Download Lecture 3: Heterostructures, Quasielectric Fields, and Quantum

Lecture 3: Heterostructures, Quasielectric Fields, and
Quantum Structures
MSE 6001, Semiconductor Materials Lectures
Fall 2006
3
Semiconductor Heterostructures
A semiconductor crystal made out of more than one material, where composition varies with position but the crystal periodicity of atoms is kept, is referred to as a “heterostructure”.
3.1
Semiconductor alloys
Single-crystal semiconductor alloys can be made that have bandgaps and other properties in between those of the end point materials. Figure 1 illustrates how the bandgap of Aly Ga1−y As, with
0 ≤ y ≤ 1, has a bandgap energy Eg in between those of AlAs and GaAs. Grading the composition, that is varying the concentration with position, gives spatially-varying semiconductor
properties.
The Aly Ga1−y As alloy system The AlAs/GaAs family of materials has been extensively developed for its use in heterostructures. These materials have the same crystal structure, the cubic zinc
blende crystal lattice, and almost exactly the same lattice constant a (the periodic repeat length in
the crystal). AlAs and GaAs are also completely miscible, and all values of y in Aly Ga1−y As are
stable.
3.2
Quasi electric fields.
Consider a GaAs wafer that has a surface layer of Aly Ga1−y As that is graded from y = 0.3 at the
surface where x = 0 to y = 0 at x = xo . This composition profile and the resulting spatially
GaAs
AlyGa1-yAs
AlAs
EC
EC
EC
EV
EV
EV
F IGURE 1: Bandgap energies for the materials GaAs, Aly Ga1−y As, and AlAs.
3-1
y
0.3
0
x0
x
E
∆EC = 0.263 eV
EC (x)
EV(x)
F IGURE 2: Linearly graded material layer.
varying bandgap are given in Fig. 2.
The slope in the conduction band edge EC (x) corresponds to a quasi electric field, E ∗ . Recall
that for a real electric field E,
E=
1 dEC
1 dEV
=
.
q dx
q dx
(1)
The units of the electric field are [V/cm]. In the absence of an applied or internal electric field, the
conduction band quasi electric field due to composition changes is defined as,
EC∗ =
1 dEC
.
q dx
(2)
For linear grading of composition, as in Fig. 2,
dEC
∆EC
=
.
(3)
dx
∆x
Note that for the AlAs/GaAs system, the conduction band and valence band quasi electric fields
have opposite signs. In our example above, ∆EC = −0.263 eV for ∆y = 0.3. If ∆x is taken as
100 nm, then the quasi electric field is EC∗ = −2.63 × 104 V/cm. Conduction band electrons would
experience this field and be accelerated in the positive x-direction. Note that the quasi electric field
for the valence band will in general have a different magnitude and can have a different sign. In
this case, the valence band quasi electric field has a value EV∗ = +1.32 × 104 V/cm. [For material
parameters, see Tiwari and Frank, APL 60 630 (1992).]
3-2
electronegativity and is continuous across the interface if
the electronegativity difference is equal to the interface dipole. This energy also correlates with dangling bond energies at a semiconductor interface and with the energies of
transition impurities. It should be noted that there are
other models” that also predict this correlation between
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aa
Es
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GaP
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continuities as a function of the lattice constant, and simultaneously provides an estimate of the Schottky barrier
heights. The figure also includes unstrained Gel-$3, alloys, a material system in which there is considerable current interest. The band alignment of any two latticematched alloys can be obtained by observing the relative
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FIG. 1. The conduction band edge and valence band edge energies plotted as a function of the lattice constant of semiconductors. The circles indicate
the band edges of the binary semiconductors and the lines show the band edges of the ternary alloys. The two endpoints of each ternary line are the binary
constituents of that ternary. Discontinuities between two lattice matched or nearly matched semiconductor alloys may be found from the difference in
energy between their band-edge energies. The zero energy point represents the approximate gold Schottky barrier position in the band gap of any given
alloy.
Vol. 60,
No. 5, lattice
3 February
1992
S. Tiwari
D. J. Frank
631
F631
IGUREAppl.
3: Phys.
BandLett.,
edges
versus
constant
for III-V semiconductors. [Tiwari
and and
Frank,
APL 60 630
Downloaded 06 Oct 2003 to 155.99.175.44. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp
(1992).]
3.3
Materials Families
Only small lattice constant mismatches may be accommodated by strain in a heterostructure. Figure 3 displays band edges versus lattice constant for most of the III-V semiconductors. A “family”
of materials refers to the alloy compositions that may be grown on a given substrate, such as InAs.
Lattice constants generally need to match within about 1%, depending on layer thicknesses.
3.4
Quantum heterostructures
Heterostructure layers may be thin enough that the conduction band electrons and valence band
holes are quantum-mechanically confined to a layer with lower bandgap. This confinement changes
the electron and hole energies, giving a bandgap that depends on the well width: a “quantum well”.
(Fig. 4) A periodically repeating sequence of quantum wells and barriers is called a “superlattice”.
For a deep quantum well, where the energy of the confined state is well below the top of the confining barrier, the confinement energy of the nth energy level in the conduction band well, ECn
may be estimated with the expression for an infinitely-deep 1D quantum well of width l,
n2 h̄2 π 2
.
(4)
2mn l2
This energy of the lowest-confined state energy, EC1 , defines the slightly raised, lower edge of
the conduction band in the quantum well material. The electron mass mn is the conduction band
effective mass of the quantum well material. The confinement in valence band, EV n , depends
En =
3-3
E
EC (x)
EC1
l
x
ψ(x)
x
F IGURE 4: Semiconductor quantum well.
on the effective mass of the valence band holes, m∗h , and the highest-confined state energy, EV 1 ,
defines the slightly lowered, upper edge of the valence band in the quantum well material. For
GaAs, the effective masses are m∗e = 0.067me and m∗h = 0.082me . The new bandgap of the
quantum well material becomes
Eg(QW ) = Eg + EV 1 + EC1 .
(5)
Quantum wells are used, for example, to engineer the lasing energy (light color) of semiconductor lasers. They are also used to engineer the electronic transport, for example, by doping with
donors in the barrier of the well, from which the electrons fall into the well to give a 2D sheet of
charge. The 2D electron gas (2DEG) can move in the sheet with very high conductivity because
there are no dopant atoms in the sheet to interrupt the motion. Such manipulation of the optical
and electrical properties with heterostructures is refered to as “bandgap engineering”.
3.5
Semiconductor nanostructures
Quantum well confinement effects become important for layers that are on-the-order of 10 nm or
thinner in most materials. For most of the column-IV and III-V semiconductors, the cubic lattice
constants lay between 0.45 nm and 0.65 nm, which corresponds to a thickness of two molecular
layers, giving quantum well thicknesses of some 40 molecular layers or less. Figure 5 gives a cross
sectional image formed with scanning-tunneling microscopy (STM) of the atoms in a super lattice
made with InAs and GaSb layers.
If instead of a thin sheet of material, a thin wire of semiconductor is formed, then it is referred
to as a “quantum wire” or a semiconductor “nano wire.” Figure 6 gives a SEM micrograph of a
silicon nanowire grown at the University of Utah by Sun-Gon Jun and a diagram of a nanowire transistor under development by Justin Jackson. If only a small volume with 10-nm-scale dimensions
3-4
is formed, then is is referred to as a “quantum dot”. Figure 7 displays an atomic force microscope
(AFM) image of strained InAs quantum-dot islands grown in etched GaAs trenches and a transmission electron microscope (TEM) cross-sectional image through strained GaAs nanowires on a
GaP substrate.
InSb –like
InSb –like
[001]
GaSb
GaSb
InAs
GaSb
InAs
GaSb
InAs
M. Weimer Texas A&M University
F IGURE 5: STM image of the cleaved edge of an InAs/GaSb superlattice.
3-5
Nanowire
MOSFET
nanowire channel
wrap around
poly Si gate
n-poly Si drain
spin on glass,
PECVD SiO2
n-Si source
p-Si substrate
(a)
~ 100 nm
(b)
F IGURE 6: An electron micrograph of silicon nanowires (a) and a schematic cross section through a
nanowire transistor (b). The smallest lithographic features needed are ∼ 100 nm for the catalyst and gate
etch. (S. G. Jun and M. S. Miller, U Utah).
dot
chain
1 µm
(b)
(a)
F IGURE 7: Strained quantum dots and quantum wires. (a) An electron micrograph of InAs quantum dot
islands in GaAs trenches, and (b) TEM of GaAs wires on GaP substrate. (B. Ohlsson, Lund U., and M. S.
Miller, U Utah).
3-6