. Sci. Technol. 13, 249 (1998).
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Transition energies as a function of the electric
field.
Photocurrent spectra (vertically offset for
clarity) for selected values of the dc electric
field applied in the growth direction of the
superlattice. Labels at the peaks give the
Wannier-Stark ladder index.
F. Agulló-Rueda, et al, Phys. Rev. B 40, 1357
(1989).
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An impressive confirmation of the oscillation of
the Bloch wave packets was the observation of THz
emission of the wave packet. In this experiment,
an optical pulse impinging onto the sample
creates the Bloch wave packets. The oscillating
dipole moment associated with the wave packet
performing BOs causes radiation in the THz
regime, which is detected with a small antenna
which is gated with optical pulses producing a
short in a fast photoconductor..
THz emission from a semiconductor superlattice.
Shown is the measured THz field amplitude as a
function of delay time for various electric
fields.
C. Wischke, et al, Phys. Rev. Lett. 70, 3319
(1993).