PHYSICS OF SOLAR CELLS:
PART I
Solar Cells Instructor: Dr. Alessia Polemi
OUTLINE: • Structure of semiconductors • Energy band structure and carrier concentration • pn junction (very short review)
INTRO
Semiconductor solar cells devices 1) Semiconductors absorb light and deliver a portion of the energy of the absorbed photons to carriers: e- and h+. 2) A semiconductor diode separates and collects the carriers and conducts the generated electrical current in a specific direction.
à a solar cell is simply a semiconductor diode designed to absorb and convert light energy from the sun into electrical energy.
Conventional solar cell
• Sunlight is incident from the top
• Metallic grid • Antireflective coating • n-type and p-type are
brought together to form a junction.
• The diode’s other electrical contact is formed by a metallic layer on the back of the solar cell.
STRUCTURE OF SEMICONDUTORS
Solar cells can be fabricated from various semiconductor materials • most commonly silicon (Si) – crystalline, polycrystalline, and
amorphous • GaAs • GaInP • Cu(InGa)Se2 • CdTe
Properties of semiconductors
Solar cell materials are chosen on the basis of: • how well their absorption characteristics match the solar
spectrum • their cost of fabrication
Silicon is the most popular choice • its absorption characteristics are a fairly good match to the
solar spectrum • Silicon fabrication technology is well developed (electronics
industry) à IC
Properties of semiconductors
Structure of Solids The solids used in photovoltaics can be broadly classified as crystalline, polycrystalline, or amorphous. Crystalline à single-crystal materials; Polycrystalline à materials with crystallites (crystals or equivalently grains) separated by disordered regions (grain boundaries); Amorphous àmaterials that completely lack long-range order
Crystalline: - long-range order represented by the lattice and a basic building block (the unit
cell) - Fixed positions (there are only 14 crystal lattices possible in a three-dimensional
universe)
Structure of Solids Polycrystalline: - composed of many single-crystal regions = grains, that exhibit long-range order - the transition regions between the grains are termed grain boundaries - grain boundaries have a significant influence on physical properties (they can
getter dopants or other impurities, store charge in localized states arising from bonding defects)
- grain boundaries are broadly classified as open or closed. An open boundary is easily accessible to gas molecules; a closed boundary is not.
- Diffusion coefficients are generally an order of magnitude larger along such boundaries than those observed in single-crystal material
Amorphous solids: - disordered materials that contain large numbers of structural and bonding
defects - no long-range structural order (there is no unit cell and lattice) - composed of atoms or molecules that display only short-range order, at best - there is no uniqueness in the amorphous phase, e.g. there are a myriad of
amorphous silicon-hydrogen (a-Si:H) materials that vary according to Si defect density, hydrogen content, and hydrogen-bonding details.
Structure of Solids
Phonon spectra
• Because of the interactions among its atoms, a solid has vibrational modes. The quantum of vibrational energy is called phonon.
• Phonons can be involved in heat transfer, carrier generation (thermal or in conjunction with light absorption), carrier scattering, and carrier recombination processes.
• They behave like particles, e.g. when an electron in a solid interacts with a vibrational mode, the event is best viewed as an interaction between two types of particles, electrons and phonons
Phonon spectra
• Phonons have a dispersion relationship Epn(k) [k wave vector of the vibrational mode]. Analogous to the dispersion relationship for light (Ept=hc|k|/2π)
• Epn(k) is more complicated and gives what is called phonon
spectrum or phonon energy bands in a solid. In the case of both phonons and photons, k has the interpretation of particle momentum.
Phonon spectrum for Si and GaAs
Phonon bands in two crystalline solids: (a) silicon, and (b) gallium arsenide
• O refers to optical branches (these modes can be strongly involved in optical properties)
• A refers to acoustic branches (so called because frequencies audible to the human ear)
• T and L refer to the transverse and longitudinal modes
SINGLE-CRYSTAL, MULTICRYSTALLINE, AND MICROCRYSTALLINE SOLIDS
Phonon spectra
Si
1) Of the order of 108 cm-1
2) Notice! Superimpose the plot for all photon energies Ept<3eV (solar spectrum) versus k onto phonon spectrum
• the momentum of the photons constituting the majority of the solar spectrum is very small compared to the momentum of phonons
• phonon energies are of the order of 10-2eV,10-1eV. Photon energies, at least those in the NIR, VIS, NUV range, are of the order of 1eV
Phonon spectra
NANOPARTICLES AND NANOCRYSTALLINE SOLIDS
• As particle or grain size becomes smaller, the surface-to-volume ratio increases and surface-stress effects on bulk and surface phonon modes become more important
AMORPHOUS SOLIDS
• There is no Brillouin zone in reciprocal space because there is no unit cell in real space (there is no crystal lattice)
• It is difficult to distinguish between acoustic and optical phonons • Phonons play the same critical roles in electron transport, heat
conduction, etc., as they do in crystalline solids
Phonon spectra
ENERGY BAND STRUCTURE AND
CARRIER CONCENTRATION
Energy Band Structure We have already looked at how by solving SE for periodic crystalline structures we obtain electronic properties
∇2ψ +2m!2
E −V( )ψ = 0
Motion of the e- in the crystal is like that of an e- in free space if its mass, m, is replaced by an effective mass m∗
E
Energy Band Structure
Frequently, the detailed E(k) plot is not needed. In many applications:
• BandGap EG, • CB edge EC, • VB edge EV,
• and local vacuum level EVL (energy needed to escape the material) or equivalently electron affinity χ = energy to promote
an e- from the bottom of the CB to the vacuum level
Software wxAMPS
Single crystal
Even polycrystalline and amorphous materials exhibit a similar band structure (over short distances, the atoms are arranged in a periodic manner and an electron wavefunction can be defined). However, there are a large number of localized energy states within the mobility gap that complicate the analysis.
Energy Band Structure
Localized states: - acceptor-like - donor-like - amphoteric in nature
Software wxAMPS
Energy Band Structure Acceptor states: neutral when empty and negative when occupied by an electron (ionized) Donor states: neutral when occupied by an electron and positive when empty (ionized)
Software wxAMPS
Amphoteric gap states: can be occupied by none, one , o r two va lence electrons. Their charge state depends on their occupancy
Localized (gap) states can serve as sources of carriers for the bands = doping It is possible to have so many gap states present in a material that they form a band within the energy gap. Such a band within the energy gap is termed an intermediate band (IB).
Software wxAMPS
Energy Band Structure
∇2ψ +2m!2
E −V( )ψ = 0
Energy Band Structure
Very resourceful equation!
Excitons are multi-electron solutions that may be viewed as an electron bound to a hole via Coulombic attraction
Possible solutionà Excitons: can be involved in the light absorption process in solar cell materials
Energy Band Structure
1) Binding energy is dictated by the Coulombic attraction à materials that polarize more have lower binding energies (i.e., the binding energy correlates inversely with the dielectric constant)
2) They can be created by photon absorption
3) They can be mobile in a solid. When they move, energy moves, but not net charge. Since they are not charged, they can only move by diffusion.
4) If excitons are produced by light absorption in a solar cell
and are to be utilized, then some process must also be present to convert the exciton into at least one free negative–positive charge carrier pair
Energy Band Structure Where are e- and h+ located in terms of energy bands?
Density of states
Carrier Concentration
Density of states in the CB gC E( ) =mn* 2mn
* E −Ec( )π 2!3
cm−3eV −1"# $%
Density of states in the VB gV E( ) =mp* 2mp
* EV −E( )π 2!3
cm−3eV −1"# $%
@thermal equilibrium (i.e. at a uniform temperature with no external injection or generation of carriers), probability of finding an e- (or h+) is given by the Fermi function
Carrier Concentration
T=0
f (E) = 1
1+ exp E −EF
kBT"
#$
%
&'
kB = Boltzmann constant
EF =!2
2m3π 2NV
!
"#
$
%&
2/3
EF
for e-
Carrier Concentration
n0 = gC(E) f (E)Ec
Ec+χ∫ dE
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−≈
TkEEEf
B
Fexp)(gC(E) = 4π (2mn* )3/2h−3(E −Ec )
1/2
= A(E −Ec )1/2
Density of states in the CB Fermi-Dirac function ≈ Boltzmann function
Carrier Concentration
n0 = Electron concentration in the CB Nc = Effective density of states at the CB edge Ec = Conduction band edge, EF = Fermi energy kB = Boltzmann constant, T = Temperature (K)
n0 = NC exp −(EC −EF )kBT
"
#$
%
&'
Effective Density of States at CB Edge
Nc = Effective density of states at the CB edge, me* = Effective mass of the electron
in the CB, k = Boltzmann constant, T = Temperature, h = Planck’s constant
NC = 22πmn
*kBTh2
!
"#
$
%&
3/2
Carrier Concentration
Carrier Concentration
p0 = Hole concentration in the VB Nv = Effective density of states at the VB edge
Ev = Valence band edge, EF = Fermi energy kB = Boltzmann constant, , T = Temperature (K)
p0 = NV exp −(EF −Ev )kBT
"
#$
%
&'
Effective Density of States at VB Edge
Nv = Effective density of states at the VB edge, mh* = Effective mass of a hole in
the VB, k = Boltzmann constant, T = Temperature, h = Planck’s constant
NV = 22πmh
*kBTh2
!
"#
$
%&
3/2
n0p0 = ni2 = NCNV exp −
EG
kBT"
#$
%
&'
Mass Action Law
The np product is a “constant”, ni2, that depends on the material properties NC, NV,
Eg, and the temperature. If somehow n is increased (e.g. by doping), p must decrease to keep np constant.
Mass action law applies
in thermal equilibrium
and
in the dark (no illumination)
ni = intrinsic concentration
Intrinsic semiconductors
In un-doped (intrinsic) semiconductor, n. of electrons in CB and n. of holes in VB are equal
no = po = ni (ni =intrinsic carrier concentration)
ni = NCNV exp −EG
2kBT"
#$
%
&'
The Fermi energy in intrinsic semiconductor
Ei =EV +EC
2+kBT2ln NV
NC
!
"#
$
%&
typically very close to the middle of the bandgap
Intrinsic semiconductors
ni: very small compared with the densities of states and typical doping densities (ni ≈ 1010 cm−3 in Si) and intrinsic semiconductors behave very much like insulators à they are not good conductors of electricity
ni ∝ exp −EG
2kBT#
$%
&
'(
Intrinsic concentration.swf
Intrinsic semiconductors
n. of e- and h+ in their respective bands, and hence the conductivity, can be controlled through the introduction of specific impurities, or dopants, called donors and acceptors
Extrinsic semiconductors
All impurities introduce additional localized electronic states into the band structure, often within the bandgap. - If the energy of the state ED introduced by a donor atom is
sufficiently close to the conduction band edge (within a few kBT), there will be sufficient thermal energy to allow the extra e- to occupy a state in the CB. The donor state will then be positively charged (ionized).
- Similarly, an acceptor atom will introduce a negatively charged (ionized) state at energy EA.
Extrinsic semiconductors
Usually donors and acceptors are assumed to be completely ionized so that no≈ND in n-type material and po≈NA in p-type material. The Fermi energy: EF = Ei + kBT ln
ND
ni, EF = Ei − kBT ln
NA
ni
Extrinsic semiconductors
Extrinsic semiconductors
- Very large concentration of dopants à the dopants can no longer be thought of as a minor perturbation to the system, i.e. effect on the band structure à bandgap narrowing (BGN) and thus increase of intrinsic carrier concentration
- BGN is detrimental to solar cell performance; solar cells are typically designed to avoid this effect
- BGN may be a factor in the heavily doped regions near the solar cell contacts
Bandgap narrowing
Extrinsic Semiconductors: n-Type
(a) The four valence electrons of As allow it to bond just like Si but the fifth electron is left orbiting the As site. The energy required to release to free fifth-electron into the CB is very small. (b) Energy band diagram for an n-type Si doped with 1 ppm As. There are
donor energy levels just below Ec around As+ sites.
As has 5 electrons
Extrinsic Semiconductors: n-Type
edhd
ied eN
NneeN µµµσ ≈⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
2
Nd >> ni, then at room temperature, the electron concentration in the CB will nearly be equal to Nd, i.e. n ≈ Nd
A small fraction of the large number of electrons in the CB recombine with holes in the VB so as to maintain np = ni
2
n = Nd and p = ni2/Nd
np = ni2
Extrinsic Semiconductors: p-Type
(a) Boron doped Si crystal. B has only three valence electrons. When it substitutes for a Si atom one of its bonds has an electron missing and therefore a hole. (b) Energy band
diagram for a p-type Si doped with 1 ppm B. There are acceptor energy levels just above Ev around B- sites. These acceptor levels accept electrons from the VB and
therefore create holes in the VB.
B has 3 electrons
Extrinsic Semiconductors: n-Type
haea
iha eN
NneeN µµµσ ≈⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
2
Na >> ni, then at room temperature, the hole concentration in the VB will nearly be equal to Na, i.e. p ≈ Nd
A small fraction of the large number of holes in the VB recombine with electrons in the CB so as to maintain np = ni
2
p = Na and n = ni2/Na
np = ni2
Semiconductor energy band diagrams
Energy band diagrams for (a) intrinsic (b) n-type and (c) p-type semiconductors. In all cases, np = ni
2. Note that donor and acceptor energy levels are not shown. CB = Conduction band, VB = Valence band, Ec = CB
edge, Ev = VB edge, EFi = Fermi level in intrinsic semiconductor, EFn = Fermi level in n-type semiconductor, EFp = Fermi level in p-type semiconductor. χ is the electron affinity. Φ, Φn and Φp are the work functions for the intrinsic, n-
type and p-type semiconductors
Compensation Doping Compensation doping describes the doping of a semiconductor with both donors and acceptors to control the properties.
ad NNn −=ad
ii
NNn
nnp
−==
22
iad nNN >>−More donors than acceptors
More acceptors than donors ida nNN >>−
da NNp −=da
ii
NNn
pnn
−==
22
PN JUNCTION:
DARK CURRENT
(very short review)
Basic of pn junction Remember: this is the simplest solar cell possible
Properties of the pn junction
(a) The p- and n- sides of the pn junction before the contact.
(b) The pn junction after contact, in equilibrium and in open circuit (depletion region, SCL). Built-in field E0 tries to drift holes back into p and electrons into n.
(c) Carrier concentrations along the whole device: at all points, npoppo = nnopno = ni
2
pn Junc&on
Depletion Widths NaWp = NdWn
Donor concentration
(d) Net space charge density ρnet across the pn junction. Charge neutrality.
Acceptor concentration
(e) The electric field across the pn junction is found by integrating ρnet in (d).
pn Junc&on
ερ )(net x
dxd
=E
Field (E) and net space charge density Net space charge density
Eo = −eNdWno
ε=eNaWpo
ε
Emax =−eNaNdWo
ε Na + Nd( )
Field in depletion region
∫−=x
Wpdxxx )(1)( netρ
εE
(f) The potential V(x) across the device. Contact potentials are not shown at the semiconductor-metal contacts now shown.
(g) Hole and electron potential energy (PE) across the pn junction. Potential energy is charge Î potential = q V
pn Junc&on
E = − dVdx
V x( ) = E x( )dx =∫
Vo x ≥Wn
V0 −eNd
2εx +Wn
2( ) 0 ≤ x ≤Wn
eNa
2εx −Wp
2( ) Wp ≤ x ≤ 0
0 x ≤Wp
%
&
'''
(
''''
eV0 ≈ EG
eV0 = kBT lnNaNd
ni2
Some algebra
pn Junc&on
Depletion region width
( ) 2/12
⎥⎦
⎤⎢⎣
⎡ +=
da
odao NeN
VNNW ε
where Wo = Wn+ Wp is the total width of the depletion region under a zero applied voltage
By establishing continuity at the junction:
LEFT: Consider p- and n-type semiconductor (same material) before the formation of the pn junction, separated from each and not interacting.
ENERGY BANDS
Open circuit pn-junction 1) Same EF 2) Away from M, in the n-type, Ec-EFn
is the same 3) Away from M, in the p-type, EFp-
Ev is the same 4) The bandgap must be the same Ec-
Ev in the two materials 5) Ec and Ev must bend in the SCL
a) e- diffuse from n to p, they deplete the n-side near the junction à Ec must move away from Efn
b) h+ diffuse from p to n, they deplete the p-side near the junction à Ev must move away from Efn
Energy band diagrams for a pn junction under (a) open circuit and (b) forward bias
P0 = e−qVo /kBT P = e−q Vo−V( )/kBT
Energy band diagrams for a pn junction under reverse bias (Shockley model)
Bias.swf
Forward biased pn junction and the injection of minority carriers. The negative polarity of the supply reduces the potential barrier V0 by V. Potential barrier against diffusion is reduced to V0-Vàmore injection of minority carriers
Forward Biased pn Junction
- Vo + + V -
Forward Bias: Diffusion Current
⎥⎦
⎤⎢⎣
⎡−⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛= 1exp
2
hole, TkeV
NLneDJ
Bdh
ihD
Einstein relation:
Dh = µhkBT/e,De = µekBT/e
Assuming Boltzmann statistics + some algebra
Hole diffusion current in n-side in the neutral region
Similar expression for JD,elec in the p-region.
The total current anywhere in the device is constant. Just outside the depletion region it is due to the diffusion of minority carriers.
Forward Bias: Total Current
Forward Bias: Diffusion Current
Ideal diode (Shockley) equation
J = Jo expeVkBT
!
"#
$
%&−1
(
)*
+
,-
Jo =eDh
LhNd
!
"#
$
%&+
eDe
LeNa
!
"#
$
%&
'
()
*
+,ni
2
Reverse saturation current
Jo depends strongly on the material (e.g. bandgap) and temperature
Forward Bias: Diffusion Current
J = J0 expeVηkBT!
"#
$
%&−1
(
)*
+
,-
Diode (Shockley) equation
η =1÷ 2Ideality factor
Schematic sketch of the I-V characteristics of Ge, Si, and GaAs pn junctions.
V
I
0.2 0.6 1.0
Ge Si GaAs
Ge, Si and GaAs Diodes: Typical Characteristics
1 mA
Forward Bias: Recombination Current
Some minority carries also recombine in the SCLà supply provides also for the electrons and holes lost in SCLà Recombination Current
Typical I-V characteristics of Ge, Si and GaAs pn junctions as log(I) vs. V. The slope indicates e/(ηkBT)
Typical I-V characteristics of Ge, Si and GaAs pn junctions
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