Lecture #4. Nondegenerately Doped Semiconductor

Transcription

1 Lecture #4 ANNOUNCEMENTS Prof. King will not hold office hours this week, but will hold an extra office hour next Mo (2/3) from AM-2:30PM Quiz # will be given at the beginning of class on Th 2/6 covers material in Chapters & 2 (HW# & HW#2) closed book; one page of notes allowed OUTLINE Drift (Chapter 3.)» carrier motion» mobility» resistivity EE30 Lecture 4, Slide Nondegenerately Doped Semiconductor Recall that the expressions for n and p were derived using the Boltzmann approximation, i.e. we assumed Ev + 3kT EF Ec 3kT The semiconductor is said to be nondegenerately doped in this case. EE30 Lecture 4, Slide 2

2 Degenerately Doped Semiconductor If a semiconductor is very heavily doped, the Boltzmann approximation is not valid. In Si at T=300K: E c -E F < 3kT if N D >.6x0 8 cm -3 E F -E v < 3kT if N A > 9.x0 7 cm -3 The semiconductor is said to be degenerately doped in this case. Ev + 3kT EF Ec 3kT EE30 Lecture 4, Slide 3 Band Gap Narrowing If the dopant concentration is a significant fraction of the silicon atomic density, the energy-band structure is perturbed the band gap is reduced by E G N = 0 8 cm -3 : N = 0 9 cm -3 : EE30 Lecture 4, Slide 4

3 Free Carriers in Semiconductors Three primary types of carrier action occur inside a semiconductor: drift diffusion recombination-generation EE30 Lecture 4, Slide 5 Electrons as Moving Particles F = (-q) = m o a F = (-q) = m n *a where m n * is the electron effective mass EE30 Lecture 4, Slide 6

4 Carrier Effective Mass In an electric field,, an electron or a hole accelerates: electrons holes Electron and hole conductivity effective masses: Si Ge GaAs m n /m m p /m EE30 Lecture 4, Slide 7 Thermal Velocity Average electron or hole kinetic energy 3 = kt = m * v th v th 9 3kT ev (.6 0 J/eV) = = 3 m * kg = m/s = cm/s EE30 Lecture 4, Slide 8

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: ~ K Other scattering mechanisms: deflection by ionized impurity atoms deflection due to Coulombic force between carriers carrier-carrier scattering only significant at high carrier concentrations The net current in any direction is zero, if no electric field is applied. EE30 Lecture 4, Slide electron Carrier Drift When an electric field (e.g. due to an externally applied voltage) is applied to a semiconductor, mobile chargecarriers will be accelerated by the electrostatic force. This force superimposes on the random motion of electrons: electron Electrons drift in the direction opposite to the electric field current flows Because of scattering, electrons in a semiconductor do not achieve constant acceleration. However, they can be viewed as quasi-classical particles moving at a constant average drift velocity v d EE30 Lecture 4, Slide 0

6 Electron Momentum With every collision, the electron loses momentum m n *v d Between collisions, the electron gains momentum (-q) τ mn where τ mn = average time between scattering events EE30 Lecture 4, Slide Carrier Mobility m n *v d = (-q) τ mn v d = q τ mn / m n * = µ n µ n [qτ mn / m n *] is the electron mobility Similarly, for holes: v d = q τ mp / m p * µ p µ p [qτ mp / m p *] is the hole mobility EE30 Lecture 4, Slide 2

7 Electron and Hole Mobilities µ has the dimensions of v/ : cm/s V/cm 2 cm = V s Electron and hole mobilities of selected intrinsic semiconductors (T=300K) Si Ge GaAs InAs µ n (cm 2 /V s) µ p (cm 2 /V s) EE30 Lecture 4, Slide 3 Example: Drift Velocity Calculation Find the hole drift velocity in an intrinsic Si sample for = 0 3 V/cm. What is τ mp, and what is the distance traveled between collisions? Solution: EE30 Lecture 4, Slide 4

8 Mobility Dependence on Doping Mobility (cm 2 V - s - ) Electrons Holes = τ τ phonon = µ µ phonon + τ impurity + µ impurity 0 E4 E5 E6 E7 E8 E9 E20 Total Doping Impurity Concentration Concenration N(atoms -3 A + N D (cm -3 )) EE30 Lecture 4, Slide 5 Drift Current v d ta= volume from which all holes cross plane in time t pv d t A = # of holes crossing plane in time t q p v d t A = charge crossing plane in time t qpv d A = charge crossing plane per unit time = hole current Hole current per unit area J = q pv d EE30 Lecture 4, Slide 6

9 Conductivity and Resistivity J n,drift = qnv = qnµ n J p,drift = qpv = qpµ p J drift = J n,drift + J p,drift = σ =(qnµ n +qpµ p ) Conductivity of a semiconductor is σ qnµ n + qpµ p Resistivity ρ / σ (Unit: ohm-cm) EE30 Lecture 4, Slide 7 Resistivity Dependence on Doping For n-type mat l: ρ qnµ n n-type p-type For p-type mat l: ρ qpµ p Note: This plot does not apply for compensated material! EE30 Lecture 4, Slide 8

10 W Electrical Resistance I + V _ homogeneously doped sample t L Resistance R V I L = ρ Wt (Unit: ohms) where ρ is the resistivity EE30 Lecture 4, Slide 9 Example Consider a Si sample doped with 0 6 /cm 3 Boron. What is its resistivity? Answer: N A = 0 6 /cm 3, N D = 0 (N A >> N D p-type) p 0 6 /cm 3 and n 0 4 /cm 3 ρ = qn µ + qpµ qpµ = n p 9 6 [(.6 0 )(0 )(450)] =.4 Ω cm p EE30 Lecture 4, Slide 20

11 Example: Dopant Compensation Consider the same Si sample, doped additionally with 0 7 /cm 3 Arsenic. What is its resistivity? Answer: N A = 0 6 /cm 3, N D = 0 7 /cm 3 (N D >>N A n-type) n 9x0 6 /cm 3 and p.x0 3 /cm 3 ρ = qn µ + qpµ qnµ = n p 9 6 [(.6 0 )(9 0 )(600)] = 0.2 Ω cm n EE30 Lecture 4, Slide 2 Summary Electrons and holes moving under the influence of an electric field can be modelled as quasi-classical particles with average drift velocity v d = µ The conductivity of a semiconductor is dependent on the carrier concentrations and mobilities σ = qnµ n +qpµ p Resistivity ρ = = σ qn µ + qpµ n p EE30 Lecture 4, Slide 22