Lecture #5 OUTLINE Intrinsic Fermi level Determination of E F Degenerately doped semiconductor Carrier properties Carrier drift Read: Sections 2.5, 3.1

Spring 2007EE130 Lecture 5, Slide 2 Intrinsic Fermi Level, E i To find E F for an intrinsic semiconductor, use the fact that n = p:

Spring 2007EE130 Lecture 5, Slide 3 n(n i, E i ) and p(n i, E i ) In an intrinsic semiconductor, n = p = n i and E F = E i :

Spring 2007EE130 Lecture 5, Slide 4 Example: Energy-band diagram Question: Where is E F for n = cm -3 ?

Spring 2007EE130 Lecture 5, Slide 5 Dopant Ionization Consider a phosphorus-doped Si sample at 300K with N D = cm -3. What fraction of the donors are not ionized? Answer: Suppose all of the donor atoms are ionized. Then Probability of non-ionization 

Spring 2007EE130 Lecture 5, Slide 6 Nondegenerately Doped Semiconductor Recall that the expressions for n and p were derived using the Boltzmann approximation, i.e. we assumed The semiconductor is said to be nondegenerately doped in this case. EcEc EvEv 3kT E F in this range

Spring 2007EE130 Lecture 5, Slide 7 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 1.6x10 18 cm -3 E F -E v 9.1x10 17 cm -3 The semiconductor is said to be degenerately doped in this case. Terminology: “n+”  degenerately n-type doped. E F  E c “p+”  degenerately p-type doped. E F  E v

Spring 2007EE130 Lecture 5, Slide 8 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 = cm -3 :  E G = 35 meV N = cm -3 :  E G = 75 meV

Spring 2007EE130 Lecture 5, Slide 9 Mobile Charge Carriers in Semiconductors Three primary types of carrier action occur inside a semiconductor: –Drift: charged particle motion under the influence of an electric field. –Diffusion: particle motion due to concentration gradient or temperature gradient. –Recombination-generation (R-G)

Spring 2007EE130 Lecture 5, Slide 10 Electrons as Moving Particles F = (-q) E = m o a F = (-q) E = m n *a where m n * is the electron effective mass In vacuumIn semiconductor

Spring 2007EE130 Lecture 5, Slide 11 Carrier Effective Mass In an electric field, E, an electron or a hole accelerates: Electron and hole conductivity effective masses: electrons holes * *

Spring 2007EE130 Lecture 5, Slide 12 Average electron kinetic energy Thermal Velocity

Spring 2007EE130 Lecture 5, Slide 13 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 electron

Spring 2007EE130 Lecture 5, Slide 14 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: electron E 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

Spring 2007EE130 Lecture 5, Slide 15 Electron Momentum With every collision, the electron loses momentum Between collisions, the electron gains momentum (-q) E  mn  mn is the average time between electron scattering events

Spring 2007EE130 Lecture 5, Slide 16  p  [q  mp / m p *] is the hole mobility Carrier Mobility m n *v d = (-q) E  mn |v d | = q E  mn / m n * =  n E  n  [q  mn / m n *] is the electron mobility Similarly, for holes: |v d | = q E  mp / m p *   p E

Spring 2007EE130 Lecture 5, Slide 17 Electron and hole mobilities of selected intrinsic semiconductors (T=300K)  has the dimensions of v/ E : Electron and Hole Mobilities

Spring 2007EE130 Lecture 5, Slide 18 a) Find the hole drift velocity in an intrinsic Si sample for E  = 10 3 V/cm. b) What is the average hole scattering time? Solution: a) b) Example: Drift Velocity Calculation v d =  n E

Spring 2007EE130 Lecture 5, Slide 19 Mean Free Path Average distance traveled between collisions

Spring 2007EE130 Lecture 5, Slide 20 Summary The intrinsic Fermi level, E i, is located near midgap –Carrier concentrations can be expressed as functions of E i and intrinsic carrier concentration, n i : In a degenerately doped semiconductor, E F is located very near to the band edge Electrons and holes can be considered as quasi- classical particles with effective mass m* –In the presence of an electric field , carriers move with average drift velocity, where  is the carrier mobility