Introduction to
Electronic Band Structures and
Homo/Hetero-interfaces
in Inorganic Semiconductors
Kris T. Delaney
University of California, Santa Barbara
02/21/14
Solid-state Physics
+
Quantum mechanics
Topics covered
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Insulators vs. metals vs. semiconductors
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Electronic band structure of a crystal
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–
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Definition and formation
Allowed quantum processes
Band engineering
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Doping
Interfaces in photovoltaic devices
What is a semiconductor?
In a crystal, discrete quantum energy levels form quasicontinuum “bands”
conduction
band
conduction
band
band gap
valence
band
valence
band
Metal
Semiconductor
conduction
band
band gap
valence
band
Insulator
Semiconductor: energy gap is small; thermal excitation of a few electrons to conduction states
Electrons occupy lowest energy levels, subject to Pauli principle
What is a semiconductor?
In a crystal, discrete quantum energy levels form quasicontinuum “bands”
conduction
band
conduction
band
band gap
valence
band
valence
band
Metal
Semiconductor
conduction
band
band gap
valence
band
Insulator
Semiconductor: energy gap is small; thermal excitation of a few electrons to conduction states
Electrons occupy lowest energy levels, subject to Pauli principle
Only allowed in quantum states. Non-classical transition with added energy.
Transport: Drift vs. Diffusion
Diffusion:
Smoothing of concentration gradients (inhomogeneities)
Drift:
Average motion in (internal or applied) electric field
e-- “n” transport
Applied
Electric
Field
A Simple Model of Drift in SCs
Conduction bands
Valence bands
Position in material
A Simple Model of Drift in SCs
Conduction bands
Applied
Electric
Field
Valence bands
Position in material
Filled valence band, empty conduction band → No conductivity
“electrons can't hop to neighbors”
A Simple Model of Drift in SCs
Conduction bands
Valence bands
Position in material
Filled valence band, empty conduction band → No conductivity
“electrons can't hop to neighbors”
Excite carrier (thermal, photon, …)
A Simple Model of Drift in SCs
e-- “n” transport
Conduction bands
Applied
Electric
Field
Valence bands
“
+
”
h “p” transport
Position in material
Filled valence band, empty conduction band → No conductivity
“electrons can't hop to neighbors”
Excite carrier (thermal, photon, …)
Semiconductors have ~1010 cm-3 excited carriers at room temperature (out of ~1024)
Insulators have big gap
: ~0 carriers at room temperature
How do bands form? Chemistry view
Antibonding MO
Overlapping H 1s orbitals:
Bonding MO
Bonding
Wave
functions
Antibonding
●
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Electron
density
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Atomic orbitals hybridize
● Bonding & antibonding MOs
Electrons occupy lowest levels first
Electrons have Fermi statistics
● Pauli principle
● Max. one electron / state
Bonding vs. Anti-bonding: brief math
Assume many-body correlations are weak
Assume MOs are single-particle states
Approximate one-body molecular orbitals from Linear Combination of Atomic Orbitals (LCAO)
Variational LCAO:
Calculate the energy expectation value for any c1, c2:
Minimize the energy. Two solutions:
“bonding” MO
“anti-bonding” MO
Molecules → crystals
Consider an infinitely long chain of H atoms:
LCAO molecular orbitals:
Generalization of H2 result:
Questions:
● What values of k are allowed?
● What is the energy for each k?
● What does k mean?
with
Wave functions
Largest possible k (fastest phase oscillation)
Smaller k (periodicity of 6 unit cells)
Energies of different k states
Band is half filled (one electron per atom)
No energy gap between filled and empty
→ metallic system
Energy of bonding state (k=0)
lower E than pure antibonding
Energies of different k states
Bands vs. lattice spacing (overlap)
Band is half filled (one electron per atom)
No energy gap between filled and empty
→ metallic system
Energy of bonding state (k=0)
lower E than pure antibonding
Low mobility
High mobility
Peierls Instability
Peierls Instability
Metal-insulator transition driven by dimerization of chain
Electronic structure intricately linked to crystal structure
Band Structures: Physics Perspective
at band edge
As in free-electron case, states are
delocalized throughout crystal
Example Band Structure: Si diamond
Real-life 3D crystals with multi-orbital elements are more complicated
But the same concepts are valid
Band Engineering with Chemistry
Schematic diagram of band structures of three semiconductors:
Descending rows in periodic table → reduce size of semiconductor band gap
Quantum Processes: Photoemission
Photoemission and inverse photoemission: non-neutral excitation
→ add/remove electrons
Defines the band structure “in real life”:
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Strictly the E(k) relationship of full many-body states with correlations
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As measured in PES and inverse PES
Quantum Processes: Optics
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Optical absorption: the first step of a photovoltaic
Absorption rules and rates are derived from Fermi's golden rule
Result: k and E are both conserved
Maximum work from photocarriers
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Photoelectrons at high E thermalize in ~fs
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e-h pairs at band edges survive for ~s
Optical absorption in indirect gaps
Absorption rates (therefore intensities) are very low at the minimum E gap
Rates are high for across-gap transitions, but extra energy lost to heat
Direct vs. indirect gap: absorption
(assuming homogeneous medium)
Excitons
Binding: strong
Spatially localized
Binding: weak
Spatially delocalized
Note:
The optical gap can be very different from the band gap
Urbach tails, optical selection rules, direct vs. indirect band gaps, exciton binding energy
Quantum processes: lose e-h pairs
Spontaneous emission
Auger Recombination
Shockley-Read-Hall
Recombination
Quantum processes: lose e-h pairs
Spontaneous emission
Auger Recombination
Shockley-Read-Hall
Recombination
Impact Ionization
Produce 2 e-h pairs from one photon
Summary (Pt. 1)
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Quantum mechanics → discrete energy levels
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–
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Electrons jump non-classically between states
–
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Broaden into “bands” in crystalline solids
Band structures often complex in 3D
Near band edges, behave like free electrons of m*
Energy conservation required: often photons
Generation of e-h pairs by optical absorption
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–
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Loss of pairs by recombination, SRH, Auger
Control loss with crystal quality
Or extract carriers before loss...
Extracting photoexcited charges
Now that we understand absorption spectra, k conservation, and rapid thermalization of carriers,
it is sufficient to consider only extrema of valence and conduction bands vs. position:
Conduction bands
Valence bands
Position in material
How do we extract the photoexcited charges?
Engineered spatial asymmetry required...
Heterojunctions
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Heterojunction:
–
An interface between compositionally distinct
semiconductor films
Type I: straddling
Type II: staggered
Alignment is affected by
Chemistry
Interface states and dipoles (engineering)
Type III: broken
Type-II hetero-interface for PVs
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Common for thin-films
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CdTe, CIGS, OPV, etc.
An efficient carrier-extraction method if the carrier diffusion length is long compared
to domain size / film thickness
Fermi-Dirac Statistics
Probability of electron occupying a state
Low temperature: lowest-E states filled first
Higher temperature: thermally excite elns
Long tail is sufficient to excite 1 in ~1014 elns
at room temperature
→ “intrinsic conductivity” of semiconductor
Band engineering by doping
Deliberate implantation of impurities
Typical densities ~1017 cm-3
(roughly 1 per 1,000,000 host atoms)
Intrinsic SC
p-type SC
n-type SC
pn junction band bending
Step 1: pn junction out of equilibrium
Step 2: Chemical potential difference: electrons flow to p-type, holes flow to n-type
pn junction is a diode
Optimal depth of pn interface
Positioning of pn interface affects photovoltaic efficiency
A highly engineered interface stack
Summary (Pt. 2)
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Collection of photo-excited carriers:
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Interface required
Heterojunction stack of thin films
pn junction if suitable dopants
Conclusions
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Optically generate e-h pairs
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–
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Drive charges apart using:
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Weak exciton binding in crystalline inorganics
Easy charge separation
Type-II band alignment, or
Internal electric field set up by pn junction
Challenges:
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Absorb strongly
Carrier diffusion length vs. device layout & traps