# Semiconductor Device Physics

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1 1 Semiconductor Device Physics Lecture

2 Semiconductor Device Physics 2

3 Three primary types of carrier action occur inside a semiconductor: Drift: charged particle motion in response to an applied electric field. Diffusion: charged particle motion due to concentration gradient or temperature gradient. Recombination-Generation: a process where charge carriers (electrons and holes) are annihilated (destroyed) and created. 3

4 electron Carrier Scattering Mobile electrons and atoms in the Si lattice are always in random thermal motion. Electrons make frequent collisions with the vibrating atoms. Lattice scattering or phonon scattering increases with increasing temperature. Average velocity of thermal motion for electrons: ~1/1000 x speed of light at 300 K (even under equilibrium conditions). Other scattering mechanisms: Deflection by ionized impurity atoms. Deflection due to Coulombic force between carriers or carrier-carrier scattering. Only significant at high carrier concentrations. The net current in any direction is zero, if no electric field is applied.

5 Carrier Drift When an electric field (e.g. due to an externally applied voltage) is applied to a semiconductor, mobile charge-carriers will be accelerated by the electrostatic force. This force superimposes on the random motion of electrons F = qe 5 4 E electron Electrons drift in the direction opposite to the electric field Current flows. Due to scattering, electrons in a semiconductor do not achieve constant velocity nor acceleration. However, they can be viewed as particles moving at a constant average drift velocity v d. 5

6 Drift Current v d t v d t A p v d t A All holes this distance back from the normal plane will cross the plane in a time t All holes in this volume will cross the plane in a time t Holes crossing the plane in a time t q p v d t A Charge crossing the plane in a time t q p v d A q p v d Charge crossing the plane per unit time, Hole drift current I P drift (Ampere) Current density associated with hole drift current, J P drift (A/m 2 ) 6

7 7 Drift Velocity vs. Electric Field v v d d pe E n Linear relation holds in low field intensity, ~510 3 V/cm

8 Hole and Electron Mobility 8 has the dimensions of v/e : cm/s V/cm 2 cm V s Electron and hole mobility of selected intrinsic semiconductors (T = 300 K) Si Ge GaAs InAs n (cm 2 /V s) p (cm 2 /V s)

9 9 For holes, I J P drift P drift qpv A qpv d d Hole and Electron Mobility Hole current due to drift Hole current density due to drift In low-field limit, J v d P drift pe q pe p μ p : hole mobility Similarly for electrons, J J N drift v d N drift qnvd ne q ne n Electron current density due to drift μ n : electron mobility

10 Temperature Effect on Mobility 10 R L R I Impedance to motion due to lattice scattering: No doping dependence Decreases with decreasing temperature Impedance to motion due to ionized impurity scattering: Increases with N A or N D Increases with decreasing temperature

11 11 Temperature Effect on Mobility Carrier mobility varies with doping: Decrease with increasing total concentration of ionized dopants. Carrier mobility varies with temperature: Decreases with increasing T if lattice scattering is dominant. Decreases with decreasing T if impurity scattering is dominant.

12 12 J N drift = qnv d = q n ne J P drift = qpv d = q p pe Conductivity and Resistivity J drift = J N drift + J P drift =q( n n+ p p)e = E Conductivity of a semiconductor: = q( n n+ p p) Resistivity of a semiconductor: = 1/

13 13 Resistivity Dependence on Doping For n-type material: 1 q N For p-type material: 1 q N n p D A

14 14 Carrier Mobility as Function Doping Level

15 15 Example Consider a Si sample at 300 K doped with /cm 3 Boron. What is its resistivity? N A = cm 3, N D = 0 (N A >> N D p-type) p cm 3, n 10 4 cm 3 1 qnn qpp 1 q p p [( )(437)(10 )] cm

16 16 Example Consider the same Si sample, doped additionally with /cm 3 Arsenic. What is its resistivity now? N A = cm 3, N D = cm 3 (N D > N A n-type) n N D N A = cm 3, p n i2 /n = cm 3 1 qnn qpp 1 q n n [( )(790)(9 10 )] μ n is to be taken for the value at N = N A + N D = cm 3 μn 790 cm 2 /V s cm

17 17 Example Consider a Si sample doped with cm 3 As. How will its resistivity change when the temperature is increased from T = 300 K to T = 400 K? The temperature dependent factor in (and therefore ) is n. From the mobility vs. temperature curve for cm 3, we find that n decreases from 770 at 300 K to 400 at 400 K. As a result, increases by a factor of:

18 Increasing electron energy 18 Increasing hole energy Potential vs. Kinetic Energy Electron kinetic energy E c Hole kinetic energy E v E c represents the electron potential energy: P.E. E E c reference E ref is arbitrary

19 19 E Band Bending Until now, E c and E v have always been drawn to be independent of the position. When an electric field E exists inside a material, the band energies become a function of position. E c E v x Variation of E c with position is called band bending

20 P.E. qv The potential energy of a particle with charge q is related to the electrostatic potential V(x): Band Bending 1 V ( E ) c E reference q E V dv dx 1 de 1 de 1 de E q dx q dx q dx c v i Since E c, E v, and E i differ only by an additive constant 20

21 Diffusion Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion (Brownian Motion). 21

22 22 1-D Diffusion Example Thermal motion causes particles to move into an adjacent compartment every τ seconds.

23 23 dn qd dx dp qd dx J N diff N P diff P J Diffusion Currents n e p h Electron flow Current flow x D is the diffusion coefficient, in [cm 2 /sec] x Hole flow Current flow

24 24 Total Currents J J J dn JN JN drift JN diff qnne qdn dx dp JP JP drift JP diff qp pe qdp dx N P Drift current flows when an electric field is applied. Diffusion current flows when a gradient of carrier concentration exist.

25 25 Current Flow Under Equilibrium Conditions In equilibrium, there is no net flow of electrons or : JN 0, JP 0 The drift and diffusion current components must balance each other exactly. A built-in electric field of ionized atoms exists, such that the drift current exactly cancels out the diffusion current due to the concentration gradient. J dn q ne qd dx N n N 0

26 Current Flow Under Equilibrium Conditions Consider a piece of non-uniformly doped semiconductor: n-type semiconductor Decreasing donor concentration E c (x) F c n NCe dn NC EFEc kt de e dx kt dx n de c kt dx E F dn q E E kt c E v (x) dx kt n E Under equilibrium, E F inside a material or a group of materials in intimate contact is not a function of position 26

27 27 J Einstein Relationship between D and But, under equilibrium conditions, J N = 0 and J P = 0 dn q ne qd dx q qne qne D kt N n N 0 n N 0 Similarly, D N n DP p kt q kt q Einstein Relationship Further proof can show that the Einstein Relationship is valid for a non-degenerate semiconductor, both in equilibrium and non-equilibrium conditions.

28 28 Example: Diffusion Coefficient What is the hole diffusion coefficient in a sample of silicon at 300 K with p = 410 cm 2 / V.s? D P kt q p mev 410 cm V s 1e 2 cm mv 410 Vs cm /s ev 1 V 1 e 19 1 ev J Remark: kt/q = mv at room temperature

29 29 Recombination Generation Recombination: a process by which conduction electrons and holes are annihilated in pairs. Generation: a process by which conduction electrons and holes are created in pairs. Generation and recombination processes act to change the carrier concentrations, and thereby indirectly affect current flow.

30 30 Generation Processes Band-to-Band R G Center Impact Ionization E 1 de c q dx Release of energy E T : trap energy level Due to lattice defects or unintentional impurities Also called indirect generation E G Only occurs in the presence of large E

31 31 Recombination Processes Band-to-Band R G Center Auger Collision Rate is limited by minority carrier trapping Primary recombination way for Si Occurs in heavily doped material

32 32 Homework 2 1. (4.17) Problem 3.6, Pierret s Semiconductor Device Fundamentals. 2. (4.27) Problem 3.12, from (a) until (f), for Figure P3.12(a) and Figure P3.12(f), Pierret s Semiconductor Device Fundamentals. 3. (5.28) The electron concentration in silicon at T = 300 K is given by 16 x 3 nx ( ) 10 exp cm 18 where x is measured in μm and is limited to 0 x 25 μm. The electron diffusion coefficient is D N = 25 cm 2 /s and the electron mobility is μ n = 960 cm 2 /(Vs). The total electron current density through the semiconductor is constant and equal to J N = 40 A/cm 2. The electron current has both diffusion and drift current components. Determine the electric field as a function of x which must exist in the semiconductor. Sketch the function.

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