Semiconductor Device Physics

1. Semiconductor Device Physics Lecture 3 Dr. GauravTrivedi, EEE Department, IIT Guwahati

2. Boltzmann Approximation of Fermi Function Boltzmann Approximation of Fermi Function • The Fermi Function that describes the probability that a state at energy E is filled with an electron, under equilibrium conditions, is already given as: • Fermi Function can be approximated as: if E – EF > 3kT if EF– E > 3kT

3. Nondegenerately Doped Semiconductor • The expressions for n and p will now be derived in the range where the Boltzmann approximation can be applied: Boltzmann Approximation of Fermi Function Ec 3kT EF in this range 3kT Ev • The semiconductor is said to be nondegenerately doped(lightly doped) in this case.

4. Degenerately Doped Semiconductor Degenerately Doped Semiconductor Degenerately Doped Semiconductor • If a semiconductor is very heavily doped, the Boltzmann approximation is not valid. • For Si at T = 300 K,Ec-EF < 3kT if ND > 1.6 1018 cm–3EF-Ev < 3kT if NA > 9.1  1017 cm–3 • The semiconductor is said to be degenerately doped (heavily doped) in this case. • ND = total number of donor atoms/cm3 • NA = total number of acceptor atoms/cm3

5. Equilibrium Carrier Concentrations • Integrating n(E) over all the energies in the conduction band to obtain n (conduction electron concentration): Boltzmann Approximation of Fermi Function • By using the Boltzmann approximation, and extending the integration limit to , • NC = “effective” density of conduction band states • For Si at 300 K, NC = 3.22  1019 cm–3

6. Equilibrium Carrier Concentrations • Integrating p(E) over all the energies in the conduction band to obtain p (hole concentration): Boltzmann Approximation of Fermi Function • By using the Boltzmann approximation, and extending the integration limit to , • NV = “effective” density of valence band states • For Si at 300 K, NV = 1.83  1019 cm–3

7. Intrinsic Carrier Concentration Boltzmann Approximation of Fermi Function • Relationship between EF and n, p : • For intrinsic semiconductors, where n = p = ni, • EG : band gap energy

8. Intrinsic Carrier Concentration Boltzmann Approximation of Fermi Function

9. Alternative Expressions: n(ni, Ei) and p(ni, Ei) • In an intrinsic semiconductor, n = p = ni and EF = Ei, where Ei denotes the intrinsic Fermi level. Boltzmann Approximation of Fermi Function

10. Ec EG = 1.12 eV Ei Ev Si Intrinsic Fermi Level, Ei • To find EF for an intrinsic semiconductor, we use the fact that n = p. Boltzmann Approximation of Fermi Function • Ei lies (almost) in the middle between Ec and Ev

11. Example: Energy-Band Diagram Boltzmann Approximation of Fermi Function • For Silicon at 300 K, where is EF if n = 1017 cm–3 ? • Silicon at 300 K, ni= 1010 cm–3

12. Charge Neutrality and Carrier Concentration Boltzmann Approximation of Fermi Function • ND: concentration of ionized donor (cm–3) • NA: concentration of ionized acceptor (cm–3)? • Charge neutrality condition: • Ei quadratic equation in n

13. Charge-Carrier Concentrations • The solution of the previous quadratic equation for n is: Boltzmann Approximation of Fermi Function • New quadratic equation can be constructed and the solution for p is: • Carrier concentrations depend on net dopant concentration ND–NAor NA–ND

14. Ec 400 K 300 K EF (donor-doped) Ei EF (acceptor-doped) 400 K 300 K Net dopant concentration (cm–3) Ev 1013 1014 1015 1016 1017 1018 1019 1020 Dependence of EF on Temperature Boltzmann Approximation of Fermi Function

15. Carrier Concentration vs. Temperature Phosphorus-doped Si Boltzmann Approximation of Fermi Function ND = 1015 cm–3 • n : number of majority carrier • ND : number of donor electron • ni : number of intrinsic conductive electron

16. Carrier Action Boltzmann Approximation of Fermi Function • 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.

17. 2 3 1 electron 4 5 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. Boltzmann Approximation of Fermi Function

18. 2 3 1 4 electron 5 E 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. Boltzmann Approximation of Fermi Function F=–qE • Electrons drift in the direction opposite to the electric fieldèCurrent 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 vd.

19. Drift Current Boltzmann Approximation of Fermi Function vdt All holes this distance back from the normal plane vdtA All holes in this volume will cross the plane in a time t pvdt A Holes crossing the plane in a time t q pvdt A Charge crossing the plane in a time t qpvd A Charge crossing the plane per unit time I (Ampere) Þ Hole drift current qpvdCurrent density associated with hole drift current J (A/m2)

20. Hole and Electron Mobility • For holes, Boltzmann Approximation of Fermi Function • Hole current due to drift • Hole current density due to drift • In low-field limit, • μp : hole mobility • Similarly for electrons, • Electron current density due to drift • μn : electron mobility

21. Drift Velocity vs. Electric Field Boltzmann Approximation of Fermi Function • Linear relation holds in low field intensity, ~5103 V/cm

22. Hole and Electron Mobility Boltzmann Approximation of Fermi Function  has the dimensions ofv/E: Electron and hole mobility of selected intrinsic semiconductors (T = 300 K)

23. RL RI Temperature Effect on Mobility Boltzmann Approximation of Fermi Function • Impedance to motion due to lattice scattering: • No doping dependence • Decreases with decreasing temperature • Impedance to motion due to ionized impurity scattering: • increases with NA or ND • increases with decreasing temperature

24. Temperature Effect on Mobility Boltzmann Approximation of Fermi Function • 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.

25. Conductivity and Resistivity Boltzmann Approximation of Fermi Function JN|drift = –qnvd = qnnE JP|drift= qpvd = qppE Jdrift = JN|drift + JP|drift=q(nn+pp)E = E • Conductivity of a semiconductor:  =q(nn+pp) • Resistivityof a semiconductor:  =1 / 

26. Resistivity Dependence on Doping • For n-type material: Boltzmann Approximation of Fermi Function • For p-type material:

27. Example • Consider a Si sample at 300 K doped with 1016/cm3 Boron. What is its resistivity? Boltzmann Approximation of Fermi Function • NA = 1016/cm3 , ND = 0 • (NA >> NDp-type) • p 1016/cm3, n 104/cm3

28. Example • Consider a Si sample doped with 1017cm–3 As.How will its resistivity change when the temperature is increased from T = 300 K to T = 400 K? Boltzmann Approximation of Fermi Function • The temperature dependent factor in  (and therefore ) is n. • From the mobility vs. temperature curve for 1017cm–3, we find that n decreases from 770 at 300 K to 400 at 400 K. • As a result, increases by a factor of: 770/400 = 1.93

29. Assignment Boltzmann Approximation of Fermi Function