Solid State Device Fundamentals

4. lectrons and Holes Solid State Device Fundamentals NS 45 Lecture Course by Alexander M. Zaitsev alexander.zaitsev@csi.cuny.edu Tel: 718 982 2812 4N101b 1

4. lectrons and Holes Free electrons and holes in semiconductors A small number of electrons from the valence band is excited into the conduction band forming this way free holes and free electrons. The excitation may occure in different ways, e.g. via thermal agitation,, or via absorption of light. 2

in c re a sin g e le c tr o n e n e rg y i n c re a sin g h o le e n e rg y Solid State Device Fundamentals 4. lectrons and Holes Thermalization of free electrons and holes c v Both excited electrons and holes tend to seek their lowest energy positions. Free electrons fall down to the bottom in the conduction band. Free holes float up the valence band like bubbles in water.

ffective mass of electrons in crystal 4. lectrons and Holes Interaction of moving free electrons and free holes with crystal atoms can be presented as a change of their mass. If the interaction retard motion, the effective mass increases. If the interaction promotes motion, the effective mass decreases. ffective mass, m* d 2 2 / dk 2 4

4. lectrons and Holes Calculated energy band structure of silicon Calculated energy band structure of GaAs Directions in k-space: X [100], K [110], L [111] 5

ffective electron mass in semiconductors 4. lectrons and Holes In an electric field,, acceleration of electrons and holes a is determined by their effective mass: a= e m n a= e m p electrons holes lectron and hole effective masses is common semiconductors Si Ge GaAs InAs AlAs m n /m 0 0.26 0.12 0.068 0.02 2 m p /m 0 0.9 0. 0.5 0. 0. 6

Density of states 4. lectrons and Holes Density of states g c () is the number of states per unit energy and unit volume the electrons can occupy in crystal. g c ( ) number of states in volume 1 ev cm Crystal can be modeled as an infinite quantum well of size L, in which electrons with effective mass, m *, are free to move. Motion of free electrons in conduction band and free holes in valence band is described by Schroedinger wave: Calculation of the number of states with wavenumber less than k. 7

Density of states in semiconductors 4. lectrons and Holes D c c c D v v D v Density of states in conduction band: g c 8 mn 2mn c 8 mp 2mp v ( ) h Density of states in valence band: g v ( ) h 8

4. lectrons and Holes Real density of states in silicon 9

Density of states in silicon 4. lectrons and Holes Calculate density of states in conduction and valence bands of silicon for electrons and holes of thermal energy at room temperature. m n * = 0.26 m e m p * = 0.9 m e h = 6.6 10-4 m 2 kg / s m e = 9.1 10-1 kg k B = 1.8 10-2 m 2 kg s -2 K -1 g 8 m ) 2m p p v ( h v g 8 m ) 2m n n c ( h c 10

Thermal equilibrium and the Fermi function 4. lectrons and Holes 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. f ( ) 1 1 e F kt 11

College of 4. Staten lectrons Island / Holes CUNY Dependence of Fermi function on temperature The Fermi-Dirac distribution function, also called Fermi function, provides the probability of occupancy of energy levels by electrons. The Pauli exclusion principle postulates that only one electron can occupy a single quantum state. Therefore, as electrons are added to an energy band, they fill the available states with ever higher energy, just like sand fills a bucket. At absolute zero temperature (T = 0 K), the energy levels are filled up to a maximum energy, which is called the Fermi level and no states above the Fermi level are filled. At higher temperature, the transition between completely filled states and completely empty states is gradual. Free electrons in VB Free holes in VB The Fermi function at three different temperatures. ight of the 24 possible configurations in which 20 electrons can be placed having a total energy of 106 ev. 12 College College of Staten of Staten Island Island / CUNY / CUNY

4. lectrons and Holes lectrons of high and low energy For electrons of high energy (greater than k B T), the Fermi function can be simplified to Boltzmann distribution function: f f f + kt + 2kT f kt f ( ) e f kt Boltzmann approximation f + kt f f kt f f 2kT kt f kt f ( ) 1 e f kt 0.5 1 f() 1

4. lectrons and Holes 1. What is the probability for an electron to have an excess of energy of k B T? f ( ) 1 1 e F kt 2. Show that at an energy over k B T the Fermi function can be approximated by Boltzmann distribution function. 14

Charge carrier density versus energy College of 4. Staten lectrons Island / Holes CUNY The density of electrons in a semiconductor is related to the density of available states and the probability that each of these states is occupied. The density of occupied states per unit volume and energy, n(), ), is the product of the density of states in the conduction band, g c () and the Fermi function, f(): Since holes correspond to empty states in the valence band, the probability of having a hole equals the probability that a particular state is not filled, so that the hole density per unit energy, p(), equals: 15 College College of Staten of Staten Island Island / CUNY / CUNY

Density of electrons and holes College of 4. Staten lectrons Island / Holes CUNY Density of electrons, n, and holes, p, in a semiconductor is obtained by integrating the product of the density of states and the probability density function over all possible states. For electrons in the conduction band the integral is taken from the bottom of the conduction band, c, to the top of the conduction band: n n top of conduction band c 8m n h 2m n c f ( ) g c c e ( ) d f kt d For holes in the valence band the integral is taken from the top of the valence band, v, to the bottom of the valence band: p p 8m p h bottom of valence band V 2m p c f ( ) g V [1 e V ( ) d f kt ] d 16 16

n N e c ( c ffective density of states f )/ kt p N e v ( f v 4. lectrons and Holes With a good accuracy, the total electron and hole concentrations can be found by the following formulae: )/ kt N c 2m nkt 2 2 h 2 N v 2m pkt 2 2 h 2 N c is the effective density of states of the conduction band. For silicon at room temperature N c = 2.8 10 19 cm -. N v is the effective density of states of the valence band. For silicon at room temperature N V = 1.04 10 19 cm -. The closer f to the bottom of the conduction band the larger the electron concentration. The closer f to the top of the valence band the larger the hole concentration. 17

Solid State Device Fundamentals 4. lectrons and Holes 18 Intrinsic Semiconductors * * ln 4 ln 2 2 ) ( ) ( ) ( p n F C V V C F kt V kt C m m kt midgap N N kt e N e N V F F C For every electron in the CB there is a hole in the VB: p n Fermi level is in middle of bandgap if effective masses not too different for electrons and holes.

4. lectrons and Holes Charge carrier concentration and Fermi level in Si Find concentration of electrons and holes in silicon at the "conventional" limit temperatures of operation of silicon bipolar diode ( 65 C to +125 C). n N C e kt g N V e kt g p F midgap kt 4 m ln m n p * * m n * = 0.26 m e m p * = 0.9 m e k B T (00K) = 26 mev N C [cm - ] = 0.54 10 17 T 1.5 N V [cm - ] = 0.2 10 17 T 1.5 19

4. lectrons and Holes Intrinsic charge carrier concentration Since for intrinsic semiconductors n = p, the concentration of electron and hole can be expressed as the intrinsic charge carrier concentration n i : i C V ( C V ) G 2kT NC NV e kt 2 np n N N e For silicon, intrinsic charge carrier concentration can be found using the following formula: g 2kT n 17 i[cm ] 0.10 e C 66 20

College of 4. Staten lectrons Island / Holes CUNY Intrinsic carrier concentration versus temperature Intrinsic carrier density versus temperature is calculated with (solid lines) and without (dotted lines) the temperature dependence of the energy bandgap. For silicon at room temperature, n i is about 10 10 cm - 21 College College of Staten of Staten Island Island / CUNY / CUNY