Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 2 SEMICONDUCTOR BANDSTRUCTURE HOLES IN SEMICONDUCTORS: HOW DO HOLES MOVE? Holes behave as if they carry a positive charge. t2 > t1>0 t=0 t = t1 Ε Ε Fx F E D G C B A t = t2 F kx D H I J K Ε Fx E G C B A kx H I J K E D G C B A kx H I J K (c) (b) (a) Fx F F ve e je vh h jh (d) The movement of an empty electron state, i,e,. a hole under an electric field. The electrons move in the direction opposite to the electric field so that the hole moves in the direction of the electric field thus behaving as if it were positively charged, as shown in (a), (b), and (c). (d) The velocities and currents due to electrons and holes. The current flow is in the same direction, even though the electron and holes have opposite velocities. The electron effective mass in the valence band is negative, but the hole behaves as if it has a positive mass. © Prof. Jasprit Singh www.eecs.umich.edu/~singh FREE CARRIERS IN SEMICONDUCTORS: INTRINSIC CARRIERS In semiconductors, at finite temperatures, there are electrons in the conduction band and holes in the valence band. METALS Evac ENERGY EF Ec Electrons in the conduction band can carry current (a) SEMICONDUCTORS Evac ENERGY – – Electrons in the conduction band (density, n) carry current Ec Mobile carrier density = n + p + + + EV Holes in the valence band (density, p) carry current Valence band (b) (a) A schematic showing allowed energy bands in electrons in a metal. The electrons occupying the highest partially occupied band are capable of carrying current. (b) A schematic showing the valence band and conduction band in a typical semiconductor. In semiconductors only electrons in the conduction band holes in the valence band can carry current. For small electron (n), hole (p) densities we can use Boltzmann approximation: n = Nc exp [(EF– Ec)/kBT] where Nc = 2 p=2 ( m*ekBT 3/2 2πh2 ) m*hkBT 3/2 exp [(Ev– EF)/kBT] 2πh2 ( ) = Nv exp [(Ev– EF)/kBT] Intrinsic case: ni = pi = 2 kBT 3/2 (m*em*h)3/4 exp (–Eg/2kBT) 2πh2 ( ) © Prof. Jasprit Singh www.eecs.umich.edu/~singh INTRINSIC CARRIER DENSITIES FOR SOME SEMICONDUCTORS 1019 1000 500 T(°C) 200 100 27 0 –20 INTRINSIC CARRIER DENSITY ni (cm–3) 1018 1017 1016 Ge 1015 1014 Temperature dependence of ni, pi in Si, Ge, GaAs Si 1013 1012 11 10 10 10 109 GaAs 10 8 10 7 106 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1000/T (K–1) MATERIAL CONDUCTION BAND EFFECTIVE DENSITY (NC ) VALENCE BAND EFFECTIVE DENSITY (NV ) INTRINSIC CARRIER CONCENTRATION Si (300 K) 2.78 x 1019 cm–3 9.84 x 1018 cm–3 1.5 x 1010 cm–3 Ge (300 K) 1.04 x 1019 cm–3 6.0 x 1018 cm–3 2.33 x 1013 cm–3 GaAs (300 K) 4.45 x 1017 cm–3 7.72 x 1018 cm–3 1.84 x 106 cm–3 (ni = pi) Effective densities and intrinsic carrier concentrations of Si, Ge and GaAs. The numbers for intrinsic carrier densities are the accepted values even though they are smaller than the values obtained by using the equations derived in the text. © Prof. Jasprit Singh www.eecs.umich.edu/~singh DOPING OF SEMICONDUCTORS: DONORS AND ACCEPTORS If an impurity atom replaces a host semiconductor atom in a crystal it could donate (donor) an extra electron to the conduction band or it could accept (acceptor) an electron from the valence band producing a hole. All 4 outer electrons go into the valence band EC EV Silicon host atom = + + Electron-ion Coulombic Pentavalent donor impurity = Silicon-like + attraction A schematic showing the approach one takes to understand donors in semiconductors. The donor problem is treated as the host atom problem together with a Coulombic interaction term. The silicon atom has four “free” electrons per atom. All four electrons are contributed to the valence band at 0 K. The dopant has five electrons out of which four are contribted to the valence band, while the fifth one can be used for increasing electrons in the conducton band. E Si Si Si Ec Excess Donor Si positive level charge – Si Si Ev Excess electron + As Si n-type silicon Arsenic (As) atom donates one electron to the conduction band to produce an n-type silicon Conduction band Ed AAAA AAAA AAAA ( )( εεso )2 eV m* Ed = Ec –13.6 m o Valence band Charges associated with an arsenic impurity atom in silicon. Arsenic has five valence electrons, but silicon has only four valence electrons. Thus four electrons on arsenic form tetrahedral covalent bonds similar to silicon, and the fifth electron is available for conduction. The arsenic atom is called a donor because when ionized it donates an electron to the conduction band. © Prof. Jasprit Singh www.eecs.umich.edu/~singh FREE CARRIERS IN DOPED SEMICONDUCTORS If electron (hole) density is measured as a function of temperature in a doped semiconductor, one observes three regimes: Freezeout: Temperature is too small to ionize the donors (acceptors), i.e., kBT < EC – ED (kBT< ED – EV). Saturation: Most of the donors (acceptors) are ionzed. Intrinsic: Temperature is so high that ni > doping density. TEMPERATURE (K) 10 17 500 1000 300 200 100 75 50 Si ELECTRON DENSITY n (cm–3) Nd = 1015cm–3 Intrinsic range 10 16 Saturation range 10 15 10 Freeze-out range 14 ni 10 13 0 4 8 12 16 20 1000/T (K–1) Electron density as a function of temperature for a Si sample with donor impurity concentration of 1015 cm–3. It is preferable to operate devices in the saturation region where the free carrier density is approximately equal to the dopant density. It is not possible to operate devices in the intrinsic regime, since the devices always have a high carrier density that cannot be controlled by electric fields. every semiconductor has an upper temperature beyond which it cannot be used in devices. The larger the bangap, the higher the upper limit. © Prof. Jasprit Singh www.eecs.umich.edu/~singh