Band Alignment and Graded
Heterostructures
Guofu Niu
Auburn University
Outline
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Concept of electron affinity
Types of heterojunction band alignment
Band alignment in strained SiGe/Si
Cusps and Notches at heterojunction
Graded bandgap
Impact of doping on equilibrium band diagram
in graded heterostructures
Reference
• My own SiGe book – more on npn SiGe HBT base
grading
• The proc. Of the IEEE review paper by Nobel
physics winner Herb Kromer – part of this lecture
material came from that paper
• The book chapter of Prof. Schubert of RPI – book
can be downloaded online from docstoc.com
• http://edu.ioffe.ru/register/?doc=pti80en/alfer_e
n.tex - by Alfreov, who shared the noble physics
prize with Kroemer for heterostructure laser work
3
Band alignment
• So far I have intentionally avoided the issue of band
alignment at heterojunction interface
• We have simply focused on
– ni^2 change due to bandgap change for abrupt junction
– Ec or Ev gradient produced by Ge grading
• We have seen in our Sdevice simulation that the final
band diagrams actually depend on doping
– A Ec gradient favorable for electron transport is obtained
for forward Ge grading in p-type (npn HBT)
– A Ev gradient favorable for hole transport is obtained for
forward Ge grading in n-type (pnp HBT)
Electron Affinity – rough picture
• Neglect interface between vacuum and semiconductor,
vacuum level is drawn to be position independent
• The energy needed to move an electron from Ec to
vacuum level is called electron affinity.
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Electron Affinity Model
• The electron affinity model is the oldest model
invoked to calculate the band offsets in
semiconductor heterostructures (Anderson,
1962).
• This model has proven to give accurate
predictions for the band offsets in several
semiconductor heterostructures, whereas the
model fails for others. We first outline the basic
idea of the electron affinity model and then
discuss the limitations of this model.
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Semiconductor vacuum interface
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The band diagram of a semiconductor-vacuum
interface is shown.
Near the surface, the n-type semiconductor is
depleted of free electrons due to the pinning of the
Fermi level near the middle of the forbidden gap at
the semiconductor surface. Such a pinning of the
Fermi level at the surface occurs for most
semiconductors.
The energy required to move an electron from the
semiconductor to the vacuum surrounding the
semiconductor depends on the initial energy of the
electron in the semiconductor.
Promoting an electron from the bottom of the
conduction band to the vacuum beyond the reach
of image forces requires work called the Electron
Affinity. Lifting an electron from the Fermi level
requires work called the Work Function, which is
defined the same way in semiconductors as it is in
metals. Finally, raising an electron from the top of
the valence band requires the ionization energy .
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Interface of Two Semiconductors
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Next consider that two semiconductors are brought into physical contact.
The two semiconductors are assumed to have an electron affinity of chi1 and
chi2 and a bandgap energy of Eg1 and Eg2 , respectively, as illustrated below.
Near Surface bending and image force has been neglected – which actually can
change the conclusion (this effect is typically neglected though in practice)
Energy balance of moving an electron from vacuum to semiconductor “1”, from
“1” to “2”, and from “2” to vacuum must be zero, that is
Interface of 2 semiconductors: Delta Ev
• The valence band discontinuity naturally follows:

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By convention delta E_v
definition is Ev1 – Ev2.
Limitations of electron affinity
model
• The delta E_c and delta E_v equations from electron
affinity model are valid only if the potential steps caused
by atomic dipoles at the semiconductor surfaces and the
heterostructure interfaces can be neglected.
• In this case, the knowledge of the electron affinities of
two semiconductors provides the band offsets between
these two semiconductors.
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Band Alignment Types
• Abrupt “heterojunction” band diagram – the abrupt
changes in Ec / Ev are determined by Electron Affinity
and bangap changes
• Three distinct band alignments are possible – type I, II,
and III
Straddling, e.g.
SiGe/Si,
AlGaAs/GaAs
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Staggering,
e.g. InP/InSb
Broken-gap,
GaSb/InAs
Strained SiGe/Si band alignment
• Type 1
• Mostly valence band offset
• To first order
– Delta Ev = 0.74 * xMole eV
– Delta Ec \approx 0
Connecting Hetero Materials of
Opposite Doping
• Electrons / holes
will move around,
like in
homojunction
• Potential drops will
develop, until Fermi
level is the same on
both sides
• Far away from
interface, potentials
are flat
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Cusps / Notches at Heterojunction
• Some cusp or notch must form in the conduction
or valence band, depending on the details of the
system.
• Exactly what happens and what the cusps look
like depends on many details, you must solve the
Poisson equation properly for a specific case.
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General Band Alignment PN
Heterojunction
• Assuming material 1 is p-type, material 2 is ntype – for drawing purpose, chi1 > chi2
• We will make these assumptions to allow
“quantitative” drawing
P-SiGe/n-Si Heterojunction (EB, CB
junction of NPN HBT)
• Before connection
P-SiGe/n-Si Heterojunction (EB
junction of NPN HBT)
• After connection
P-Si / n-SiGe (EB, CB of PNP SiGe HBT)
forward Ge grading in p-type
Retrograding of Ge in p-type
Forward Ge grading in n-type
Retrograding of Ge in n-type
p-SiGe/p-Si abrupt heterojunction
N-SiGe/n-Si abrupt heterojunction
Connecting Materials
• Electrons / holes
will move around,
like in
homojunction
• Potential drops will
develop, until Fermi
level is the same on
both sides
• Far away from
interface, potentials
are flat
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Cusps / Notches at Heterojunction
• Some cusp or notch must form in the conduction
or valence band, depending on the details of the
system.
• Exactly what happens and what the cusps look
like depends on many details, you must solve the
Poisson equation properly for a specific case.
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• POINT, APEX: as a : a point of transition (as from one
historical period to the next) : TURNING POINT; also :
EDGE, VERGE <on the cusp of stardom> b : either horn
of a crescent moon c : a fixed point on a mathematical
curve at which a point tracing the curve would exactly
reverse its direction of motion d : an ornamental
pointed projection formed by or arising from the
intersection of two arcs or foils e (1) : a point on the
grinding surface of a tooth (2) : a fold or flap of a
cardiac valve
- cus·pate /'k&s-"pAt, -p&t/ adjective
- cusped /'k&spt/ adjective
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Isotype Heterojunction (n-n or pp)
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Biased Isotype Heterojunction
• Apply bias U -> Fermi
level / total potential
changes by qU
• Majority (electrons here)
carriers conduct current
• It is easier for electrons to
move from left to right
than from right to left
• We may use an isotype
heterojunction to inject
majority carriers from the
wide band gap material
into the small band gap
material
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Strained SiGe on Unstrained Si
bulk
SiGe
Si
strained
SiGe
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Graded Bandgap Structures
• In regular semiconductor heterostructures, the
chemical transition from one semiconductor to
another semiconductor structure is abrupt.
• In the preceding discussion, we have seen that
the periodic potential and the band diagram are
nearly as abrupt as the chemical transition. That
is, the transition of the periodic potential and of
the band diagram occur within a few atomic
layers of a chemically abrupt semiconductor
heterostructure.
• In graded heterostructures, the chemical
transition from one semiconductor to another
semiconductor is intentionally graded.
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Assuming two semiconductors “A” and “B” are
chemically miscible, the mixed compound, also called
semiconductor alloy, is designated by the chemical
formula A1-xBx, where x is the mole fraction of
semiconductor A in the mixed compound.
 The mole fraction is also designated as the chemical
composition of the compound A1-xBx . Most
semiconductors of practical relevance are completely
miscible.
 Assume further that the gap energy of A and B are
different, and that the bandgap energy depends on the
composition. The dependence of Eg on the composition
is usually expressed in terms of a parabolic (linear plus
quadratic) dependence. The Eg of the alloy A1-xBx is then
given by


where the first two summands describe the linear
dependence of the gap and the summand (1 ) b
describes the quadratic dependence of the gap. The
parameter b is called the Bowing parameter.
 For some semiconductor alloys, . (AlAs) (GaAs) , the
bowing parameter is vanishingly small. The bandgap of
the alloy is then given by

Equations (17.6) and (17.7) are valid for homogeneous
bulk semiconductors. However, the validity of the
equations is not limited to bulk semiconductors. They
also apply to the local bandgap of graded structures. We
have seen in the preceding section that the atomic
potentials and the energy bands closely follow the
composition in a chemically abrupt heterojunction.
Accordingly, the band edges and the gap energy will
follow the chemical composition of graded
semiconductors.

Example of linearly graded heterostructure is shown below
 the figure shows a narrow Eg semiconductor A, a wide Eg
semiconductor B, and a linearly graded transition region
“A1-xBx”, with thickness Delta Z.
Quasi-fields and the possibility of moving electrons
and holes same direction
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Graded base bandgap transistor
vision
• Kroemer also envisioned graded-gap hetero
bipolar transistors which enhance the minority
carrier transport through the base.
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Graded SiGe Base NPN HBT –
Equilibrium Status
• Constant Fermi level -> Ec slope instead of Ev slope as base is p-type,
• Electric field will develop such that holes will not move (zero current)
• End result is: Quasi-field + electrostatic field = 0, no net hole “drift”,
diffusion is zero due to uniform doping
• Electrons “drift” due to the e-field, but drift current is cancelled by
diffusion, as electron concentration n is higher near collector
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Graded SiGe Base NPN HBT – with
Bias
• Electron concentration at end of base is lower, as the EB junction is
forward biased and will inject electrons into the base, so electron
diffusion and drift will be along the same direction
• Hole quasi-fermi level gradient must be small due to high p (uniform
p), Jp is always much less than Jn (beta >>1 by design)
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Lattice constant of semiconductor
alloys (unstrained or relaxed)
• In graded semiconductor structures, the composition
of the semiconductor is varied. This variation in
chemical composition is not only accompanied by a
change of the bandgap energy, but also (in general,
but not necessarily) by a change in the lattice
constant. The change in lattice constant is, for all
semiconductor alloys, governed by Vegard’s law.
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lattice-matched graded
semiconductors
• For most graded semiconductor structures, it is
imperative that the lattice constant does not change as
the composition of the alloy is varied. Such structures
are called lattice-matched graded semiconductors, e.g.
in our SiGe HBT case, strained SiGe on unstrained Si
substrate.
• If semiconductors are not lattice matched, microscopic
defects occur when the composition is varied. These
defects degrade the quality, e.g. the radiative
efficiency, of the semiconductor.
• Lattice matching required for low defect density
– This is particularly important for minority carrier devices
(HBTs, lasers)
– This is not so important for majority carrier devices
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Relaxation of strain
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Semiconductor heterostructures
• Ideal: Heterostructures are formed by
semiconductors with the same crystal structure
and the same lattice constant: An example is Al
Ga As on GaAs
• Often: Mismatched structures result in misfit
dislocations defects which act as recombination
centers. An example is GaN on sapphire
• Diagrams of energy gap-versus-lattice-constant
for of different semiconductors
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Kromer’s Central Design Principle of
hetero structure Devices
46Proc
IEEE 1982 review paper
Double Heterojunction Laser
Shockley’s 1948 Patent on Hetero
transistor
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Abstract of 1982 Kromer’s classic
HBT review paper
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