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Non-Equilibrium Carrier Concentrations

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Non-Equilibrium Carrier Concentrations 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.
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