Intrinsic and Extrinsic Semiconductors

Home , Donor (semiconductors), Shallow donor

1 MTLE-6120: Advanced Electronic Properties of Materials

Intrinsic and extrinsic semiconductors

Contents:

I Band-edge density of states

I Intrinsic Fermi level and carrier concentrations

I Dopant states and ionization

I Extrinsic Fermi level and carrier concentrations

I Mobility: temperature and carrier density dependence

I Recombination mechanisms Reading:

I Kasap: 5.1 - 5.6 2 Band structure and conduction

I Metals: partially ﬁlled band(s) i.e. bands cross Fermi level

I Semiconductors / insulators: each band either ﬁlled or empty (T = 0) eτ I Drude formula applicable, mobility µ = m∗ ∗ 2 ~ −1 I Eﬀective mass m = ~ [∇~k∇~kEn(k)] tensorial in general R ∗ −1 I Filled band does not conduct: eτ dk(m ) = 0 for each band 2 2 I Metals conduct due to carriers near Fermi level σ = g(EF )e vF τ/3 I Semiconductors: g(EF ) = 0 (will show shortly) ⇒ no conduction at T = 0 3 Band structure of silicon (diamond-cubic semiconductor) 6

E c E 0 v

E in eV −6

−12 LΛΓΔ ΧΣΓ

I HOMO = Valence Band Maximum (VBM) with energy Ev and LUMO = Conduction Band Minimum (CBM) with energy Ec

I HOMO-LUMO gap Eg = Ec − Ev ≈ 1.1 eV

I HOMO and LUMO at diﬀerent ~k ⇒ indirect band gap

I Diamond: similar band structure, much larger gap (≈ 5.5 eV) ⇒ insulator 2 2 I Valence electrons/cell = 8 (even), conﬁguration: 3s 3p (two Si/cell) 4 Band structure of GaAs (zinc-blende semiconductor)

I HOMO-LUMO gap Eg = Ec − Ev ≈ 1.4 eV

I HOMO and LUMO at same ~k (Γ) ⇒ direct band gap 2 1 2 3 I Valence electrons/cell = 8 (even), conﬁguration: Ga(4s 4p ), As(4s 4p ) 5 Density of states: silicon

I Can calculate numerically from band structure

I Parabolic band approximation valid for narrow energy range near gap 6 Density of states: parabolic-band semiconductor

g(E) [eV-1nm-3] Valence band, m* = -0.3 Conduction band, m* = 0.5 4

0 -3 -2 -1 0 1 2 3 4 5 E [eV]

I Parabolic bands near each band edge, with diﬀerent eﬀective masses

I Overall DOS reduces with reduced eﬀective mass magnitude

I Set Ev = 0 conventionally (overall energy not well-deﬁned)

I Conduction band edge Ec = Eg

I Where is the Fermi level? 7 Where is the Fermi level?

I At T = 0, valence band fully occupied ⇒ f(Ev = 0) = 1 ⇒ EF > 0

I At T = 0, conduction band fully empty ⇒ f(Ec = Eg) = 0 ⇒ EF < Eg

I Therefore, at T = 0, 0 < EF < Eg i.e. Fermi level is in the band gap

I Chemical potential µ → EF as T → 0

I In semiconductor physics, typically refer to EF (T ) instead of µ(T )

I Therefore, Fermi functions will be 1 f(E,T ) = exp E−EF (T ) + 1 kB T 8 Where is the Fermi level at T > 0?

I Given Fermi level EF and density of states g(E) R ∞ I Number of electrons in conduction band is Ne = dEg(E)f(E) Eg R 0 I Number of holes in valence band is Nh = −∞ dEg(E)(1 − f(E)) I Total number of electrons cannot change with T ⇒ Ne = Nh

Z 0 Z ∞ dEg(E)(1 − f(E)) = dEg(E)f(E) −∞ Eg Z 0 exp E−EF Z ∞ 1 dEg(E) kB T = dEg(E) E−EF E−EF −∞ exp + 1 E exp + 1 kB T g kB T Z 0 Z ∞ dEg(E)e−(EF −E)/(kB T ) ≈ dEg(E)e−(E−EF )(kB T ) −∞ Eg ∞ ∞ Z ε Z ε −EF /(kB T ) − k T (EF −Eg )/(kB T ) − k T e dεg(−ε)e B ≈ e dεg(Eg + ε)e B 0 0 | {z } | {z } ≡Nv ≡Nc

Assuming EF ,Eg − EF kBT , ε ≡ energy from band edge 9 Band edge eﬀective density of states

I Given density of states as a function of energy away from band edge

∞ Z ε − k T Nc/v ≡ dεgc/v(ε)e B 0

√ 3 2m∗ √ I In parabolic band approximation g(ε) = 4π ε for both bands 2π~ ∗ ∗ (but with different m ; for tensor m , above defines DOS meff)

I Therefore band-edge eﬀective of density states:

3 q ∗  ∞ 2m Z c/v √ ε − k T Nc/v ≡ dε   4π εe B 0 2π~

3 3 q ∗  q ∗  2mc/v 2πmc/vkBT 3/2 =   4πΓ(3/2)(kBT ) = 2   2π~ 2π~

∗ 3/2 I Nc/v ∝ (mc/v) (steeper g(ε) parabola) 3/2 I Nc/v ∝ T (climb higher up the g(ε) parabola) 10 Fermi level for T > 0

I Charge neutrality imposes

−EF /(kB T ) (EF −Eg )/(kB T ) Nve = Nce

I Solve for Fermi level position:

Eg kBT Nv EF (T ) = + ln 2 2 Nc

I At T → 0, EF is exactly at the middle of the band gap

I At ﬁnite T , EF moves away ∼ kBT Eg (still close to gap center)

I Which way does the Fermi level move with increasing T ?

I For electrons in metals (and classical gases), µ ↓ with T ↑

I For semiconductors, EF (T ) ↓ with T ↑ iﬀ Nc > Nv (more DOS in positive m∗ band; negative m∗ pulls µ other way) 11 Electron and hole concentrations

−(Eg −EF )/(kB T ) I Number density of electrons n ≡ Ne = Nce

−EF /(kB T ) I Number density of holes p ≡ Nh = Nve

I Which one is larger? So far, they are equal: charge neutrality!

−Eg /(kB T ) 2 I Note product np = NcNve ≡ ni , independent of EF I Neutral pure semiconductor, n = p = ni, intrinsic carrier density

I If EF ↑, then n ↑ and p ↓ (more electrons than holes)

I If EF ↓, then n ↓ and p ↑ (more holes than electrons) 2 I But np = ni , constant in all these cases + − −14 2 I This is an equilibrium constant, eg. [H ][OH ] = 10 M in water

I How do you change EF ? Doping! (Also later, gating) 12 Intrinsic semiconductor thermodynamics and transport

I Fermi level far from band edges ⇒ Boltzmann statistics in both bands

I Velocity distribution: Maxwell-Boltzmann distribution (classical gases)

I Internal energy of electrons n · (Eg + 3kBT/2)

I Internal energy of holes −p · (−3kBT/2) (holes are missing electrons!)

I Net internal energy n · (Eg + 3kBT/2) − p · (−3kBT/2)

I Drude theory conductivity σ = neµe + peµh 13 Intrinsic semiconductors: typical values at T = 300 K

∗ ∗ -3 -3 -3 Eg [eV] me/me mh/me Nc [cm ] Nv [cm ] ni [cm ] Ge 0.66 0.04,0.28 1.64,0.08 1.0 × 1019 6.0 × 1018 2.3 × 1013 Si 1.10 0.16,0.49 0.98,0.19 2.8 × 1019 1.2 × 1019 1.0 × 1010 GaAs 1.42 0.082 0.067 4.7 × 1017 7.0 × 1018 2.1 × 106

I Note that meff for Nc/v is an average of longitudinal / transverse values 1/3 2/3 (meff = mL mT ; for values see Table 5.1 in Kasap) I Nc and Nv increase with meff

I ni drops exponentially with increasing Eg 14 Diamond-cubic structure: sp3 bonding

3 I Valence s and three p orbitals ⇒ four sp hybrid orbitals

I Orbitals point towards vertices of regular tetrahedron

I Si, C, Ge: 4 valence electrons each

I Form covalent bonds with four neighbours (8 shared electrons/atom)

I Bonding orbitals → valence band, anti-bonding orbitals → conduction band

I Tetrahedral network: FCC lattice with two atoms per cell 15 Zinc-blende structure: sp3 bonding

3 I Valence s and three p orbitals ⇒ four sp hybrid orbitals

I Orbitals point towards vertices of regular tetrahedron

I Combine Ga,In (3 electrons) with As,Sb (5 electrons)

I Form covalent bonds with four neighbours (8 shared electrons/atom)

I Bonding orbitals → valence band, anti-bonding orbitals → conduction band

I Tetrahedral network: FCC lattice with two atoms per cell

I With Al and N, tend to form closely related Wurtzite structure (FCC to HCP cell) 16 Doping: acceptors and donors

I Extra / impurity Group III atoms: one less electron per atom

I Extra / impurity Group V atom: one extra electron per atom

I Covalent bonding theory: atoms want 8 (ﬁlled-shell) of shared electrons

I Group III ‘acceptor’: pick up electron from solid ⇒ hole in valence band

I Group V ‘donor’: give electron to solid ⇒ electron in conduction band Simple picture of doping:

I Density Na of acceptor atoms: charge −eNa

I Density Nd of donor atoms: charge +eNd

I Charge neutrality −en + ep − eNa + eNd = 0 ⇒ n − p = Nd − Na 2 I Change in n and p due to shift in EF , but np = ni

Eg kB T nNv kB T n I Solve for n and p, then ﬁnd EF = + ln = EF 0 + ln 2 2 pNc 2 p I Even simpler picture: usually Nd,Na ni ⇒ either p n or n p 17 Doping: p-type and n-type

n-type semiconductor:

I Donor impurities dominate Nd > 0 (Na = 0 or < Nd) 2 I Typically Nd − Na ni ⇒ n p (since p = ni /n) 2 I Therefore n ≈ Nd − Na, p ≈ ni /(Nd − Na) kB T n Nd−Na I EF = EF 0 + ln = EF 0 + kBT ln (shifted ↑ towards CBM) 2 p ni I Current predominantly carried by electrons p-type semiconductor:

I Acceptor impurities dominate Na > 0 (Nd = 0 or < Na) 2 I Typically Na − Nd ni ⇒ p n (since n = ni /p) 2 I Therefore p ≈ Na − Nd, n ≈ ni /(Na − Nd) kB T n Na−Nd I EF = EF 0 + ln = EF 0 − kBT ln (shifted ↓ towards VBM) 2 p ni I Current predominantly carried by holes 18 Doping: a more complete picture

I Simple picture: donor atom donates an electron, becomes positively charged

I Positively charged donor ion can bind electrons: like a hydrogen atom −2 m∗ I Binding energy of pseudo-hydrogenic atom Eb = Ryd ∼ 0.05 eV r me I Donor level: Ed = Ec − Eb (electrons bound relative to CBM)

I Exact argument for acceptors and holes, with charges swapped

I Acceptor level: Ea = Ev + Eb (holes bound relative to VBM)

I Levels in Si: note some impurities introduce multiple levels

Physics of Semiconductor Devices, S. M. Sze 19 Dopant levels in Ge and GaAs

I For GaAs, Group II or Group VI are shallow dopants

I For GaAs, Group IV can be donor and acceptor dopants: how?

Physics of Semiconductor Devices, S. M. Sze 20 Donor charge density

I For each donor atom, degenerate donor levels typically with gd = 2 (spin)

I Electron occupation: zero or one for the whole atom (repulsions)

I Probability of occupation zero ∝ 1

EF −Ed I Probability of occupation one ∝ gd exp kB T I Normalized probability of ionized donor (occupation zero): 1 P + = d EF −Ed 1 + gd exp kB T

I Therefore number density of ionized donors: N N + = d d EF −Ed 1 + gd exp kB T

(which is ≈ Nd as long as EF several kBT below Ed) 21 Acceptor charge density

I For each acceptor atom, degenerate acceptor levels typically with ga = 4 (two for spin, two for dgenerate hole bands)

I Hole occupation: zero or one for the whole atom (repulsions)

I Probability of hole occupation zero ∝ 1

Ea−EF I Probability of hole occupation one ∝ ga exp kB T (tricky: ﬂip energy axis when thinking in terms of holes)

I Normalized probability of ionized donor (hole occupation zero): 1 P − = a Ea−EF 1 + ga exp kB T

I Therefore number density of ionized acceptors: N N − = a a Ea−EF 1 + ga exp kB T

(which is ≈ Na as long as EF several kBT above Ea) 22 Charge neutrality

0 = ρ(EF ) + − = −en + ep + eNd − eNa   E −E − g F − EF Nd Na = e − N e kB T + N e kB T + −  c v EF −Ed Ea−EF  | {z } | {z } kB T kB T n p 1 + gde 1 + gae

I If Na − Nd ni, then p n (p-type)

I If Nd − Na ni, then n p (n-type) I Previous simple analysis holds if:

I Net doping is stronger than ni (one of the two regimes above), and I Doping is small enough that EF is far above Ea and far below Ed Nd/a (remember EF moves up/down ∼ kB T ln ) ni 23 ρ(EF ): intrinsic semiconductor

Na = 0, Nd = 0 + - 1020

1018

] 1016 -3 [cm e

/ 14 ρ 10

1012

1010 0 0.2 0.4 0.6 0.8 1 E [eV]

I Electrons increase with increaisng EF (hence ρ decreases)

I Cross-over point from + to − is neutral EF 24 ρ(EF ): moderate p doping

14 Na = 10 , Nd = 0 + - 1020

1018

] 1016 -3 [cm e

/ 14 ρ 10

1012

1010 0 0.2 0.4 0.6 0.8 1 E [eV]

I Acceptors pull down Fermi level

I At moderate doping level, far from mid-gap and acceptor levels 25 ρ(EF ): moderate n doping

14 Na = 0, Nd = 10 + - 1020

1018

] 1016 -3 [cm e

/ 14 ρ 10

1012

1010 0 0.2 0.4 0.6 0.8 1 E [eV]

I Donors pull up Fermi level

I At moderate doping level, far from mid-gap and donor levels 26 ρ(EF ): high n doping

19 Na = 0, Nd = 10 + - 1020

1018

] 1016 -3 [cm e

/ 14 ρ 10

1012

1010 0 0.2 0.4 0.6 0.8 1 E [eV]

I At high n doping level, approach / cross donor level

I Donors may be partially ionized (simple model no longer works) 27 ρ(EF ): high p doping

19 Na = 10 , Nd = 0 + - 1020

1018

] 1016 -3 [cm e

/ 14 ρ 10

1012

1010 0 0.2 0.4 0.6 0.8 1 E [eV]

I At high p doping level, approach / cross acceptor level

I Acceptors may be partially ionized (simple model no longer works) 28 Degenerate doping

I High-enough n doping: Fermi level enters conduction band (‘n+’)

I High-enough p doping: Fermi level enters valence band (‘p+’)

I One of our approximations breaks down for n+ case with EF > Eg, −(Eg −EF )/(kB T ) n 6≈ Nce but instead !3 1 p2m∗(E − E ) n ≈ F g 3π2 ~

I Similarly, for p+ case with EF < 0: !3 1 p2m∗(−E ) p ≈ F 3π2 ~

√ ∗ I These are the Fermi theory expressions with kF = 2m εF /~ (where εF is Fermi energy relative to band edge)!

I Important: partial donor / acceptor ionization in this regime 29 Partial donor ionization

I Consider a p-type material (Nd = 0) with EF = Ea (we cross this point as we increase p-doping before getting to p+)

I Around this EF , acceptors are partially ionized

I When exactly does this occur?

I Since EF far from conduction band, neglect n p

I Charge neutrality yields

EF N − k T a Nve B = Ea−EF 1 + ga exp kB T Ea N − k T a ⇒ Nve B = 1 + ga Ea ⇒ kBT = log (1+ga)Nv Na

I Therefore, this happens for high Na when Ea kBT

I But also, for kBT lower than Ea: dopant freeze-out

I Importance of shallow donor/acceptor levels! (for donors, replace Ea → Eg − Ed) 30 Carrier density in ionization regime

I If T much smaller than ionization threshold, neutrality:

EF N − k T a Nve B = Ea−EF 1 + ga exp kB T EF N Ea−EF − k T a − k T → Nve B ≈ e B ga Ea kBT gaNv ⇒ EF = + log 2 2 Na

I Very similar to intrinsic case, except Na/ga replaces Nc

I Eﬀective gap between valence band and acceptor level!

2 − −Ea I In this regime, p = p · N = (NvNa/ga) exp a kB T I Similar behavior for frozen-out donors in ionization regime for n-type 31 Temperature dependence of EF

1.0

0.8

12 Na/d = 10

] 14 V 0.6 Na/d = 10 e [

16 F Na/d = 10 E 18 Na/d = 10 0.4

0.2

0.0 1/1000 1/300 1/100 1/50 1/T [K 1]

I Doping dominates at low T , approach intrinsic level at high T

I At low T , Fermi level decided by donor/acceptor level

I Note using Ea/d = 0.1 eV from band edges to exaggerate eﬀect 32 Temperature dependence of carrier concentration

1020 12 Na/d = 10 14 Na/d = 10 18 16 10 Na/d = 10 18 Na/d = 10

1016 ] 3

Intrinsic m c [

n Extrinsic 1014

Ionization 1012

1010 1/1000 1/300 1/100 1/50 1/T [K 1]

I Ionization regime at low T upto threhsold which increases with Nd/a

I Constant concentration in extrinsic regime; threshold increases with Nd/a

I At high T , dopants don’t matter: intrinsic regime 33 Intrinsic mobility

∗ I Drude theory: mobility µ = eτ/m ∗ I Typically semiconductor m ∼ 0.1 − 1 me, so expect higher µ than metals e 2 2 I At room temperature, µi-Si ∼ 1400 cm /(Vs) and µAg ∼ 60 cm /(Vs) I Eﬀective mass alone does not explain it! −1 I Remember τe-ph ∝ g(E)T I For metals, g(E) → g(EF ) since most carriers near Fermi level I For semiconductors,√ carriers within few kBT of band edge where g(E) ∝ E (and much smaller than metals) √ I Averaged over carriers, g(E) ∝ T −1 3/2 −3/2 I Therefore, τe-ph ∝ T and µ ∝ τ ∝ T 34 Impurity scattering

I Doped semiconductor contains ionized donors / acceptors

I Charged impurities cause electron PE ∝ 1/r near them

I Electrons with KE PE not scattered signiﬁcantly

I Electrons with KE PE scattered most strongly 2 I Eﬀective cross-section ∝ rc , where PE(rc) ∼ KE ∼ kBT −1 −2 I Therefore rc ∝ T and cross-section σcs ∝ T −1 I Scattering time τI = (Na/dσcsv) √ I Average velocity v ∝ T −1 3/2 I Therefore τI ∝ Na/dT 35 Extrinsic mobility

−3/2 I Intrinsic scattering time τe-ph ∝ T −1 3/2 I Dopant / impurity scattering time τI ∝ Na/dT 3/2 −3/2−1 I Net scattering time τ ∝ T + Na/dT (Mathiessen rule) 3/2 −3/2−1 I Therefore mobility µ ∝ T + Na/dT

I At high T , e-ph scattering dominates (intrinsic regime)

I At high doping concentration (or low T ), impurity scattering dominates

Fig. 5.18 and 5.19 from Kasap 36 Extrinsic conductivity

I Conductivity σ = neµe + peµh: similar dependence as n (exponentials dominate over polynomial)

I µ eﬀect visible mainly in extrinsic regime where n is constant

Fig. 5.20 from Kasap 37 Recombination

E c E 0 v

E in eV −6

−12 LΛΓΔ ΧΣΓ

I Excite electrons and holes in semiconductor to higher energy

I e-ph scattering brings electrons and holes to band edges 2 I If np = ni , equilibrium: nothing further happens I What if you make more electron-hole pairs (eg. using light) 2 such that np 6= ni ? I Electrons and holes will recombine to restore equilibrium 38 Recombination: direct vs indirect

E c E 0 v

E in eV −6

−12 LΛΓΔ ΧΣΓ

I Direct gap: electrons and holes at band edges at same k

I Indirect gap: band edge carriers at diﬀerent k

I Which will recombine faster?

I Direct gap: momentum conservation, recombine and emit light (usually)

I Indirect gap: cannot directly recombine: momentum not conserved 39 Recombination mechanisms

2 I Recombination rate proportional to np − ni I Radiative / direct recombination (direct gap materials)

I Trap-assisted (Shockley-Read-Hall recombination)

I Trap level in gap captures electron (hole), becoming − (+) charged

I Later captures hole (electron), becoming neutral

I Energy from recombination emitted to phonons

I Probability of each capture ∝ Boltzmann factor of trap depth from band edge 2 Et−Eg /2 I Net rate ∼ sech (trap level Et) 2kB T I Strongest for mid-band-gap states!

I Auger recombination

I Energy and momentum of e-h pair go to excite another e or h

I Need e or h to excite, so rate ∝ n, p

I Dominates at very high carrier concentrations 2 I Recombination rate = α(np − ni ) I Minority carrier lifetime = 1/(max(n, p)α)