SEMICONDUCTORS

Eg

characterized by:

1) a band gap (Eg) < 3.5 eV 2) increasing conductivity with increasing temperature ne2 metals: n is ~ constant; τ decreases with increasing T   SCs: τ also decreases; n increases exponentially with T m E /2kT SCs: ne gB 623 sp3 SEMICONDUCTORS

624 sp3 SEMICONDUCTORS

GaAs band structure

625 PARABOLIC BAND PICTURE

GaAs band structure near valleys, bands can always be approximated as parabolas:

h Parabolic Band Approximation

22  k EEC * 2me 22 Eg  k EEV * 2mh

good approximation for SCs • electrons near CBM • holes near VBM 626 INTRINSIC and EXTRINSIC SEMICONDUCTORS

pure SC: intrinsic doped SC: extrinsic

thermally impurity promoted states in electrons bandgap EF E g empty ED EF states - “holes”

• negligible impurities • impurities form states in gap • electrons thermally promoted (e- donors or acceptors) across gap • electrons easily excited into

• n = p = ni conduction band from donor. • n > p ≠ ni DOPING IN SEMICONDUCTORS Adding foreign atoms (dopants) of Group V or Group III to a Group IV semiconductor produces n-type or p-type material.

dopant type: donor acceptor majority carrier: electrons holes 628 DONOR and ACCEPTOR LEVELS

629 ELECTRONIC CHARGE CARRIERS electron empty state Conduction band

e‐

• Electrons move amid empty states • Negative charges drift against field

hole Valence band

e‐ e‐ e‐ e‐ h+ e‐ e‐ e‐

• Electrons move by effective motion of positive holes • Positive charges drift with field Filled band e‐ e‐ e‐ e‐ e‐ e‐ e‐ e‐

• All wavevectors fully occupied, no net motion • Filled bands are inert 630 HOLES

Electrons are the only charge carriers. However, we may, whenever it is convenient, consider the current to be carried entirely by fictitious particles of positive charge that fill all those levels in the band that are unoccupied by electrons. The fictitious particles are called holes.

Applied field causes electrons and holes to move in opposite directions, but the current is in the same direction (opposite charges moving in opposite dir.)

It is convention to consider CB currents to be carried by electrons and VB currents to be carried by holes. Be careful not to mix the two concepts in a single band! 631 FERMI LEVEL SHIFTS WITH DOPING

The Fermi level moves up when electrons are added and down when electrons are removed, such that

f(EF) = 1/2.

With sufficient doping, the Fermi level can move into a band, giving a degenerate semiconductor (metallic). 632 EQUILIBRIUM CARRIER CONCENTRATION equilibrium  single temp, no applied fields (optical, electric, magnetic)

simplest case: direct gap, intrinsic semiconductor (n = p = ni)

concentration of electrons in CB:  nT() g () E f (,) ET dE  ee EC

number of occupancy [electrons] states at each dE

concentration of holes in VB: 0 p()TgEfETdE ()[1 (,)]  ee  633  0 nT() g () E f (,) ET dE p()TgEfETdE ()[1 (,)]  ee  ee EC  in the parabolic approximation:

* 3/2 * 3/2 1 2meds 1/2 1 2mhds 1/2 gEeg()22 ( E E ) gEe () 22 E E  0 2  2 

for E  Eg

1 fEe () exp[(EkT  ) /B ] 1

for Eg > 0.15 eV and |E - μ| >> kBT the Fermi function reduces to the exponential Boltzmann distribution: fEeB() exp[( E )/ kT ]  1 fEhB() exp[( E )/ kT ]  1

634 substituting these forms of g(E) and f(E) into the integrals gives:

3/2 * Emax 1 2meds ()/EkT 1/2 nT() e ( E Eg ) dE 2 22   Ec

* 3/2 0 1 2m ()/EkT 1/2 and p()TeEdE hds 2 22   Emin the results are:

3/2 * mkTeds B ()/EkTCC  ()/ EkT nT() 22 e NC () T e 2  3/2 * mkThds B ()/EkTVV  ()/ EkT pT() 22 e NV () T e 2  635 ()/EkTC  nT() NC () T e NC and NV : effective densities of states at bottom of CB, top of VB ()/ EkTV pT() NV () T e

mathematical tool

EEEgCV

  Eg / 2 EE EECV  CF [(EkTC )/ ]  2 (/2)EkTg kT nT() NTeCCC () NTe () NTe ()

The concentrations of both electrons and holes increase exponentially

with temperature with an activation energy of Eg/2. *The product n(T)p(T) is:

()/()/EkTEkTCV  nT()() pT NCV () T N () T e e

EkTg / 2 NTNTeCV ( ) ( ) n i

This product is independent of the position of the Fermi level. In other words, the product is a constant at a given temperature no matter the doping (n-type, intrinsic, or p-type).

it is an example of the LAW OF MASS ACTION heat ↔ e- + h+

Keq = [e][h] = np

For an intrinsic semiconductor: 3/2 3/4 kTB ** EkTg /2 nTii() pT () nTpT ii () ()  22  m edshds m e 2  637 FERMI LEVEL, INTRINSIC CASE

EECF  kT NC nNe CCF E E kT ln  n

EEFV subtract the  N kT V two equations pNe VFV E E kT ln  p n = p

EECV kT  N V EEFi = + ln 22NC

The Fermi level lies very close to the middle of the bandgap.

638 639 TEMPERATURE DEPENDENCE, INTRINSIC

Eg /2kTB nNNeiCV

Eg (eV) GaAs 1.42 Si 1.12 Ge 0.67

intrinsic (undoped)

640 TEMPERATURE DEPENDENCE, EXTRINSIC

Example: n-type silicon with 15 donor concentration ND ~ 10

EEAg /2 Promotion of electrons from donor to CB results in donor ionization (temp dependent)   EEEA  CD D De

extrinsic (doped)

641 TEMPERATURE DEPENDENCE of n, p

Donor ionization occurs when electrons are promoted to conduction band: D → D+ + e- Acceptor ionization occurs when electrons are promoted from the valence band to the acceptor state: A + e- → A-

zero ionization energy CB “saturation” - all dopants ionized

VB EgB/2kT dopants nei  neutral ‐1000/T TEMPERATURE DEPENDENCE of MOBILITY

vE =  ;  em / * mobility is the sum of drift scattering processes, often with one process dominant Si 1 11   ...  lattice impurities lattice scattering • ionized impurity scattering impurity 3/2 scattering i  T • acoustic phonon scattering 3/2 l  T other scattering mechanisms are possible (surface, defects)

643 RT MOBILITIES of SOME SEMICONDUCTORS

644 TEMPERATURE DEPENDENCE of SIGMA

intrinsic case: Ge extrinsic – depends on  en()eh p doping

1 E /2kT  e gB 

extrinsic (n-type) case:

  nee >1014 fold complex variation variation of n with T at low temp

645 HALL EFFECT to measure n, p

A conductor that carries a current in the presence of a transverse magnetic field develops a voltage across the sample normal to both.

simplest case (n-type): JBIB FqvB m ntwn t qV w FqEHall eHallw

1 IB force balance gives: V  Hall qn t

646 DIRECT / INDIRECT GAP SEMICONDUCTORS

10

647 648

 h 649 ABSORPTION SPECTROSCOPY

650 Implications for solar cells? 651 652

 h

   h h g hE () 

 General expression: BEHAVIOR NEAR ABSORPTION EDGE NEAR BEHAVIOR FERMI’S GOLDEN RULE The rate of an optical transition from a single initial state to a final state is given by:

Transition Rate for Single State

2 2 2' if E0 fHi|| ( E f  E i h ) 

square of matrix element resonance condition (strength of coupling) (energy conservation)

2 • E0 is light intensity • hν is the photon energy • in the dipole approximation, H’ = -er∙E • derived w/ time-dependent perturbation theory fHi||'*'  H d  fi 653 Transition Rate for Given Wavelength 2 2   EfHi2'|| ( EEh    ) fi,,if fi 0 f i  2 2 ≈ E 2'fHi | | ( E E h ) 0  fi, fi  Let’s assume the state f and i are conduction and valence band states. Then < f |H’| i > → < c |H’| v >. We can define the joint density of states as 2 ()hEkEkhdk   ( () () )3 CV8 3  C V • vertical (direct) transitions only

2 2'2 EEkEkh0 H(()())CV CV C  V    654 22  k Ek() E 22 CC* k 2me  EkCV() Ek () E g* 22 2m  k r EkVV() E * 2mh reduced mass of 2 electron-hole pair (EHP) ()hEkEkhdk   ( () () )3 CV8 3  C V 2 22k  (Ehdk ) 3 3* g 82mr 22  k Let XE g  Eh E  * 2mr

integral vanishes unless X = 0 E = hν -Eg

* 3/2 1 2m 1/2 parabolic bands ()hhE r free electron DOS: CV22 g 2   656

 h

  h h OPTICAL ABSORPTION EDGES

for allowed transitions in the parabolic approximation:

*3/2 1/2 Direct gap:  ()Amdr (2)  h E g

2 Indirect gap: ()  Ahi ( E g E phonon )

the prefactors Ad and Ai contain the matrix elements and fundamental constants

657 OPTICAL SPECTRA

d = 100 nm

1 I IRe(1 )  x absorption d  0 length:  SEMICONDUCTOR BANDGAPS 660 661 EXCITONS • The annihilation of a photon in exciting an electron from the valence band to the conduction band in a semiconductor can be written as an equation: hv  e- + h+ • Since there is a Coulomb attraction between the electron and hole, the photon energy required is lower than the band gap by this attraction (giving bound states). • To correctly calculate the absorption coefficient we have to introduce a two-particle state consisting of an electron attracted to a hole, known as an exciton (a quasiparticle).

662 WANNIER and FRENKEL EXCITONS

SCs of large dielectric constant solids of small dielectric constant – Excitons represent the elementary excitation of a semiconductor. In the ground state the semiconductor has only filled or empty bands. The simplest excitation is to excite one electron from a filled band to an empty band, so creating an electron and a hole – Excitons are neutral overall but carry an electric dipole moment and therefore can be excited by either a photon or an electron 663 HYDROGENIC MODEL OF EXCITONS Bohr atom picture, modified with effective mass and dielectric constant of crystal Hydrogen Atom: Circular orbit: centripetal force = Coulombic force mv2 e2 e  2 22 rr40 4 0n  2 rnan 0 Angular momentum: mvre  n 2 mee 1 e2 2 r1 = Bohr radius = a0 = 0.529 Å Emve () 240r 22 1 ee 4  ( ) me R E  e 2400rr 4 n 22 22 2 2(160 ) nn 1 e2  R (Rydberg) = 13.606 eV 24 0r 664 HYDROGENIC MODEL OF EXCITONS Excitons: 1. Use dielectric constant of crystal e2 Ur() 2. Use effective reduced mass 40r 111 *** mmmreh

Exciton Bohr Radius: 22 2 4  0nnm e ran *2 * 0 merr m

me rex  * 0.529 A mr Binding Energy: * mr R ௑ En 22 ଶ 665 mne TOTAL EXCITONIC ENERGY

if referenced to the top of the valence band: 22 *  K mr R EKEnlm() g 22 2M mne

exciton kinetic energy binding energy (translational motion of neutral quasiparticle)

666 Exciton Parameters for Several SCs

Semiconductor E (eV) ε(0)* E (meV) r (nm) g mmre / B ex ** (/mmee ; mm he/) Si 1.11 11.8 0.190 18.6 3.28

Ge 0.67 16 0.132 7.01 6.41

GaAs 1.42 13.2 0.0616 4.81 11.3 (0.067, 0.76) InSb 0.163 17.7 0.0135 0.586 69.4

CdSe 1.74 (0.13, 0.45) 15 5.2

Bi 0 0.001 small > 50

ZnO 3.4 (0.27, ?) 59 3

GaN 3.4 (0.19, 0.60) 25 11

667 668 = exciton binding energy binding energy = exciton

X   h h R

 h FREE EXCITON ABSORPTION FREE EXCITON 669

 h EXCITONS in BULK GaAs EXCITONS FRENKEL EXCITONS

670 QUANTUM CONFINEMENT IN NANOSTRUCTURES Materials with at least one dimension on the scale of the exciton diameter are said to be quantum confined. The electronic wavefunction, energy levels, and DOS will depend on the dimension(s).

Confinement in 1 dimension = quantum well 2 dimensions = quantum wire 3 dimensions = quantum dot

well wire dot “Particle in a box” model – energy level separation increases → discrete states 2 2 nx h well: ELgx() E g0 *2 8mLrx

bulk gap 1D confinement 671 2 2 2 2 h1 n ny n ELLL(, ,) E x z dot: gxyz g0 *2 2 2 8mLrx L y L z

bulk gap 3D confinement

assumptions: parabolic bands, independent electrons, infinite barriers

672 673 674 675 676 677 1/2 hkT / B hE g e BUILDING BLOCKS OF SEMICONDUCTOR DEVICES

• p-n junctions

• metal-semiconductor junctions

• metal-insulator-semiconductor (MIS) capacitors

Reading: AM Ch. 29

679 THE MOSFET (metal-oxide-semiconductor field-effect transistor)

amplification, switching, logic

680 BASIC EQUATIONS FOR DEVICE PHYSICS

in 1D:

dd2E q Poisson’s Equation:  p NnN   2  DA dx dx ss  dn Drift-Diffusion JqnqD E Equations: nn ndx dp JqpqD E pp pdx

n J p J Continuity GU n GU p Equations: tqnn tqpp CONTINUITY

n J GU n tqnn

Un

682 THE PN (HOMO)JUNCTION

before contact after contact, equilibrium

B. Van Zeghbroeck, 2011. 683 ABRUPT PN JUNCTIONS

dd2 E  2 dx dx  s

integrate

WDn ()xxdx  () E s  WDp integrate, flip sign   ()xxdx  E () WDp flip sign Ex()  q () x LINEARLY-GRADED PN JUNCTIONS

more realistic doping profile, same basic result and device physics SIMPLE DERIVATION OF CURRENT-VOLTAGE CHARACTERISTICS

two electron currents:

1) generation current (Jgen,e) - electrons from p to n side

2) recombination current (Jrec,e) - electrons from n to p side rec q()/ V kT Jee  at equilibrium (V = 0): rec gen q / kT JJeee rec gen qV/ kT so at any voltage V: JJeee rec gen gen qV/ kT total electron current: JJee  J e Je e(1)  total current (e + h): gen gen qV/ kT JJ eh J()(1) J e  J h e  DIODE AT EQUILIBRIUM

 qV  Shockley ideal diode equation (1949): JJ 0 exp 1  nkT 

V = 0

Jdiff,n = Jdrift,n rectification: current flows preferentially in one direction Jdiff,p = Jdrift,p Zero net current REVERSE BIAS

 qV  JJ 0 exp 1  nkT 

V < 0 (reverse bias)

-

Jdiff,n < Jdrift,n Jdiff,p < Jdrift,p FORWARD BIAS

 qV  JJ 0 exp 1  nkT 

V > 0 (forward bias)

+

Jdiff,n > Jdrift,n Jdiff,p > Jdrift,p Only the diffusion current changes significantly with bias. Diffusion dominates in forward bias. DETAILED DERIVATION OF CURRENT-VOLTAGE CHARACTERISTICS derivation of the Shockley ideal diode equation (1949):

EEFi EEiF 2 at equilibrium: nn i exp pn i exp pn ni kT kT

EEFn i EEiFp with bias V: nn i exp pn i exp qV EFnFp E kT kT

EE pn n2 for V 0 pn n2 expFn Fp i i  2 kT pn ni for V  0

qV hole density at x = W : pWnDnno() p exp Dn kT boundary conditions qV electron density at x = -W : nWpDppo() n exp Dp kT DETAILED DERIVATION CONTINUED E dpdE d2 p pp continuity on n-side: UpDnn 0 with, U  nno ppnpdx dx dx2  p

2 dpn in the neutral region (no field): UDp 0 dx2

qV xW px ( ) p p exp  1 exp  Dn LD  solution ( Wx Dn  ): nnono p pp kT Lp

dpn qDpno p  qV  hole diffusion current at x = W : JqDpp exp  1 Dn dx L kT  WDn p  

qDnpo n  qV  electron diffusion current at x = -WDp: J n  exp 1 LkTn  

 qV  qDpno p qD npo n total current: JJpn J J0 exp  1 with, J 0   kT  LLpn Illuminated pn junction

• Absorbed photons generate excess minority carriers

• Minority carriers within a diffusion length of the depletion region are swept across the junction by the electric field  electrons and holes are separated and collected

n-dopedSCR p-doped

EFn

EFp

Lp WD Ln

photocurrent JL for uniform JqGLWL () generation rate G: LpDn Current‐voltage characteristics

“Current superposition”: JL causes downward shift of J-V curve

 qV  dark: JJ 0 exp 1  nkT 

qV light: JJ 0 exp 1 JL nkT Operating modes

1st quadrant (dark): LED or LD

3rd quadrant: 4th quadrant: photodiode solar cell Solar cell external parameters

P JV J V  max mm FF SCOC PPin in P in

-2 Pin (normal sunlight): ~100 mW cm

JJqGLLWSC L() n  p D  maximize light absorption (G)  maximize W & diffusion lengths

kT J L J mmV VOC ln 1 FF  qJ0 J SCV OC  minimize saturation current  reduce recombination  maximize photocurrent  reduce series resistance Silicon pn junction cells

JV   FF SC OC Pin parameters of good lab cells -2 JSC = 42 mA cm VOC = 0.72 V FF = 0.80 η = 24%

Eg (eV) Example of Si cell spectral response

parameters 500 nm 450 μm 19 -3 ND = 5 x 10 cm 16 -3 NA = 1 x 10 cm Ln = 51 μm Lp = 23 μm 4 Sp = 10 cm/s

m Jq()[1 RSRd ()]() L  0 Losses in ideal and real cells

Ideal cells (SR = 1)

1) transparency 2) relaxation (major loss) 3) thermodynamic loss

max η: ~33% at 1 sun

Real cells

1) incomplete absorption (lowers JSC)

2) parasitic dark currents (lowers VOC, FF)

3) bulk, junction, and surface recombination (JSC, VOC, FF) 4) series resistance and leakage currents (FF) Recombination processes

1) Band-to-band E • Radiative EC • Auger E 2) Trap assisted t EV

Dominant mechanism(s) depend on semiconductor, cell design, and processing. •Surface recombination •Depletion region recombination •Bulk recombination • Recombination at metal semiconductor contacts Beyond silicon: PV design rules

Silicon is successful, deploying rapidly, but still expensive. (indirect gap semiconductor requires thick, high-purity, $ layers).

Alternative (thin film) technologies must:

1) Be much more economical, at MW-TW scales 2) Absorb sunlight and collect charges with ~100% efficiency carrier diffusion length > device thickness > absorption length 1 ()LLnp d ()

3) Collect the carriers at a large voltage (VOC > Eg/q – 0.5 V)

Eg = 1-1.5 eV, large Δ in EF, low dark current PV technologies

Technologies 1. Crystalline Silicon  Monocrystalline (m-Si)  Poly- or multicrystalline (poly-Si or mc-Si) m-Si mc-Si 2. Thin Film  Cadmium Telluride (CdTe)  Copper-Indium Gallium diSelenide (CIGS)  Amorphous Silicon (a-Si)  Thin-Film Silicon (TF-Si) 3. Multijunction Concentrators a-Si  Lattice-Matched (LM)  Metamorphic (MM)  Inverted Metamorphic (IMM) 4. Emerging Technologies  Dye-Sensitized (DSC)  Organic (OPV) nc-CIGS Organic  Copper Zinc Tin Sulfide (CZTS)  Other earth-abundant materials  3rd generation concepts (QDs, IB) PV CONVERSION EFFICIENCIES

702 FIN

703