 P-N Junction Diodes  Current Flowing through a Diode I-V Characteristics Quantitative Analysis (Math, math and more math)

p-n Junction I-V Characteristics In Equilibrium (no bias) Total current balances due to the sum of the individual components no net current! Electron Drift Current Electron Diffusion Current Hole Drift Current Hole Diffusion Current

p-n Junction I-V Characteristics In Equilibrium (no bias) Total current balances due to the sum of the individual components n vs. E p-Type Material n-Type Material - qVBI EC EV EF Ei + + + + + + + + + + + + + + + + + + + EC EV EF Ei p vs. E no net current!

p-n Junction I-V Characteristics Lowering of potential hill by VA Forward Bias (VA > 0) Electron Diffusion Current IN Electron Drift Current Current flow is proportional to e(Va/Vref) due to the exponential decay of carriers into the majority carrier bands surmount potential barrier Lowering of potential hill by VA VA Hole Diffusion Current Hole Drift Current IP Current flow is dominated by majority carriers flowing across the junction and becoming minority carriers I

p-n Junction I-V Characteristics Increase of potential hill by VA Reverse Bias (VA < 0) Increase of potential hill by VA Electron Drift Current Current flow is constant due to thermally generated carriers swept out by E fields in the depletion region Electron Diffusion Current negligible due to large energy barrier Hole Diffusion Current negligible due to large energy barrier Current flow is dominated by minority carriers flowing across the junction and becoming majority carriers Hole Drift Current

p-n Junction I-V Characteristics Where does the Reverse Bias Current come from? Generation near the depletion region edges “replenishes” the current source.

p-n Junction I-V Characteristics Putting it all together -I0 for Ideal diode Vref = kT/q

p-n Junction I-V Characteristics Diode Equation  : Diode Ideality Factor

Quantitative p-n Diode Solution Assumptions: 1) Steady state conditions 2) Non- degenerate doping 3) One- dimensional analysis 4) Low- level injection 5) No light (GL = 0) Current equations:

Quantitative p-n Diode Solution Application of the Minority Carrier Diffusion Equation Quisineutral Region minority carrier diffusion eq. minority carrier diffusion eq. Since electric fields exist in the depletion region, the minority carrier diffusion equation does not apply here.

Quantitative p-n Diode Solution Quisineutral Region quasi-Fermi levels formalism

Equilibrium Non-Equilibrium Quasi - Fermi Levels The Fermi level is meaningful only when the system is in thermal equilibrium. The non-equilibrium carrier concentration can be expressed by defining Quasi-Fermi levels Fn and Fp . Equilibrium Non-Equilibrium

Quantitative p-n Diode Solution Quisineutral Region quasi-Fermi levels formalism

Quantitative p-n Diode Solution Quisineutral Region ?

Quantitative p-n Diode Solution Quisineutral Region x”=0 x’=0 Approach: Solve minority carrier diffusion equation in quasineutral regions. Determine minority carrier currents from continuity equation. Evaluate currents at the depletion region edges. Add these together and multiply by area to determine the total current through the device. Use translated axes, x  x’ and -x  x’’ in our solution.

Quantitative p-n Diode Solution Quisineutral Region x”=0 x’=0 Holes on the n-side

Quantitative p-n Diode Solution Quisineutral Region x”=0 x’=0 Holes on the n-side

Quantitative p-n Diode Solution Quisineutral Region x”=0 x’=0 Similarly for electrons on the p-side…

Quantitative p-n Diode Solution Quisineutral Region Depletion Region Continuity equation Negligible thermal R-G implies Jn and Jp are constant throughout the depletion region. Thus, the total current can be define in terms of only the current at the depletion region edges.

Quantitative p-n Diode Solution Continuity Equations

Quantitative p-n Diode Solution Quisineutral Region x”=0 x’=0 Total on current is constant throughout the device. Thus, we can characterize the current flow components as… J -xp xn

pn-junction diode structure used in the discussion of currents pn-junction diode structure used in the discussion of currents. The sketch shows the dimensions and the bias convention. The cross-sectional area A is assumed to be uniform.

Hole current (solid line) and recombining electron current (dashed line) in the quasi-neutral n-region of the long-base diode of Figure 5.5. The sum of the two currents J (dot-dash line) is constant.

Hole density in the quasi-neutral n-region of an ideal short-base diode under forward bias of Va volts.

The ratio of generation-region width xi to space-charge-region width xd as a function of reverse voltage for several donor concentrations in a one-sided step junction.

The current components in the quasi-neutral regions of a long-base diode under moderate forward bias: J(1) injected minority-carrier current, J(2) majority-carrier current recombining with J(1), J(3) majority-carrier current injected across the junction. J(4) space-charge-region recombination current.

(d) Adapted from [8]. Current-voltage characteristic for a diode near the boundary between (a) and (c), showing diffusion current at lower voltages and a transition to thermionic-emission current at higher biases.

Current in a Heterojunction

(a) Transient increase of excess stored holes in a long-base ideal diode for a constant current drive applied at time zero with the diode initially unbiased. Note the constant gradient at x = xn as time increases from (1) through (5), which indicates a constant injected hole current. (Circuit shown in inset.) (b) Diode voltage VD versus time.

(a) Transient decay of excess stored holes in a long-base ideal diode (a) Transient decay of excess stored holes in a long-base ideal diode. In the case shown, the initial forward bias applied through the series resistor is abruptly changed to a negative bias at time t = 0. (Circuit shown in inset.) (b) Diode current ID versus time.

Junction and Free-Carrier Storage

Quantitative p-n Diode Solution J e l c x n - r g i o = + h S C L M t y a d f u s j T W – p The total current anywhere in the device is constant. Just outside the depletion region it is due to the diffusion of minority carriers.

Quantitative p-n Diode Solution Thus, evaluating the current components at the depletion region edges, we have… J = Jn (x”=0) +Jp (x’=0) = Jn (x’=0) +Jn (x”=0) = Jn (x’=0) +Jp (x’=0)  Ideal Diode Equation Shockley Equation Note: Vref from our previous qualitative analysis equation is the thermal voltage, kT/q

Current-Voltage Characteristics of a Typical Silicon p-n Junction

Quantitative p-n Diode Solution Examples Diode in a circuit