1. President UniversityErwin SitompulSDP 3/1 Dr.-Ing. Erwin Sitompul President University Lecture 3 Semiconductor Device Physics http://zitompul.wordpress.com

2. President UniversityErwin SitompulSDP 3/2 Chapter 2Carrier Modeling 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 – E F > 3kT if E F – E > 3kT

3. President UniversityErwin SitompulSDP 3/3 EcEc EvEv 3kT E F in this range Chapter 2Carrier Modeling Nondegenerately Doped Semiconductor The expressions for n and p will now be derived in the range where the Boltzmann approximation can be applied: The semiconductor is said to be nondegenerately doped (lightly doped) in this case.

4. President UniversityErwin SitompulSDP 3/4 Chapter 2Carrier Modeling Degenerately Doped Semiconductor If a semiconductor is very heavily doped, the Boltzmann approximation is not valid. For Si at T = 300 K, E c  E F 1.6  10 18 cm –3 E F  E v 9.1  10 17 cm –3 The semiconductor is said to be degenerately doped (heavily doped) in this case. N D = total number of donor atoms/cm 3 N A = total number of acceptor atoms/cm 3

5. President UniversityErwin SitompulSDP 3/5 Chapter 2Carrier Modeling Equilibrium Carrier Concentrations Integrating n(E) over all the energies in the conduction band to obtain n (conduction electron concentration): By using the Boltzmann approximation, and extending the integration limit to , N C = “effective” density of conduction band states For Si at 300 K, N C = 3.22  10 19 cm –3

6. President UniversityErwin SitompulSDP 3/6 Chapter 2Carrier Modeling Equilibrium Carrier Concentrations Integrating p(E) over all the energies in the conduction band to obtain p (hole concentration): By using the Boltzmann approximation, and extending the integration limit to , N V = “effective” density of valence band states For Si at 300 K, N V = 1.83  10 19 cm –3

7. President UniversityErwin SitompulSDP 3/7 Chapter 2Carrier Modeling Intrinsic Carrier Concentration Relationship between E F and n, p : For intrinsic semiconductors, where n = p = n i, E G : band gap energy

8. President UniversityErwin SitompulSDP 3/8 Chapter 2Carrier Modeling Intrinsic Carrier Concentration

9. President UniversityErwin SitompulSDP 3/9 In an intrinsic semiconductor, n = p = n i and E F = E i, where E i denotes the intrinsic Fermi level. Chapter 2Carrier Modeling Alternative Expressions: n(n i, E i ) and p(n i, E i )

10. President UniversityErwin SitompulSDP 3/10 EcEc EvEv E G = 1.12 eV Si EiEi Chapter 2Carrier Modeling Intrinsic Fermi Level, E i To find E F for an intrinsic semiconductor, we use the fact that n = p. E i lies (almost) in the middle between E c and E v

11. President UniversityErwin SitompulSDP 3/11 Chapter 2Carrier Modeling Example: Energy-Band Diagram For Silicon at 300 K, where is E F if n = 10 17 cm –3 ? Silicon at 300 K, n i = 10 10 cm –3

12. President UniversityErwin SitompulSDP 3/12 Chapter 2Carrier Modeling Charge Neutrality and Carrier Concentration N D : concentration of ionized donor (cm –3 ) N A : concentration of ionized acceptor (cm –3 )? Charge neutrality condition: E i quadratic equation in n + –

13. President UniversityErwin SitompulSDP 3/13 Chapter 2Carrier Modeling Charge-Carrier Concentrations The solution of the previous quadratic equation for n is: New quadratic equation can be constructed and the solution for p is: Carrier concentrations depend on net dopant concentration N D –N A or N A –N D

14. President UniversityErwin SitompulSDP 3/14 Chapter 2Carrier Modeling Dependence of E F on Temperature

15. President UniversityErwin SitompulSDP 3/15 Phosphorus-doped Si N D = 10 15 cm –3 Chapter 2Carrier Modeling Carrier Concentration vs. Temperature n:number of majority carrier N D :number of donor electron n i :number of intrinsic conductive electron

16. President UniversityErwin SitompulSDP 3/16 Chapter 3Carrier Action 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. President UniversityErwin SitompulSDP 3/17 1 2 3 4 5 electron Chapter 3Carrier Action 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.

18. President UniversityErwin SitompulSDP 3/18 1 2 3 4 5 electron E F = –qEF = –qE Chapter 3Carrier Action 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. 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.

19. President UniversityErwin SitompulSDP 3/19 v d tAll holes this distance back from the normal plane v d t A All holes in this volume will cross the plane in a time t p v d t A 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 ACharge crossing the plane per unit time I (Ampere)  Hole drift current q p v d Current density associated with hole drift current J (A/m 2 ) Chapter 3Carrier Action Drift Current

20. President UniversityErwin SitompulSDP 3/20 Chapter 3Carrier Action Hole and Electron Mobility For holes, In low-field limit, Similarly for electrons, Hole current due to drift Hole current density due to drift Electron current density due to drift μ p : hole mobility μ n : electron mobility

21. President UniversityErwin SitompulSDP 3/21 Drift Velocity vs. Electric Field Chapter 3Carrier Action Linear relation holds in low field intensity, ~5  10 3 V/cm

22. President UniversityErwin SitompulSDP 3/22 Electron and hole mobility of selected intrinsic semiconductors (T = 300 K)  has the dimensions of v/ E : Chapter 3Carrier Action Hole and Electron Mobility

23. President UniversityErwin SitompulSDP 3/23 RLRL RIRI 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 Chapter 3Carrier Action Temperature Effect on Mobility

24. President UniversityErwin SitompulSDP 3/24 Temperature Effect on Mobility Chapter 3Carrier Action 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. President UniversityErwin SitompulSDP 3/25 J P|drift = qpv d = q  p p E J N|drift = –qnv d = q  n n E J drift = J N|drift + J P|drift =q(  n n+  p p) E =  E Chapter 3Carrier Action Conductivity and Resistivity Resistivity of a semiconductor:  = 1 /  Conductivity of a semiconductor:  = q(  n n+  p p)

26. President UniversityErwin SitompulSDP 3/26 Chapter 3Carrier Action Resistivity Dependence on Doping For n-type material: For p-type material:

27. President UniversityErwin SitompulSDP 3/27 Chapter 3Carrier Action Example Consider a Si sample at 300 K doped with 10 16 /cm 3 Boron. What is its resistivity? N A = 10 16 /cm 3, N D = 0(N A >> N D  p-type) p  10 16 /cm 3, n  10 4 /cm 3

28. President UniversityErwin SitompulSDP 3/28 Chapter 3Carrier Action Example Consider a Si sample doped with 10 17 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 10 17 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:

29. President UniversityErwin SitompulSDP 3/29 Chapter 2Carrier Action Homework 2 1.a. (4.2) Calculate the equilibrium hole concentration in silicon at T = 400 K if the Fermi energy level is 0.27 eV above the valence band energy. Deadline: 2 February 2011, at 10:30 (Wednesday). 1.b. (E4.3) Find the intrinsic carrier concentration in silicon at: (i) T = 200 K and (ii) T = 400 K. 1.c. (4.13) Silicon at T = 300 K contains an acceptor impurity concentration of N A = 10 16 cm –3. Determine the concentration of donor impurity atoms that must be added so that the silicon is n-type and the Fermi energy level is 0.20 eV below the conduction band edge. 2. (4.17) Problem 3.6, Pierret’s “Semiconductor Device Fundamentals”.