ECE 531 Semiconductor Devices Lecture Notes – 06 Dr. Andre Zeumault Outline 1 Overview 1 2 Revisiting Equilibrium Considerations1 2.1 What is Meant by Thermal Equilibrium?...... 1 2.2 Non-Equilibrium Carrier Concentrations...... 3 3 Recombination and Generation3 3.1 Generation...... 3 4 Continuity Equations5 4.1 Minority Carrier Diffusion Equations...... 5 5 Quasi-Fermi Levels8

1 Overview

So far, we have only discussed charge carrier concentrations under equilibrium conditions. Since no current can flow in thermal equilibrium, this particular case isn’t much interest to us in trying to create electronic devices. Therefore, in this lecture, we discuss the situations and governing equations in which a charge carrier responds under non-equilibrium conditions. Topics to be covered include:

• Recombination and Generation: An increase (generation) or decrease (recombination) in charge carrier concentrations away from or towards equilibrium. (e.g. due to absorption of light)

• Current Continuity: Charge conservation, applied in the context of drift/diffusion.

• Minority Carrier Diffusion Equations: Minority carrier diffusion in the presence of a large majority carrier concentration.

• Quasi-Fermi Levels: Treating carrier densities under non-equilibrium conditions.

2 Revisiting Equilibrium Considerations

2.1 What is Meant by Thermal Equilibrium? When we discussed equilibrium charge carrier concentrations, it was argued that the production of charge carriers was thermally activated. This was shown to be true regardless of whether or not the semiconductor was intrinsic or extrinsic. When a semiconductor is intrinsic, the thermal energy necessary for charge carrier production is the bandgap energy EG, which corresponds to the energy needed to break a Silicon-Silicon covalent bond. For an extrinsic semiconductor, this new amount of thermal energy is the dopant ionization energy, but it is still thermal energy! (See Figure1).

At thermal equilibrium, charge carrier production is governed by thermal energy.

When we discussed transport, we assumed that the magnitude(s) of any applied electric field(s) was low enough so as not to change the equilibrium carrier concentrations. We spoke generally,

1 ECE 531 Semiconductor Devices Lecture Notes – 06 Dr. Andre Zeumault

Figure 1: (a) The bandgap energy is the minimum energy for electron-hole production in an intrinsic semiconductor (b) The dopant ionization energy is the minimum energy for electron (donor) or hole (acceptor) production in an extrinsic semiconductor.

Figure 2: An intrinsic semiconductor in thermal equilibrium (Neamen 4th edition). The rate of generation exactly balances the rate of recombination. referring to the equilibrium carrier concentrations as n, p. We need to refine this notation to differentiate between equilibrium and non-equilibrium conditions.

The charge carrier concentrations at thermal equilibrium are denoted n0, p0

Carrier concentrations at equilbrium exist due to a balance between the rate at which they are produced and the rate at which they are destroyed. We refer to these as the rate of generation and recombination respectively. Thus, it is more appropriate to think of thermal equilibrium as a steady-state condition wherein the rate of generation is equal to the rate of recombination.

At thermal equilbrium, the rate of generation equals the rate of recombination

We can represent this visually in Figure2 for an intrinsic semiconductor at room temperature.

2 ECE 531 Semiconductor Devices Lecture Notes – 06 Dr. Andre Zeumault

2.2 Non-Equilibrium Carrier Concentrations Under non-equilibrium conditions, an excess or deficiency of charge carriers exists in a semiconduc- tor. Equivalently, when an excess of charge carriers exists in a semiconductor, the semiconductor will be in a non-equilibrium state. We define the excess charge carrier concentrations ∆n ∆p as follows:

∆n = n − n0

∆p = p − p0

Thus, it can be seen that when ∆n, ∆p > 0 an excess of carriers exists whereas when ∆n, ∆p < 0 a deficiency of carriers exists.

∆n, ∆p > 0 =⇒ excess charge carriers ∆n, ∆p < 0 =⇒ deficiency of charge carriers

We can choose to discuss non-equilibrium conditions in terms of the excess carrier concentra- tions or the total carrier concentrations interchangeably. It turns out that the behavior of some devices is governed almost exclusively by the excess charge carrier concentrations. In such cases, we will work exclusively with excess charge carrier concentrations. The corresponding total carrier concentrations under non-equilibrium conditions are,

n = ∆n + n0

p = ∆p + p0

From this, it is immediately evident that under non-equilibrium conditions, the np product is no 2 longer equal to ni .

2 Under non-equilbrium conditions, np 6= ni

To restore equilibrium, the semiconductor will either seek to decrease its carrier concentration through net recombination or increase its carrier concentration through net generation.

2 np > ni (net recombination)

2 np < ni (net generation)

3 Recombination and Generation

3.1 Generation There are several ways of creating an excess of charge carriers. The most familiar one is light absorption – which occurs when we shine light having higher energy than the bandgap energy onto the semiconductor. This leads to the production of excess charge carriers referred to as generation.

Ephoton > EG =⇒ light absorption

3 ECE 531 Semiconductor Devices Lecture Notes – 06 Dr. Andre Zeumault

dn 2 Generation dt > 0 will occur whenever there is a deficiency of charge carriers np < ni . There- 2 fore, we may start by assuming that ni − np is somewhat of a driving force for the generation process. The larger the difference, the greater the generation rate. Defining a proportionality constant αr, we can write: dn dn d∆n d∆n = 0 + = = α n2 − np
dt dt dt dt r i Using the expressions for the total carrier concentrations, we can write: d∆n = α n2 − (n + ∆n)(p + ∆p)
dt r i 0 0 This expression can be simplified using the fact that ∆n = ∆p. Excess electrons are generated at the same rate as excess holes. d∆n = α n2 − (n + ∆n)(p + ∆n)
dt r i 0 0 2 2
= αr ni − (n0p0 + ∆n(n0 + p0) + ∆n )

2 Since n0p0 = ni , we can further simplify to the following: d∆n = −α ∆n (n + p + ∆n) dt r 0 0 To simplify further, we make what is known as the low-level injection approximation. This assumes that we are interested in a small concentration of minority carriers, in the presence of a large concentration of majority carriers. Mathematically speaking, we make the following:

+ − n0 >> p0, ∆p << n0 (ND >> NA )

+ − p0 >> n0, ∆n << p0 (NA >> ND )

electrons are minority carriers in a p-type semiconductor holes are minority carriers in a n-type semiconductor

This approximation leads to the following expression: d∆n = −α p ∆n dt r 0 This equation has the following exponential solution in time,

∆n = ∆n(0)e−αrp0t

−αrp0t = (n(0) − n0) e

We can also define a time constant in this exponential decay τ = 1 as the minority carrier n αrp0

4 ECE 531 Semiconductor Devices Lecture Notes – 06 Dr. Andre Zeumault lifetime for electrons. This leads to the following expressions:

− t ∆n = ∆n(0)e τn − t = (n(0) − n0) e τn

Defining the generation rate G∆n = G∆p = G∆, we can write d∆n ∆n ∆p G∆ = = − = − dt τn τn Since excess carriers are always generated in pairs, the excess carrier generation rate has to equal the negative of the excess carrier recombination rate. ∆n ∆p R∆ = −G∆ = = τn τn

The corresponding expressions for the behavior of excess holes with minority carrier lifetime τp = 1 in an n-type semiconductor is given by: αrn0

− t ∆p = ∆p(0)e τp

∆p ∆n G∆ = − = − τp τp ∆p ∆n R∆ = = τp τp Figure3 summarizes the low-level injection approximations made above.

4 Continuity Equations

Since charge is conserved, the current flowing in a semiconductor must not change spatially, except for if the charge carrier concentration changes with time (e.g. under non-equilibrium conditions). This can be shown through some simple bookkeeping of the charge carrier concentrations. (Figure4)

1 dJ dp − p + G − R = e dx p p dt 1 dJ dn n + G − R = e dx n n dt These expressions are known as current continuity equations and they relate transport to generation and recombination processes in a semiconductor.

4.1 Minority Carrier Diffusion Equations When there is no electric field, and therefore no drift current, the current will be exclusively diffusive. Substituting the diffusion current expression into the current continuity equations yields

5 ECE 531 Semiconductor Devices Lecture Notes – 06 Dr. Andre Zeumault

Figure 3: (a) Low level injection of an n-type semiconductor and (b) p-type semiconductor.

Figure 4: Simple cartoon for accounting for charge conservation for the derivation of current- continuity expressions.(from ece colorado)

6 ECE 531 Semiconductor Devices Lecture Notes – 06 Dr. Andre Zeumault the following equations for electrons and holes.

d2p dp D + G − R = p dx2 p p dt d2n dn D + G − R = n dx2 n n dt

The terms Gp,Gn and Rp,Rn denote the total generation and recombination rates for electrons and holes. They are related to the equilibrium and excess generation rates as follows: dn dn d∆n G = = 0 + = G + G n dt dt dt n0 ∆n

dp dp d∆p G = = 0 + = G + G p dt dt dt p0 ∆p We know that at equilibrium, electrons generate and recombine at the same rates (otherwise there would be current flow!):

Gn0 = Rn0

We don’t know very much regarding the relationship between G∆n and R∆n. However, we note that these not be equal to each other at equilibrium. Electrons or holes can be generated externally in pairs, such that G∆n = G∆p = G. We can therefore just use a generic G accounting for the excess carrier generation rate. What about the rate of recombination? We already worked this out. The result is: ∆n ∆p G∆n = − = − τn τp Therefore, we have the following, after making the appropriate substitutions into the diffusion equation.

d2p ∆p dp Dp 2 + G − = dx τp dt d2n ∆n dn Dn 2 + G − = dx τn dt Under steady-state conditions, and in the absence of excess carrier generation, these reduce to:

d2p ∆p Dp 2 − = 0 dx τp d2n ∆n Dn 2 − = 0 dx τn When we are interested in minority carriers greatly exceeding their equilibrium values we have the following,

p = p0 + ∆p ≈ ∆p

n = n0 + ∆n ≈ ∆n

7 ECE 531 Semiconductor Devices Lecture Notes – 06 Dr. Andre Zeumault

This leads to the minority carrier diffusion equations at steady state.

d2∆p ∆p Dp 2 − = 0 dx τp d2∆n ∆n Dn 2 − = 0 dx τn This has solutions of the form:

− x ∆p = ∆p(x = 0)e Lp − x ∆n = ∆n(x = 0)e Ln

Where the quantity Lp is referred to as the minority carrier diffusion length. √ The minority carrier diffusion length of electrons is Ln = Dnτn p The minority carrier diffusion length of holes is Lp = Dpτp

5 Quasi-Fermi Levels

The Fermi level is defined at thermal equilibrium. It does not apply generally under non-equilibrium conditions.

EF −Ei k T n0 = nie b

Ei−EF k T p0 = nie b

Thus, we can deduce that the following must be true:

EF −Ei k T n = n0 + ∆n 6= nie b

Ei−EF k T p = p0 + ∆p 6= nie b

So what do we do? It turns out that it is very simple to handle behavior under non-equilibrium conditions. We just redefine Fermi levels into quasi-Fermi levels. Now, we split our concept of a single Fermi level into two.

EF,n is the Quasi-Fermi level for electrons. EF,p is the Quasi-Fermi level for holes.

EF,n−Ei k T n = n0 + ∆n 6= nie b

Ei−EF,p k T p = p0 + ∆p 6= nie b

Quasi Fermi levels can change dramatically for minority carriers when excess carriers are generated. Since an n-type semiconductor has a much larger concentration of electrons than holes, it is easy to exceed the equilibrium hole concentration by creating excess carriers. The same is true for electrons

8 ECE 531 Semiconductor Devices Lecture Notes – 06 Dr. Andre Zeumault

Figure 5: (a) Fermi level at equilbrium. (b) Quasi-Fermi levels with additional 1 × 1013 of excess carriers. in a p-type semiconductor. This is illustrated in Figure5. In this case the number of excess holes dramatically changes the quasi-Fermi level for holes but only slightly modifies the quasi-Fermi level for electrons.

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