Technische Universität Graz Institute of Solid State Physics Intrinsic Semiconductors
ermi function f(e) is the probability that a state at energy E is occupied. f( E) 1 E E 1 exp kt B
ermi energy The ermi energy is implicitly defined as the energy that solves the following equation. n Here n is the electron density. D( E) f( E) de The density of states, the total number of electrons and the temperature are given. To find the ermi energy, guess one and evaluate the integral. If n turns out too low, guess a higher E and if n turns out too high, guess a lower E.
free electrons (simple model for a metal) 2 2 1 2 p 2 2 2 E( k) 2 mv kx ky k 3-d density of states z 2m 2m E dispersion relation m 3/2 k x k y DE ( ) 2 0 for E 0 2 2 3 E for E 0
Silicon band structure E c = bottom of the conduction band E g = E c -E v E v = top of the valence band E k x k y [111] k=0 [100] Near the bottom of the conduction band, the band structure looks like a parabola.
Effective mass p E( k) k k k mv 2m 2m 2 de( k ) kx dkx m 2 2 d E( k) 2 dk m 2 2 2 2 2 1 2 x y z 2 x Effective mass 2 2 This effective mass is used to describe the response of electrons to external forces in the particle picture. m * x d E( k) dk 2 x k x E * ee m a
GaN
Anisotropic effective mass in silicon 100 The electrons seem to have different masses when the electric field is applied in different directions.
E 2 2 kx 0.85 2 2 2 2 a k y kz 2m 2m 2m 2 l t t E c
Holes When all states in a band are occupied, the band does not contribute to the current. There are as many left-moving electrons as right-moving electrons. I ev occupied k I ev ev all k I k empty k empty k ev k k k
valence band, holes In the valence band, the effective mass is negative. m * 2 2 d E k ( ) dk 2 x 0
Holes Charge carriers in the valence band can be considered to be positively charged holes. The number of holes in the valence band is the number of missing electrons. m* h = effective mass of holes m * h 2 2 d E k ( ) dk 2 x ee m a * h
Silicon density of states D( E) Dc E Ec T = 300 K T = 300 K 10 12 electrons in the conduction band
Boltzmann approximation f( E) 1 E E exp 1 kt B E E exp kt B E E kt B 3 E E kt B
Density of electrons in the conduction band E E n D( E) f( E) de Dcexp EEcdE kt E E B c c x E E c 0 x 2 xexp dx kbt kt B 3/2 E 2 E c Dc ( )exp 3/2 E E c n Nc T kbt exp kt B kt B N c * 2D 3/2 c mkt B kbt 2 2 2 3/2 = effective density of states
Density of electrons in the conduction band n 3/2 * mkt B E E c 2 exp 2 2 kt B L X n 3/2 T E E c Nc exp 300 kt B
Density of holes in the valence band D( E) Dv Ev E 1 E E 1 f( E) 1 exp E E kt B 1 exp kt B E k
Boltzmann approximation E E exp kt B 1 f( E) 1 1 E E exp 1 kt B E E kt B 3 E E kt B
Density of holes in the valence band E v v E E p DE ( ) 1 f( E) de Dv exp Ev EdE kt B E 2 v E Dv exp 3/2 Ev E p Nv kbt exp kt B kt B E N v * mkt h 2 2 B2 3/2 = Effective density of states in the valence band
Density of holes in the valence band p * 3/2 h B v 2 kt B mkt E E 2 exp 2 p 3/2 T Ev E Nv exp 300 kt B
Law of mass action E E c Ev E np Ncexp Nvexp kt B kt B E c np Eg NcNvexp kt B E v E g or intrinsic semiconductors (no impurities) Eg n p ni NcNv exp 2 kt B intrinsic carrier density
Intrinsic carrier concentration log 10 (n i ) cm -3 GaAs Si Ge n N N i v c 3 T Eg exp 300 2kT B 1/T 300 K Silicon has ~ 5 10 22 atoms/cm 3 Good for thermometer, bad for designing circuits.
ermi energy of an intrinsic semiconductor E E c Ev E n p Ncexp Nvexp kt B kt B N v E Ec Ev E exp Nc kbt 2E Ec E v N v ln kt B kt B Nc E Ec Ev kt B N v ln 2 2 Nc
Temperature dependence of E Si GaAs E Ec Ev kt B N v ln 2 2 Nc
http://lamp.tu-graz.ac.at/~hadley/ss1/semiconductors/intrinsic_so.php
Technische Universität Graz Institute of Solid State Physics Extrinsic semiconductors The introduction of impurity atoms that can add electrons or holes is called doping. n-type : donor atoms contribute electrons to the conduction band. Examples: P, As in Si. p-type : acceptor atoms contribute holes to the valence band. Examples: B, Ga, Al in Si.
Technische Universität Graz Institute of Solid State Physics n and p n The electron density and hole density are: E E c Nc exp p kt B Ev E Nv exp kt B The law of mass action: 2 np ni NvNcexp E B g kt
Ionization of dopants Easier to ionize a P atom in Si than a free P atom E n 4 me 8 hn 2 2 2 0 Ionization energy is smaller by a factor: * m m 0 r 0 2 Ionization energy ~ 25 mev
acceptors in Si donors in Si
Crystal growth Czochralski Process add dopants to the melt images from wikipedia
Crystal growth loat zone Process Neutron transmutation 30 Si + n 31 Si + 31 Si 31 P + image from wikipedia
Gas phase diffusion AsH 3 (Arsine) or PH 3 (phosphine) for n-doping B 2 H 6 (diborane) for p-doping. http://www.microfab.de/foundry/services/diffusion/index.html
Chemical vapor deposition Epitaxial silicon CVD SiH 4 (silane) or SiH 2 Cl 2 (dichlorosilane) PH 3 (phosphine) for n-doping or B 2 H 6 (diborane) for p-doping. image from wikipedia
Ion implantation Implant at 7º to avoid channeling
http://www.synopsys.com