Lecture Number 4: Charge Transport and Charge Carrier Statistics Chem 140a: Photoelectrochemistry of Semiconductors

Review from Last Time The quantum mechanics of wave packets gave us: m* inversely proportional to curvature of E vs. k diagram In a totally symmetric lattice, k x = k y =k z, so there is one value of m*. In general, m* varies with crystallographic direction. Light Heavy GaAs k x =k y =k z Si k x ≠k y =k z

Review from Last time Carrier Concentrations: Intrinsic semiconductor:

Doping Review E CB E VB EiEi EFEF n-type Sample: At room temperature: Temperature dependence:

Charge Conduction No Field - Random Brownian motion: Constant Applied Field: v time ideal v sound limit Real limit Real limit: Phonon scattering ionized impurity scattering neutral impurities carrier-carrier Piezo-electric (ZnO, TiO 2 )

Charge Conduction Due to scattering:mean free time,  m mean free path, l m Average Drift Velocity: In vector form: Where we define the mobility, , as: There are separate mobilities for electrons and holes:

Mobility and Doping

Mobility and Temperature

Drift Current Flux: e-e- Field J = current density  = conductivity

Conductivity Electrons: Holes: Total: n-type sample:

Resistivity  = resisitivity R = resistance Remember: For n-type Si: A

Measuring Resistivity 2-point probe: I V Equivalent Circuit: Measured resistance includes contact resistance, which is large for metal contacts to Si, as we’ll see later.

Measuring Resistivity 4-point probe: Equivalent Circuit: I I I I I + i i i Measured Voltage: I >> i 4-point probe gets rid of the contact resistance. I V s s s

Measuring Resistivity Actually measure sheet resistance, R s (Sze, page 31) Different paths give different resistance, so total resistivity depends on the width, w, of the sample: w CF=4.54 for d >> s d s

Hall Measurement Carrier Concentrations: Mobility:

Diffusion Current (E-field) (conc. grad.) e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- p+p+ p+p+ p+p+ p+p+ p+p+ p+p+ p+p+ p+p+ p+p+ p+p+ Even in the absence of fields, carriers move by diffusion: Fick’s First law of diffusion:

Current Einstein Relationships: Total current (electrons): In the x-direction: Electrons and holes:

n-type Si/M before equilibration: Electrons flow from Si to metal until E F is the same everywhere. Analogous to reaction of Na with Cl Na Cl + Na + Cl - e-e- E CB E F,Si E VB E F,M Vac. n-Simetal e-e- V bi Semiconductor-Metal Junction

Much larger density of states for metal than semiconductor: hole Metal S.C. E CB E F,Si E VB E F,M n-Si metal e-e- M SC E E CB E F,Si E VB E F,M n-Si metal Charge Equilibration

Depletion Approximation: Only dopants can be ionized in the S.C., and they are completely ionized for a width W. S.C. M W x = 0 x = W NDND N D + (x) E F,Si E F,M n-Si metal V bi Depletion Region

Before equilibration: After equilibration: E CB E F,Si E VB E F,M Vac. n-Simetal V bi w Si VnVn wMwM E CB E F,Si E VB E F,M Vac. Bulk V bi w Si wMwM VnVn Depletion Metal n-Si Band-Bending

Four important Equations: Poisson’s Equation 1) 2) 3) 4)

Charge Density Interesting region Bulk Semiconductor Metal Poisson’s Equation: Integrating:

Electric Field Integrate Electric Field:

Electric Potential Calculate Depletion Width: If we had started with: We would find:

Electric Potential Energy Electric potential energy is electric potential scaled by q. Band-bending is the same in CB, VB and Vac because E g and electron affinity (EA) are the same everywhere E CB E F,Si E VB E F,M Vac. V bi VnVn Metal n-Si E EA EgEg

n-type vs. p-type E CB E F,Si E VB V bi,n VnVn Metal n-Si E F,M E CB E F,Si E VB E F,M V bi,n VnVn Metal n-Si E CB E F,Si E VB V bi,p VpVp Metal p-Si E F,M E CB E F,Si E VB V bi,p Metal p-Si E F,M VpVp Before Equilibration: After Equilibration:

Solution vs. Metal Contact D.O.S. is still much higher for a solution than for Si, so bucket in ocean analogy still holds. E(A/A - ) doesn’t move. E F,Si E(A/A - ) N D : cm -3 W:  m Q=N D W Q max =10 11 e - /cm 2 Semiconductor: Solution: 1.0 mM = 6 x molecules/cm 3 ~2 nm solution depth has molecules/cm 2