Carriers Concentration in Semiconductors - V 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India http://folk.uio.no/ravi/semi2013
Motion and Recombination of Electrons and Holes Thermal Motion for meff=0.26mo Average electron or hole kinetic energy 3 2 kt 1 2 2 mv th v th 3kT m eff 23 31.38 10 JK 0.26 9.110 1 31 300K kg 2.310 5 m/s 2.310 7 cm/s
Example An n-type silicon bar: L=3 mm, A (rectangular)= 50*100 μm AT 300 K, donor concentration is 5*10 14 cm -3 determine the electron and hole concentrations, the conductivity and V across the bar when I=1 μa exists in the bar., ni=1.45*10 10 ; n = 1.5*10 3 cm 2 V -1 s -1
Solution n N D 510 14 cm 3 10 2 ( 1.45 10 ) 5 3 4.210 14 p cm 510 qn n 1.610 19 510 14 1.510 3 0.12( cm) 1 J IL V bar L 0. 05 A V
Example: Find intrinsic carrier concentration and conductivity of silicon at 300K. Eg = 1.12eV, me * = 0.97me, mh * =0.16me, n=0.15m 2 V -1 s -1, and p=0.06m 2 V -1 s -1 N C * 2me kt 2 2 h 2.39x10 25 m 3 3/ 2 2 2x3.1415x0.97x9.11x10 (6.63x10 31 34 x1.38x10 2 ) 23 x300 3/ 2 N V * 3/ 2 31 23 3/ 2 2 2m 2 3.1415 0.16 9.11 10 1.38 10 300 hkt x x x x x x x 2 2 34 2 (6.63 10 ) h x =1.60x10 24 m 3 P.Ravindran, PHY02E Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors Chapter - V Six 5
n i N C N V exp( E g 2kT ) n 2.39x10 2.4x10 15 8.07x10 i q n 5 25 x1.60x10 x1.602x10 p q 1 i m 1 p 24 19 19 1.12x1.602x10 exp( ) 23 2x1.38x10 x300 n q( ) i n P x(0.15 0.06) Resistivity = 1/ = 1.24x10 4 m 2.4x10 15 m 3
Definition for DOS & Fermi Dirac function Density of states. Nos of states per unit energy interval level per unit volume 3/ 2 2m g( ) 4 2 h 1/ 2 g() Fermi Dirac statistics Probability of occupancy at a specific energy level g( ) f ( ) f ( ) 1 1 E exp kt F f() Chapter Six
8 Carrier concentration for intrinsic semiconductor n i p n Fi F E E V C g E E E Electron concentration n C C E F C e E e e kt E d E h m d g f n ) exp( 1 2 4 ) ( ) ( 2 3/ 2 * fe() ge() EC
Terminology Compensated Material N D = N A n-type Material N D > N A (n dominates p: n > p) p-type Material N A > N D (p dominates n: p > n)
Both electron and hole doping It is possible to add donors to a p-type crystal or, conversely, to add acceptors to n-type material. If equal concentrations of donors and acceptors permeate the semiconductor, the semiconductor remains intrinsic. If the concentration of donor atoms add to a p-type semiconductor exceeds the acceptor concentration (N D > N A ), the specimen is changed from a p-type to an n-type semiconductor. Conversely, the addition of a sufficient number of acceptors to an n-type sample results in a p- type semiconductor.
n-type Semiconductor Antimony, phosphorus, and arsenic donate excess electron carriers and are referred to as donor, or n-type, impurities The number of electrons increases and the number of holes decreases below that which would be available in the intrinsic semiconductor. The number of holes decreases because the larger number of electrons present causes the rate of recombination of electrons with holes to increase The dominant carriers are the electrons
p-type Semiconductor Boron, Aluminum, and Gallium are trivalent atoms that provide electrons to fill only three covalent bonds. The vacancy that exists in the fourth bond constitutes a hole. This type of impurity makes positive carriers available because it creates holes which can accept electrons. Called acceptors and form p-type semiconductors in which holes are the predominant carriers
Semiconductors Movement of Charges Charge carriers in a semiconductor can be positive, negative, or both. When an electron moves into the conduction band, it leaves behind a vacant site, called a hole.
Semiconductors Movement of Charges, cont. The holes act as charge carriers. Electrons can transfer into a hole, leaving another hole at its original site. The net effect can be viewed as the holes migrating through the material in the direction opposite the direction of the electrons. The hole behaves as if it were a particle with charge +e.
Doped Semiconductors Impurities can be added to a semiconductor. This process is called doping. Doping Modifies the band structure of the semiconductor Introduce carriers. Modifies its resistivity Can be used to control the conductivity of the semiconductor
Thermal Equilibrium Equilibrium No external forces (voltages, electric fields, temp.gradients) Thermal equilibrium is a dynamic situation in which every process is balanced by its inverse process. Thermal equilibrium means that time can run toward the past as well as into the future. E2 E1
Mass-Action Law Electron-hole pairs: generation rate = recombination rate Generation: G = f 1 (T) f 1 : determined by crystal physics and T Recombination: R = npf 2 (T) Electrons and holes must interact to recombine At equilibrium, G = R npf ( T) f ( T) 2 1 f ( T) 1 2 np f3( T ) ni f2( T) Intrinsic case (all carriers result from excitation across the forbidden gap): n = p = n i
Heavy Doping Light doping: impurity atoms do not interact with E each other impurity level Heavy doping: perturb the band structure of the host crystal reduction of bandgap Ec Ed Ev Eg (E)