Semiconductor Device Physics Lecture 3 Dr. Gaurav Trivedi, EEE Department, IIT Guwahati

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 Boltzmann Approximation of Fermi Function

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. EcEc EvEv 3kT E F in this range Nondegenerately Doped Semiconductor

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  cm –3 E F  E v 9.1  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 Degenerately Doped Semiconductor

Boltzmann Approximation of Fermi Function 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  cm –3 Equilibrium Carrier Concentrations

Boltzmann Approximation of Fermi Function 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  cm –3 Equilibrium Carrier Concentrations

Boltzmann Approximation of Fermi Function Relationship between E F and n, p : For intrinsic semiconductors, where n = p = n i, E G : band gap energy Intrinsic Carrier Concentration

Boltzmann Approximation of Fermi Function Intrinsic Carrier Concentration

Boltzmann Approximation of Fermi Function In an intrinsic semiconductor, n = p = n i and E F = E i, where E i denotes the intrinsic Fermi level. Alternative Expressions: n(n i, E i ) and p(n i, E i )

Boltzmann Approximation of Fermi Function EcEc EvEv E G = 1.12 eV Si EiEi 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 Intrinsic Fermi Level, E i

Boltzmann Approximation of Fermi Function For Silicon at 300 K, where is E F if n = cm –3 ? Silicon at 300 K, n i = cm –3 Example: Energy-Band Diagram

Boltzmann Approximation of Fermi Function 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 Charge Neutrality and Carrier Concentration

Boltzmann Approximation of Fermi Function 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 Charge-Carrier Concentrations

Boltzmann Approximation of Fermi Function Dependence of E F on Temperature

Boltzmann Approximation of Fermi Function Phosphorus-doped Si N D = cm –3 n:number of majority carrier N D :number of donor electron n i :number of intrinsic conductive electron Carrier Concentration vs. Temperature

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. Carrier Action

Boltzmann Approximation of Fermi Function electron 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. Carrier Scattering

Boltzmann Approximation of Fermi Function electron E F = –qEF = –qE 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  Cu rrent 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. Carrier Drift

Boltzmann Approximation of Fermi Function 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 ) Drift Current

Boltzmann Approximation of Fermi Function 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 Hole and Electron Mobility

Boltzmann Approximation of Fermi Function Linear relation holds in low field intensity, ~5  10 3 V/cm Drift Velocity vs. Electric Field

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

Boltzmann Approximation of Fermi Function 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 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. Temperature Effect on Mobility

Boltzmann Approximation of Fermi Function 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 Resistivity of a semiconductor:  = 1 /  Conductivity of a semiconductor:  = q(  n n+  p p) Conductivity and Resistivity

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

Boltzmann Approximation of Fermi Function 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 Example

Boltzmann Approximation of Fermi Function 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: 770/400 = 1.93 Example

Boltzmann Approximation of Fermi Function Assignment