ssp_21 - 6 Continued 6-11 Analogous calculation for valence...

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Unformatted text preview: 6 Continued 6-11 Analogous calculation for valence band ⇒ 3 2 2πm kT E − EF E − EF C exp − C n = 2 = N eff exp − C h kT kT * n 2 (6.8) 3 2 2πm kT V exp − EV − E F = N eff exp − EV − E F p = 2 h kT kT * p 2 (6.9) V C N eff , N eff : effective densities of state (∝ T3/2 !) C V n ⋅ p = N eff ⋅ N eff e − Eg kT 3 Eg kT * * 2 − kT (mn m p ) e = 4 2 2πh 3 (6.10) Eg = EC - EV, similarities with law of mass action. Intrinsic semiconductor n =p C V ni = pi = N eff N eff ⋅ e − Eg 2 kT kT = 2 2πh 2 3/ 2 3 4 * (mn m* ) e p − Eg 2 kT (6.11) Eg [eV] ni [cm-3] Ge 0.67 2.4 x 1013 Si 1.1 1.5 x 1010 GaAs 1.43 5 x 107 Cu / 8.5 x 1022 6-12 T=300K (6.8), (6.9) ⇒ C n = p = N eff ⋅ e EF kT ⇒e e 2 EF kT − EC kT = V = N eff ⋅ e V N eff N C eff e − EF kT e EV kT EV + EC kT V EV + EC kT N eff ln C = ⇒ EF = + 2 2 N eff mp EL + EV 3 + kT ln * 2 4 m 14 4 n 23 * = 0 for m * = m * p n * m * ≠ mn ⇒ p weak T-dependence of EF 6-13 (6.12) 6.2 Doping n i, p i: far too small for sufficient current densities, strongly T-dependent. Unwanted doping by impurities ⇒ n > ni. ni = 5 x 107 cm-3 E.g., GaAs (300K) n ≈ 1016 cm-3 for high-purity single crystals (1990) Donnors: donate electrons to conduction band ⇒ n increases. E.g., group V atoms in Si and Ge: P, As, Sb Si: 3s2 3p2 → sp3 hybride Donor: s2 p3 → sp3 hybride +e- Tetrahedral bonding preserved by donors and acceptors. Acceptors: accept electrons from valence band in sp3 hybride ⇒ p increases. E.g., group III atoms in Si, Ge: B, Al, Ga, In 6-14 Fig. 6.9 Qualitative positionof the ground state levels of donors and acceptors relative to the minimum of the conduction band EC and the maximum of the valence band EV. The quantities Ed and Ea are ionization energies of the donor and acceptor, respectively. Fig. 6.10 Schematic representation of the effect of a donor (a) and an acceptor (b) in a silicon lattice. The valence-five phosphorus atom is incorporated in the lattice at the site of a silicon atom. The fifth valence electron of the phosphorus atom is not required for bonding and is thus only weakly bound. The binding energy can be estimated by treating the system as a hydrogen atom embedded in a dielectric medium. The case of an acceptor (b) can be described similarly: the valence-three boron accepts an electron from the silicon lattice. The hole that is thereby created in the valence band orbits around the negatively charged impurity. The lattice constant and the radius of the defect center are not drawn to scale. In reality the first Bohr radius of the ″impurity orbit″ is about ten times as large as the lattice parameter. 6-15 Estimate of the ionization energy Ed of a donor H atom Donor Energy levels: ε0 → ε0ε me e 4 1 E n = ( −) 2( 4πε 0 h ) 2 n 2 ε describes screening Ground state (n = 1): E1 = 13.6 eV (kT ≈ 25 meV at 300K) Bohr radius: h2 rB = ε 0 = 0.053 nm πme 2 εSi = 11.7 * me → mn ≈ 0.3 me effective mass of a Si conduction electron Ed = 13.6 eV ⋅ 0.3 ≈ 30 meV 11.7 2 Ge: ε Ge = 15.8 ⇒ E d ≈ 6 meV * m n ≈ 0.12me r = rBεHL ⇒ ″smearing″ of the bound donor valence electron over ≈ 103 atoms ⇒ continum approximation for ε justified. Exited states (n > 1) between ground state and conduction band edge: - spacing decreases with increasing energy - finally join conduction band continuum. 6-16 Hydrogen model ⇒ Ed, Ea independent of group V donors and group III acceptors. Yields only order of magnitude. (Hole bound to acceptor traeted like electron bound to donor). Experimental ionization energies Donors (Ed) P [meV] Acceptors (Ea) As Sb [meV] B [meV] Ga In [meV] [meV] [meV] [meV] Si 45 49 39 Si Ge 12 12.7 9.6 Ge Donor: Al 45 57 65 16 10.4 10.2 10.8 11.2 neutral for kT << Ed, positive upon exitation. Acceptor: neutral for kT << Ea, negative upon exitation. Electrically active impurities in Si and Ge single crystals > 1012/cm-3 (1994). ⇒ Ge (ni = 2.4 x 1023 cm-3 at 300K) intrisicly obtainable Si (ni = 1.5 x 1010 cm-3 at 300K) never intrisic Many electrically inactive impurities: cannot bo ionized. 6-17 Carrier densities in doped semiconductors Fig. 6.11 Explanation of the notation commonly used for carrier and impurity concentrations in n- and ptype semiconductors: n and p are the concentrations of ″free″ electrons and holes. the total concentrations ND and NA of donors and acceptors consist of the density of 0 0 neutral, N D and N A , and + − ionised, N D or N A , donors and acceptors, respectively. Electrons in the conduction band (density n) and holes in the valence band (density p) originate either from interband exitations or from impurities. ″Mass action law″ also holds for doped semiconductors if E - EF >> 2kT: C V n ⋅ p = N eff ⋅ N eff e − Eg kT = const.(T ) (6.10) Note: n-doping ⇒ n increases ⇒ EF increases ⇒ E − EF p ∝ exp V kT decreases such that np = const. Neutrality condition: − + n + NA = p + ND 6-18 (6.13) Typical doping levels 1013 - 1017 cm-3 ⇒ interaction between doping atoms neglible ⇒ Fermi statistics applies: Occupation probability of donors by electrons: 0 ND 1 = ND 1 E − EF exp D +1 2 kT (6.14) Factor 1/2: donor can onle carry one electron (↑or ↓) due to Coulomb repulsion. ⇒ statistics modified. Occupation probability of acceptor by holes: − NA = NA 1 E − EF 2 exp A kT (6.15) +1 Factor 2: acceptor always occupied by 1 or 2 electrons: no empty state posible! (when seen as energy level) Fig. 6.12 An acceptor, e.g., B in Si or Ge, can either be occupied by one electron accepted from the valence band if the resulting hole is bound to the acceptor (top) or by two electrons if the hole is ionized (bottom). 6-19 n-type semiconductor n=N e C eff − EC − E F kT also holds for doped semiconductor (6.16) 0 + ND = ND + ND (6.17) 1 E + EF N = N D 1 + exp D kT 2 0 D −1 (6.14) ≡ (6.18) + n = ND + p (6.19) + Simplification: N D >> ni for Si: ni = 1.5 x 1010 cm-2 at 300K ⇒ already obeyed for low doping levels. ⇒ 1 + 0 n ≈ N D = N D − N D = N D 1 − 1 + 1 exp E D − E F kT 2 = ND 1 E − ED 1 + 2 exp F kT 6-20 (6.20) (6.16) = (6.20) ⇒ E − EF C n = N eff exp − C kT EF ⇒ e kT e EF kT 1 = ND E − ED 1 + 2 exp F kT (6.21) 2 + 1 E D E F 1 N D ECkTE D + e kT ⋅ e kT − e =0 C 2 2 N eff E D 1 kT 1 = − e (±) e 16 4 2 ED kT N + De C 2 N eff EC + E D kT E −E ED 1 kT N D CkT D 1+ 8 e =e − 1 C 4 N eff ⇒ ED 1 N D kT E F ≈ E D + kT ln 1 + 8 C e − 1 N eff 4 (6.22) in (6.21) ⇒ n≈ 2ND E D N D kT 1+ 1+ 8 C e N eff + (6.22), (6.23): approximation for N D >> ni not valid at very high temperatures. 6-21 (6.22) E d N D kT 1) ″Freeze-out range″ 8 N C e >> 1 , i.e. low temperatures eff E ⇒ −d 1 C n≈ N D N eff ⋅ e kT 2 (6.24) Sufficiently large number of donors retains their valence electrons. Cf.: V C ni = N eff N eff e − Eg kT (6.11) V N eff , E g → N D , E d 1 ND 1 1 E F = E D + E d + kT ln C 2 N 2 2 eff (6.25) C N eff ∝ T 3 / 2 C N eff < N D for T → 0 C N eff >> N D at high temp. Fig. 6.13 schematic coure of the Fermi energy EF(T)at low temperatures according to eq. 6.25. The maximum arises from the T 3/2 dependence of the C effective density of states N eff in the conduction band. 6-22 E d N D kT 2) Saturation range 8 N C e << 1 ⇒ n ≈ ND = const. eff - All atoms are ionized - Exitation of electrons from valence band still neglible. 3) Intrinsic range ni >> ND (6.22), (6.23) no longer valid! Behaviour like intrinsic semiconductor Fig. 6.14 a Qualitative temperature dependence of the concentration n of electrons in the conduction band of an ntype semiconductor for two different donor concentrations ND´> ND. The width of the forbidden band is Eg and Ed is the ionization energy of the donors; b Qualitative temperature dependence of the Fermi energy EF(T) in the same semiconductor. EC and EV are the lower edge of the valence band, respectively, ED is the position of the donor levels and Ei is the Fermi level of an intrinsic semiconductor. 6-23 Fig. 6.15 The connection n of free electrons in n-type germanium, measured using the Hall effect. For the samples (1) to (6), the donor concentration ND varies between 1018 and 1013 cm3 . The temperature dependence of the electron concentration in the intrinsic region is shown by the dashed line. n-doped Si with Nphosphorus = 3 x 1014 cm3: Saturation range 45 - 500K ⇒ constant carrier density for microelectronic applications. 6-24 Conductivity of semiconductors σ = e(nµn + pµp) (6.1) Metals: µ = µ(EF) Semiconductors: µn: average values for electrons in lower conduction band. µp: average values for holes in upper valence band Estimate of µ for n-doped semiconductors: µn = e τ * mn here τ ≠ τ (EF) ! complicated average value 1/τ ∝ Σ <v>, cf. metals: v = v(EF) E - EF >> kT ⇒ Boltzmann distribution ∞ < v >∝ ∫ v ( E )e mv 2 − 2 kT ∞ dE ∝ 0 ∞ ∫e mv 2 − 2 kT v 2e ∫ 0 ∞ 0 6-25 dv ∝T 0 ∫ ve dE mv 2 − 2 kT mv 2 − 2 kT dv Σ ph ∝< s 2 ( q ) >∝ T Scattering from accustic phonons: (scattering from optical phonons: relaxation time approximation no longer holds) µph ∝ T -3/2 ⇒ (6.26) Scattering from charged defects (ionized donors and acceptors): Σionized defects ∝ <v>-4 Rutherford scattering via Coulombs interaction µionised defects ∝ T 3/2 ⇒ (6.27) 1 1 1 ⇒ = + Addition of collision probabilities τ τ ph τ ionized defects ⇒ µ= 1 (1 / µ ph ) + (1 / µ ionized ⇒ lim µ = µ ionized T →0 defects defects ) ∝ T 3/ 2 lim µ = µ ph ∝ T −3 / 2 T →∞ Fig. 6.16 schematic temperature dependence of the mobility µ for a semiconductor in which scattering from phonons and charged inpurities occurs. 6-26 (6.28) Fig. 6.16 Experimentally determined temperature dependence of the mobility µ of free electrons. For the samples (1) to (6), the donor concentration ND varies between 1018 and 1013 cm-3. The samples are the same as those used for the measurements in Fig. 6.15. 6-27 Fig. 6.18 Experimentally measured conductivity σ of n-type germanium as a function of temperature. For the samples (1) to (6), which were also used for the measurements in Figs. 6.15 and 6.17, the donor concentration ND varies between 1018 and 1013 cm-3. T- dependence of n(T) much stronger that of µ(T) ⇒ µ(T) masked exept for saturation range where n(T) ≈ const. 6-28 ...
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