1 MTLE-6120: Advanced Electronic Properties of Materials Intrinsic and extrinsic semiconductors Contents: ◮ Band-edge density of states ◮ Intrinsic Fermi level and carrier concentrations ◮ Dopant states and ionization ◮ Extrinsic Fermi level and carrier concentrations ◮ Mobility: temperature and carrier density dependence ◮ Recombination mechanisms Reading: ◮ Kasap: 5.1 - 5.6
2 Band structure and conduction ◮ Metals: partially filled band(s) i.e. bands cross Fermi level ◮ Semiconductors / insulators: each band either filled or empty ( T = 0 ) ◮ Drude formula applicable, mobility µ = eτ m ∗ ◮ Effective mass m ∗ = � 2 [ ∇ � k )] − 1 tensorial in general k E n ( � k ∇ � � d k ( m ∗ ) − 1 = 0 for each band ◮ Filled band does not conduct: eτ ◮ Metals conduct due to carriers near Fermi level σ = g ( E F ) e 2 v 2 F τ/ 3 ◮ Semiconductors: g ( E F ) = 0 (will show shortly) ⇒ no conduction at T = 0
3 Band structure of silicon (diamond-cubic semiconductor) 6 E c E v 0 E in eV − 6 − 12 L Λ Γ Δ Χ Σ Γ ◮ HOMO = Valence Band Maximum (VBM) with energy E v and LUMO = Conduction Band Minimum (CBM) with energy E c ◮ HOMO-LUMO gap E g = E c − E v ≈ 1 . 1 eV ◮ HOMO and LUMO at different � k ⇒ indirect band gap ◮ Diamond: similar band structure, much larger gap ( ≈ 5 . 5 eV) ⇒ insulator ◮ Valence electrons/cell = 8 (even), configuration: 3 s 2 3 p 2 (two Si/cell)
4 Band structure of GaAs (zinc-blende semiconductor) ◮ HOMO-LUMO gap E g = E c − E v ≈ 1 . 4 eV ◮ HOMO and LUMO at same � k (Γ) ⇒ direct band gap ◮ Valence electrons/cell = 8 (even), configuration: Ga( 4 s 2 4 p 1 ), As( 4 s 2 4 p 3 )
5 Density of states: silicon ◮ Can calculate numerically from band structure ◮ Parabolic band approximation valid for narrow energy range near gap
6 Density of states: parabolic-band semiconductor g ( E ) [eV -1 nm -3 ] Valence band, m * = -0.3 Conduction band, m * = 0.5 4 3 2 1 0 -3 -2 -1 0 1 2 3 4 5 E [eV] ◮ Parabolic bands near each band edge, with different effective masses ◮ Overall DOS reduces with reduced effective mass magnitude ◮ Set E v = 0 conventionally (overall energy not well-defined) ◮ Conduction band edge E c = E g ◮ Where is the Fermi level?
7 Where is the Fermi level? ◮ At T = 0 , valence band fully occupied ⇒ f ( E v = 0) = 1 ⇒ E F > 0 ◮ At T = 0 , conduction band fully empty ⇒ f ( E c = E g ) = 0 ⇒ E F < E g ◮ Therefore, at T = 0 , 0 < E F < E g i.e. Fermi level is in the band gap ◮ Chemical potential µ → E F as T → 0 ◮ In semiconductor physics, typically refer to E F ( T ) instead of µ ( T ) ◮ Therefore, Fermi functions will be 1 f ( E, T ) = exp E − E F ( T ) + 1 k B T
8 Where is the Fermi level at T > 0 ? ◮ Given Fermi level E F and density of states g ( E ) � ∞ ◮ Number of electrons in conduction band is N e = E g d Eg ( E ) f ( E ) � 0 ◮ Number of holes in valence band is N h = −∞ d Eg ( E )(1 − f ( E )) ◮ Total number of electrons cannot change with T ⇒ N e = N h � 0 � ∞ d Eg ( E )(1 − f ( E )) = d Eg ( E ) f ( E ) E g −∞ � 0 exp E − E F � ∞ 1 k B T d Eg ( E ) + 1 = d Eg ( E ) exp E − E F exp E − E F + 1 E g −∞ k B T k B T � 0 � ∞ d Eg ( E ) e − ( E F − E ) / ( k B T ) ≈ d Eg ( E ) e − ( E − E F )( k B T ) E g −∞ � ∞ � ∞ ε ε e − E F / ( k B T ) d εg ( − ε ) e − ≈ e ( E F − E g ) / ( k B T ) d εg ( E g + ε ) e − kBT kBT 0 0 � �� � � �� � ≡ N v ≡ N c Assuming E F , E g − E F ≫ k B T , ε ≡ energy from band edge
9 Band edge effective density of states ◮ Given density of states as a function of energy away from band edge � ∞ ε d εg c/v ( ε ) e − N c/v ≡ kBT 0 � √ � 3 4 π √ ε for both bands ◮ In parabolic band approximation g ( ε ) = 2 m ∗ 2 π � (but with different m ∗ ; for tensor m ∗ , above defines DOS m eff ) ◮ Therefore band-edge effective of density states: � 3 2 m ∗ � ∞ 4 π √ εe − c/v ε N c/v ≡ d ε kBT 2 π � 0 � 3 � 3 2 m ∗ 2 πm ∗ c/v k B T c/v 4 π Γ(3 / 2)( k B T ) 3 / 2 = 2 = 2 π � 2 π � c/v ) 3 / 2 (steeper g ( ε ) parabola) ◮ N c/v ∝ ( m ∗ ◮ N c/v ∝ T 3 / 2 (climb higher up the g ( ε ) parabola)
10 Fermi level for T > 0 ◮ Charge neutrality imposes N v e − E F / ( k B T ) = N c e ( E F − E g ) / ( k B T ) ◮ Solve for Fermi level position: E F ( T ) = E g 2 + k B T ln N v 2 N c ◮ At T → 0 , E F is exactly at the middle of the band gap ◮ At finite T , E F moves away ∼ k B T ≪ E g (still close to gap center) ◮ Which way does the Fermi level move with increasing T ? ◮ For electrons in metals (and classical gases), µ ↓ with T ↑ ◮ For semiconductors, E F ( T ) ↓ with T ↑ iff N c > N v (more DOS in positive m ∗ band; negative m ∗ pulls µ other way)
11 Electron and hole concentrations ◮ Number density of electrons n ≡ N e = N c e − ( E g − E F ) / ( k B T ) ◮ Number density of holes p ≡ N h = N v e − E F / ( k B T ) ◮ Which one is larger? So far, they are equal: charge neutrality! ◮ Note product np = N c N v e − E g / ( k B T ) ≡ n 2 i , independent of E F ◮ Neutral pure semiconductor, n = p = n i , intrinsic carrier density ◮ If E F ↑ , then n ↑ and p ↓ (more electrons than holes) ◮ If E F ↓ , then n ↓ and p ↑ (more holes than electrons) ◮ But np = n 2 i , constant in all these cases ◮ This is an equilibrium constant, eg. [ H + ][ OH − ] = 10 − 14 M 2 in water ◮ How do you change E F ? Doping! (Also later, gating)
12 Intrinsic semiconductor thermodynamics and transport ◮ Fermi level far from band edges ⇒ Boltzmann statistics in both bands ◮ Velocity distribution: Maxwell-Boltzmann distribution (classical gases) ◮ Internal energy of electrons n · ( E g + 3 k B T/ 2) ◮ Internal energy of holes − p · ( − 3 k B T/ 2) (holes are missing electrons!) ◮ Net internal energy n · ( E g + 3 k B T/ 2) − p · ( − 3 k B T/ 2) ◮ Drude theory conductivity σ = neµ e + peµ h
13 Intrinsic semiconductors: typical values at T = 300 K N c [cm -3 ] N v [cm -3 ] n i [cm -3 ] E g [eV] m ∗ e /m e m ∗ h /m e 1 . 0 × 10 19 6 . 0 × 10 18 2 . 3 × 10 13 Ge 0.66 0.04,0.28 1.64,0.08 2 . 8 × 10 19 1 . 2 × 10 19 1 . 0 × 10 10 Si 1.10 0.16,0.49 0.98,0.19 4 . 7 × 10 17 7 . 0 × 10 18 2 . 1 × 10 6 GaAs 1.42 0.082 0.067 ◮ Note that m eff for N c/v is an average of longitudinal / transverse values ( m eff = m 1 / 3 L m 2 / 3 T ; for values see Table 5.1 in Kasap) ◮ N c and N v increase with m eff ◮ n i drops exponentially with increasing E g
14 Diamond-cubic structure: sp 3 bonding ◮ Valence s and three p orbitals ⇒ four sp 3 hybrid orbitals ◮ Orbitals point towards vertices of regular tetrahedron ◮ Si, C, Ge: 4 valence electrons each ◮ Form covalent bonds with four neighbours (8 shared electrons/atom) ◮ Bonding orbitals → valence band, anti-bonding orbitals → conduction band ◮ Tetrahedral network: FCC lattice with two atoms per cell
15 Zinc-blende structure: sp 3 bonding ◮ Valence s and three p orbitals ⇒ four sp 3 hybrid orbitals ◮ Orbitals point towards vertices of regular tetrahedron ◮ Combine Ga,In (3 electrons) with As,Sb (5 electrons) ◮ Form covalent bonds with four neighbours (8 shared electrons/atom) ◮ Bonding orbitals → valence band, anti-bonding orbitals → conduction band ◮ Tetrahedral network: FCC lattice with two atoms per cell ◮ With Al and N, tend to form closely related Wurtzite structure (FCC to HCP cell)
16 Doping: acceptors and donors ◮ Extra / impurity Group III atoms: one less electron per atom ◮ Extra / impurity Group V atom: one extra electron per atom ◮ Covalent bonding theory: atoms want 8 (filled-shell) of shared electrons ◮ Group III ‘acceptor’: pick up electron from solid ⇒ hole in valence band ◮ Group V ‘donor’: give electron to solid ⇒ electron in conduction band Simple picture of doping: ◮ Density N a of acceptor atoms: charge − eN a ◮ Density N d of donor atoms: charge + eN d ◮ Charge neutrality − en + ep − eN a + eN d = 0 ⇒ n − p = N d − N a ◮ Change in n and p due to shift in E F , but np = n 2 i ◮ Solve for n and p , then find E F = E g 2 + k B T ln nN v pN c = E F 0 + k B T ln n 2 2 p ◮ Even simpler picture: usually N d , N a ≫ n i ⇒ either p ≫ n or n ≫ p
17 Doping: p -type and n -type n -type semiconductor: ◮ Donor impurities dominate N d > 0 ( N a = 0 or < N d ) ◮ Typically N d − N a ≫ n i ⇒ n ≫ p (since p = n 2 i /n ) ◮ Therefore n ≈ N d − N a , p ≈ n 2 i / ( N d − N a ) ◮ E F = E F 0 + k B T ln n p = E F 0 + k B T ln N d − N a (shifted ↑ towards CBM) 2 n i ◮ Current predominantly carried by electrons p -type semiconductor: ◮ Acceptor impurities dominate N a > 0 ( N d = 0 or < N a ) ◮ Typically N a − N d ≫ n i ⇒ p ≫ n (since n = n 2 i /p ) ◮ Therefore p ≈ N a − N d , n ≈ n 2 i / ( N a − N d ) ◮ E F = E F 0 + k B T ln n p = E F 0 − k B T ln N a − N d (shifted ↓ towards VBM) 2 n i ◮ Current predominantly carried by holes
Recommend
More recommend
