Session 4: Solid State Physics Dopants and Carrier Concentration 1 1. Introduction 2. Crystal 3. Cubic Lattices Outline 4. Other 5. Miller Indices A B C D E F G H I J 2 1. Introduction 2. Crystal 3. Cubic Lattices Intrinsic Material 4. Other 5. Miller Indices Conduction band Free electron Intrinsic = pure 14P electron/hole pair Thermal ̿ is generated excitation hole ̿ 14P 14P 14P Valance band ͢ Ɣ ͤ Ɣ ͢$ʚ͎ʛ = # of e-/cm 3 ͢ 14P = # of h +/cm 3 ͤ = intrinsic carrier concentration ͢$ ͥͤ ͯͧ note that totally there are ͦͧ ͯͧ electrons ͢$ͯ$ ʺ Ɣ 1.5 Ɛ 10 ͗͡ Ͱͧͤͤ° 2 Ɛ 10 ͗͡ 1 out of bond is broken! 10 ͥͧ 3 1. Introduction 2. Crystal 3. Cubic Lattices Intrinsic Material 4. Other 5. Miller Indices 10 22 Conduction band Ge Intrinsic = pure Si 10 18 GaAs Thermal ̿ excitation 10 14 ʟ ̿ ͧ ͯ 10 Valance band ͡ 10 ͗ ʞ $ ͢ 10 6 10 2 1 0 400 800 1200 1600 ͎ Energy Band Gap determines the intrinsic carrier concentration. ni EgGe< EgSi< EgGaAs 4 1. Introduction 2. Crystal 3. Cubic Lattices Doping 4. Other 5. Miller Indices Doping means mixing a pure semiconductor with impurities to increase its electrical conductivity Can be done in two ways: Increasing the number of electrons by Increasing the number of holes by mixing pentavalent elements such as mixing trivalent elements such as phosphorous, arsenic, antimony aluminum, boron, gallium (means (means adding donor impurities) adding acceptor impurities) Donors and acceptors are known as dopants. Dopant ionization energy ~50meV (very low). Possible dopant deactivation & defect formation -- -- 14 -3 N or P : N D or N A < 10 cm - - 14 -3 16 -3 N or P : 10 cm < ND or N A < 10 cm 16 -3 18 -3 N or P : 10 cm < N D or N A < 10 cm + + 18 -3 20 -3 N or P : 10 cm < N D or N A < 10 cm ++ ++ 20 -3 N or P : N D or N A > 10 cm 5 1. Introduction 2. Crystal 3. Cubic Lattices Donor and Acceptor in the Band Model 4. Other 5. Miller Indices Conduction band ̿ Donor ionization Donor level energy ̿ Acceptor Acceptor level ̿ Valance band ionization energy ̿ @ Ɣ 300°ͅ ͎ͅ Ɣ 26͙͐͡ Ionization energy of selected donors and acceptors in silicon Donors Acceptors Dopant Sb P As B Al In Ionization Eng Ec-Ed or 39 44 54 45 57 160 Ea-Ev (meV) 6 1. Introduction 2. Crystal 3. Cubic Lattices Donor and Acceptor in the Band Model 4. Other 5. Miller Indices Si Ionization energies of shallow DONORS(eV) As = 0.054; P = 0.045; Sb = 0.043 Ionization energies of shallow ACCEPTORS(eV) Al = 0.072; B = 0.045; Ga = 0.074; In = 0.157 Ionization energy of various impurities in Ge, Si, and GaAs at 300K. 7 1. Introduction 2. Crystal 3. Cubic Lattices Density of States 4. Other 5. Miller Indices ̿ ̾ʚ̿ʛ ̿ ̾ ̿ ̾ʚ̿ʛ = number of states per in the energy range between ̾ʚ̿ʛ̿͘ ͗ͧ͡ ̿ and near the band edges: ̿ ƍ ̿͘ number of states in ∆ ( ͥ ) ̾ʚ̿ʛ ≡ ∆ volume .(w ͡) 2͡)ʚ̿ Ǝ ̿ʛ ͛ ̿ Ɣ ̾ ̿ Ɣ ͦͧ͜ ͡+ 2͡+ʚ̿ Ǝ ̿ʛ ͛ ̿ Ɣ ̾ ̿ Ɣ ͦͧ͜ 8 1. Introduction 2. Crystal 3. Cubic Lattices Thermal Equilibrium and Fermi Function 4. Other 5. Miller Indices An Analogy for Thermal Equilibrium Sand particles Dish Vibrating Table There is a certain probability for the electrons in the conduction band to occupy high-energy states under the agitation of thermal energy. 9 1. Introduction 2. Crystal 3. Cubic Lattices Probability of a State at E being Occupied 4. Other 5. Miller Indices •There are g1 states at E1, g2 states at E2… ̿ There are N electrons, which constantly shift ••• ̿& among all the states but the average electron 1 2 3 … gk energy is fixed at 3kT/2. •There are many ways to distribute N among n1, n2, n3….and satisfy the 3kT/2 condition. ••• ••• ̿ͦ •The equilibrium distribution is the 1 2 3 … g2 distribution that maximizes the number of ••• •• ••• combinations of placing n1 in g1 slots, n2 in g2 ̿ͥ 1 2 3 … g slots…. : 1 ͢$ 1 Ɣ ʚͯ ʛ⁄ & ͛$ 1 ƍ ͙ ÿ is a constant determined by the condition ̿ ȕ ͢$ Ɣ ͈ $ 10 1. Introduction 2. Crystal 3. Cubic Lattices Fermi Function 4. Other 5. Miller Indices Particles can be classified into 3 categories: 1. Classical particles (ball) Maxwell-Boltzmann dist. 2. Bosons (photons) Bose-Einstein dist. 3. Fermions(undist + Pauli Exclution) (electrons) Fermi-Dirac dist. Probability that an available state at energy E is occupied: 1 Ef is called the Fermi energy or the Fermi level. ͚ʚ̿ʛ Ɣ There is only one Fermi level in a system at equilibrium. 1 ƍ ͙ʚͯÿʛ⁄ & 1 Boltzmann constant, 0.9 ͎ Ɣ 0°ͅ k = 1.38 × 10 −23 J/K 0.8 ͎ Ɣ 100°ͅ k = 8.62×10 -5 eV/K 0.7 ͚ʚ̿ʛ ͎ Ɣ 300°ͅ 0.6 ͎ Ɣ 500°ͅ 0.5 0.4 0.3 0.2 0.1 0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 (ev) ̿ Ǝ ̿ 11 1. Introduction Fermi Function 2. Crystal 3. Cubic Lattices 4. Other Probability of Electron Distribution 5. Miller Indices 1 Ef is called the Fermi energy or the Fermi level. ͚ʚ̿ʛ Ɣ 1 ƍ ͙ʚͯÿʛ⁄ & If we are 3kT away from the Fermi energy then we might use Boltzmann approximation: ̿ if ͚ʚ̿ʛ Ɩ ͙ͯʚͯÿʛ⁄ & ̿ Ǝ ̿ ≫ ͎͟ ͯʚęͯʛ⁄ & if ͚ ̿ Ɩ 1 Ǝ ͙ ̿ Ǝ ̿ ≪ Ǝ͎͟ ̿ ƍ 3͎͟ ̿ ƍ 2͎͟ ͯʚͯ ʛ⁄ & ̿ ƍ ͎͟ ͙ ÿ ̿ ̿ Ǝ ͎͟ "͚ Ǝ Ɩ ʚ̿ Ǝ ̿ʛ ̿ Ǝ 2͎͟ "̿ ̿ Ǝ 3͎͟ 0.5 1 ͚ʚ̿ʛ 12 1. Introduction 2. Crystal 3. Cubic Lattices Electron / Hole Concentrations 4. Other 5. Miller Indices Derivation of n and p from D(E) and f(E) Integrate n(E) over all the energies in the conduction band to obtain n Top Of Conduction Band Ϧ ͡) 2͡) ͯʚͯÿʛ⁄ & ͢ Ɣ ǹ ͚ ̿ ̾ ̿ ̿͘ Ɣ ǹ ̿ Ǝ ̿ ͙ ̿͘ ͦͧ͜ Ė Ė Ϧ ͡) 2͡) ͯʚüͯÿʛ⁄ & ͯV⁄ & ͢ Ɣ ͙ ǹ ͙ ͘ ʚ Ɣ ̿ Ǝ ̿ʛ ͦͧ͜ ͤ ͧ⁄ ͦ Nc is called the effective density 2͡)͎͟ ͯʚüͯÿʛ⁄ & ͈ ≡ 2 of states of the conduction band. ͢ Ɣ ͈ ͙ ͦ͜ ď ͤ Ɣ ǹ ʞ1 Ǝ ͚ ̿ ʟ̾ ̿ ̿͘ Top Of Conduction Band ͧ⁄ ͦ Nv is called the effective density ͯʚÿͯďʛ⁄ & 2͡+͎͟ of states of the valence band. ͤ Ɣ ͈ ͙ ͈ ≡ 2 ͦ͜ Closer Ef to Ec the larger n , closer Ef to Ev the larger p For Si: Nc =2.8×1019 cm-3 , Nv =1.04×1019 cm-3 13 1. Introduction 2. Crystal 3. Cubic Lattices Fermi Level and Carrier Concentration 4. Other 5. Miller Indices ̿ ͯʚüͯÿʛ⁄ & ͢ Ɣ ͈ ͙ ͈ ̿ Ɣ ̿ Ǝ ͎͟ ln ͢ ̿ 18 20 10 13 10 14 10 15 10 16 10 17 10 10 19 10 ͯͧ ͈ or ͈ ʚ͗͡ ʛ 14 1. Introduction 2. Crystal 3. Cubic Lattices Carrier Concentrations 4. Other 5. Miller Indices Multiply ͯʚüͯÿʛ⁄ & and ͯʚÿͯďʛ⁄ & ͢ Ɣ ͈ ͙ ͤ Ɣ ͈ ͙ ͯʚüͯďʛ⁄ & ͯĀ⁄& ͤ͢ Ɣ ͈͈ ͙ Ɣ ͈͈ ͙ Law of Mass Action ͦ ͯĀ⁄ͦ& ͤ͢ Ɣ ͢$ ͢$ ≡ ͈͈ ͙ In an intrinsic (undopped) semiconductor, ͢ Ɣ ͤ Ɣ ͢$ ͧ⁄ ͦ ͧ⁄ ͦ 2͡)͎͟ 2͡+͎͟ ͈ ≡ 2 ͈ ≡ 2 ͢$ ͎ Ɣ ⋯ ͦ͜ ͦ͜ ̿$ ≡ ̿ʺ )Ͱ+ ∗ ̿ ƍ ̿ ͎͟ ͈ ̿ ƍ ̿ 3͎͟ ͡+ ̿ ƍ ̿ → ̿$Ɣ ƍ ln Ɣ ƍ ln ∗ Ɩ 2 2 ͈ 2 4 ͡) 2 ͯʚüͯÿʛ⁄ & ʚÿͯĜʛ⁄ & ͢ Ɣ ͈ ͙ Ĝ ͢ Ɣ ͢$ ͙ ͯʚÿͯďʛ⁄ & ͤ Ɣ ͈ ͙ ʤ ʚĜͯÿʛ⁄& ͤ Ɣ ͢$ ͙ 15 1. Introduction 2. Crystal 3. Cubic Lattices Intrinsic Material 4. Other 5. Miller Indices 500 400 300 250 ͎ ʟ 10 16 ͯͧ ʞ͗͡ Ge $ ͢ 10 14 2.5 Ɛ 10 ͥͧ ͗ͯͧ͡ ͯĀ⁄ͦ& ͢$ ≡ ͈͈ ͙ Si 10 12 ͧ⁄ ͦ 2͎͟ GaAs ͥͤ ͯͧ ͢$ Ɣ 2 10 10 1.5 Ɛ 10 ͗͡ ͦ͜ ͧ/ͨ ∗ ∗ ͯĀ⁄ͦ& ͡)͡+ ͙ 10 8 2 Ɛ 10 ͪ͗ͯͧ͡ 10 6 1000 / T K -1 2 3 4 16 1. Introduction 2. Crystal 3. Cubic Lattices Density of States 4. Other 5. Miller Indices ̾ʚ̿ʛ ͚ʚ̿ʛ ̾ ̿ . ͚ʚ̿ʛ intrinsic ̿ ̾ʚ̿ʛ ̿ ̿ ̿ ̿ Ɣ ̿$ ̿ ̾ʚ̿ʛ n-type ̿ ̾ʚ̿ʛ ̿ ̿ ̿ ̿ Ɣ ̿$ ̿ ̾ʚ̿ʛ p-type ̿ ̾ʚ̿ʛ ̿ ̿ ̿ ̿ Ɣ ̿$ ̿ ̾ʚ̿ʛ 17 1. Introduction 2. Crystal 3. Cubic Lattices Density of States 4. Other 5. Miller Indices Donor ̿ ̿ ̿ ͎ Ɣ 0°ͅ ͎Ɣ50°ͅ ͎Ɣ300°ͅ Acceptor ̿ ̿ ̿ ͎ Ɣ 0°ͅ ͎ Ɣ 50°ͅ ͎ Ɣ 300°ͅ 18 1. Introduction 2. Crystal 3. Cubic Lattices Carrier Concentration vs. Temperature 4. Other 5. Miller Indices At room temperature, all the ͢ shallow dopants are ionized. (Extrinsic region) 2 ͈ When the temperature is Freeze decreased sufficiently (~100 K), out some of the dopants are not Extrinsic region ionized.