Semiconductor Energy Band & Impurity Energy Levels Eg=1.12 eV in Si EC EV Ei Conduction band Valence band Donor impurity ED EA Acceptor impurity Semiconductor

Thermal Equilibrium - physical quantities are time-invariant - physical quantities are time-invariant - every process is balanced by its inverse process - every process is balanced by its inverse process - single Fermi energy level (EF) quantifies both electron (n) & hole (p) concentration - single Fermi energy level (EF) quantifies both electron (n) & hole (p) concentration - EF is flat - EF is flat - n p=ni2 – law of mass action ; ni is the intrinsic carrier concentration - n p=ni2 – law of mass action ; ni is the intrinsic carrier concentration

Equilibrium Statistics Electrons / holes exhibiting the duality (wave-like & particle-like nature) reside in bands: EC EV Ei Conduction band Eg=1.12 eV in Si Valence band

where gn(E)dE is the number of electrons between E and E+dE with the 3-D density of states given by

is the Fermi-Dirac distribution function denoting and is the Fermi-Dirac distribution function denoting the electron occupation probability with Fermi energy level, EF, f E EF 0.5 1 T=0 K T1 T2 T3 T1<T2<T3

The integration yields with denoting the effective density of states at conduction band bottom, and the Fermi 1/2 –integral. For non-degenerate case where EC - EF  2kT

Similarly, For non-degenerate case where EF - EV  2kT with

EF in extrinsic semiconductor in equilibrium Donor atoms are incorporated as The donor energy level ED lies a few kT below EC; the degeneracy factor g in Si is 2. Similarly, for acceptor atoms The acceptor energy level EA lies a few kT above EV; the degeneracy factor g in Si is 4.

EF is determined by charge neutrality condition with ND, NA as parameters: -1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1 100 200 300 400 500 600 700 800 T (K) EF-Ei [eV] Conduction band Valence band 1014 NA=1012 ND=1012 1016 1018

Intrinsic & extrinsic (n- & p-type) semiconductor - EFi + EFi - EFi + EFi where, ni is the intrinsic carrier concentration, and EFi (=Ei) the intrinsic Fermi energy level determined by n = p, i.e., Ei and

Fermi potential (fFn) in n-type Fermi potential (fFp) in p-type EC EV Ei qfFn EFn EC EV Ei qfFp EFp Fermi potential (fFn) in n-type Fermi potential (fFp) in p-type

In non-degenerate semiconductor, n p = ni2 in thermal equilibrium (law of mass action).

Carrier Concentration f(E) N(E) n(E), p(E) n p E N-type Intrinsic P-type

Charge Transport; drift & diffusion Recall charge conservation / continuity equation; x x+dx Jn(x) Jn(x+dx) Jn consists of drift & diffusion components: drift velocity electron mobility electron diffusion coefficient Thus Continuity equation

Similarly, for holes In 3-D

Transport Coefficient Conceptually, vd is driven by E between collisions: mean collision time with electron mobility specified by Likewise

l Non-uniform n / p gives rise to diffusion flux: n(x) The net # of electrons crossing the x-plane of cross sectional area A is given in terms of the mean free path ln as Now, electron flux

The mean free path ( ln ) is associated with the thermal velocity (vT) by where the thermal velocity (vT) is given by equipartition theorem: Thus, Similarly, i.e., one obtains Einstein relation in equilibrium

EF in equilibrium a) Single S/C system Recall in 1-D that EC, EV, Ei all represent the electronic potential energy: electrostatic potential Since In equilibrium, Jn = 0 and

b) Two S/C systems in equilibrium contact S/CL S/CR FLR(E) FRL(E) Now, in view of Pauli exclusion principle vacant electron states Transfer matrix occupied electron states Thus and No net flux leads to fL(E)=fR(E), i.e. Therefore, Fermi level lines up, EFL=EFR . In equilibrium EF is flat and lines up.

Non-equilibrium and quasi-Fermi levels A system, when under bias or illumination, is driven away from equilibrium to non-equilibrium conditions. In non-equilibrium, quantifying n, p requires respective quasi-Fermi level (imref), EFn & EFp The role of EFn , EFp identical to that of EF in equilibrium;

In intrinsic S/C under irradiation Why imrefs? In intrinsic S/C under irradiation where the photogenerated component is given via recombination time t and generation rate g (∝ light intensity). When gtn >>ni, gtn >>ni, EF should be above Ei for quantifying n & below Ei for p. Impossible to meet the condition with single EF. EC EV Ei EFn EFp Splitting of two imrefs ∝ light intensity.

Shockley Hall Read Recombination: equilibrium vs. non-equilibrium Recall in equilibrium If np > ni2 : charge injection If np < ni2 : charge extraction In the presence of single intermediate trap level, Et , Et EC EV e capture e emission h capture h emission In equilibrium, # of electrons captured is trap density

Away from equilibrium, ec rate is empty trap sites occupation probability in non-equilibrium electron capture cross-section thermal velocity Likewise emission probability Similarly, for holes

en, ep can be extracted from equilibrium condition, i.e. ree= rec with Similarly

Steady state (S.S.) vs. equilibrium In S/C uniformly irradiated with denoting e/h generation rate via band to band excitation At S.S., i.e. the net recombination of e/h is same In equilibrium, a process is balanced by its inverse process, The S.S. condition is automatically satisfied.

The S.S. condition reads as Hence, the S.S. distribution function or the probability of electron being trapped at Et is given by Compare this with equilibrium distribution function,

Next, the recombination rate is given by For simplicity, take sn = sp =s and let 1/t = svTNt , obtaining In equilibrium, n p=ni2, U=0 If n p>ni2 (charge injection), U>0: net recombination If n p<ni2 (charge extraction), U<0: generation

Minority carrier lifetime & surface recombination In n-type S/C, nn>>pn , ni tp being the minority carrier lifetime. Near interface, effective thickness vR : recombination velocity

p-n JUNCTION DIODE Definition: p-n junction diode is a rectifier and constitutes a key element in semiconductor transistors. p n V I V I FORWARD REVERSE BREAKDOWN Keywords: forward & reverse bias and current, breakdown.

p- & n-type semiconductors brought in equilibrium contact to Operation: p- & n-type semiconductors brought in equilibrium contact to form p-n junction. Under bias, the junction is pushed away from equilibrium and current flows to restore the equilibrium. p n V V=VF>0 : Forward bias V=VR<0 : Reverse bias

Equilibrium junction band bending p- & n-type S/C are brought into equilibrium contact via exchange of e/h. In equilibrium, EF should line up and be flat, leading to energy band bending. p-type n-type qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qfbi = q (fFn+fFp ) No net flows of electrons and holes, Jndrift= Jndiff , Jpdrift=Jpdiff

Electrostatics in depletion approximation The band bending is supported by the space charge developed in depletion width. p e -xp xn -em f fbi Q x qNd qNa n qfbi E EF EC EV Specifically, e/h spills over from n/p to p/n regions, leaving behind uncompensated ND+/NA-, viz. space charge, r .

Junction band bending under bias x f (fbi-V) Q p n V > 0 W EC EV E q(fbi-V) EFN EFP -xp xn Under forward bias, V>0 -- p- side is lowered by qV relative to n-side, reducing band bending. Equivalently, n-side is raised by qV relative to p-side -- p- and n-bulk are preserved as in equilibrium These two necessitate introduction of EFn , EFp & bias is accommodated via splitting of EFn & EFp, qV=EFn-EFp.

Under reverse bias, V<0 EC EV Q e f E q(fbi+|V|) EFN EFP (fbi+|V|) p n -xp xn W Under reverse bias, V<0 -- p- side is raised by q|V| relative to the n-side, increasing band bending. Equivalently, n-side is lowered by qV relative to p-side -- p- and n-bulk are preserved as in equilibrium These two necessitate introduction of EFn , EFp & bias is accommodated via splitting of EFn & EFp, q|V|=EFn-EFp. Key words: ND, NA, maximum field (Emax), built-in or contact potential (fbi), depletion width (W )

Carrier concentration under non-equilibrium (non-degenerate case) EV E q(fbi+|V|) EFn EFp -xp xn p n V =VR < 0 q|V| EC EV E q(fbi-V) EFn EFp p n V =VF > 0 -xp xn qV Note that in W : Forward bias (V>0) Charge injection Reverse bias (V<0) Charge extraction inducing recombination or generation, i.e. current flows.

Ideal current-voltage characteristics: Shockley equation Shockley assumptions: The abrupt depletion-layer approximation The built-in potential & applied voltages are supported by a dipole layer with abrupt boundaries, and outside the depletion layer S/C is assumed neutral. Low injection level The injected minority carrier densities are much smaller than the majority carrier densities in equilibrium; The Boltzmann approximation It gives simplified boundary conditions No recombination/generation currents exist in the depletion layer Therefore, the electron and hole currents are constant through the depletion layer. The total current can be approximated as

The continuity equation and boundary conditions in n-region, for example, reads as where Lp is the hole diffusion length in n-region given by The general solution is given by Likewise, one can write

The respective diffusion currents are given by and total current is given by where is the saturation current.

(ii) the current is dominated by minority diffusion current p n V = VF > 0 -xp xn x n, p np(x) np0 pn0 J pn(x) Jn Jp JF Ln Lp xn -xp x np0 pn0 pn(x) np(x) |J| Jn Jp JR p n V = -VR < 0 n, p Note (i) the condition that I is constant throughout is met by respective majority drift component (ii) the current is dominated by minority diffusion current

Ideal I-V characteristics: linear & semilog plot -6 -4 -2 2 4 6 8 10 12 14 16 18 qV/kBT J/JS FORWARD REVERSE 2 4 6 8 10 10-1 100 101 102 103 104 |J/JS| q |V| /kBT FORWARD REVERSE Ideal I-V characteristics: linear & semilog plot

Non-ideal current Recombination / generation current There is recombination (V>0) or generation (V<0) throughout the depletion and the quasi-neutral region near the junction edges For V>0 For V<0 Generation of e/h in depletion occurs primarily via intermediate trap sites inducing alternating e/h emissions

with fitting parameter m between 1~2. High level injection The quasi-neutral approximation is no longer valid, and electric field exists outside the depletion region, leading to minority drift current In this case, the effective voltage drop through the depletion region is smaller than V, and the current expression becomes exp(qV/mkT) Series resistance It further reduces the junction voltage drop and the current is better approximated by exp(qV/mkT) Thus, in general, with fitting parameter m between 1~2.

p-n junction used as LED or diode laser

lasing due to feedback

p-n junction used as photodiode