EMT 272297 Semiconductor Fundamentals Chapter 3 b Nonequilibrium

EMT 272/297 Semiconductor Fundamentals Chapter 3 b Nonequilibrium Excess Carriers in Semiconductors

Outlines Carrier Generation and Recombination ◦ The Semiconductor in Equilibrium ◦ Excess Carrier Generation and Recombination Characteristics of Excess Carriers ◦ Continuity Equations ◦ Time-Dependent Diffusion Equations Quasi-Fermi Energy Levels Thermionic Emission Process Tunneling Process High Field Effects

Carrier Generation and Recombination Introduction Ø When external field (electric, thermal, optical) is applied on the semiconductor, the semiconductor is operating under non- equilibrium conductions. Ø Excess electrons in the conduction band excess holes in the valence band may exist in addition to thermal equilibrium condition Ø Excess carriers’ movements: diffusion, drift, recombination and generation 3

Carrier Generation and Recombination Ø Ø Carrier Generation and Recombination Generation Ø Generation is the process whereby electrons and holes are created Recombination Ø Recombination is the process whereby electrons and holes are annihilated Any deviation from thermal equilibrium will tend to change the electron and hole concentration in a semiconductor. (thermal exitation, photon pumping, carrier injection) When the external excitation is removed, the concentrations of electron and hole in semiconductor will return eventually to their thermal-equilibrium values

The Semiconductor in Equilibrium Carrier Generation and Recombination Thermal-equilibrium concentrations of electron and hole in conduction and valence bands are independent of time. Since the net carrier concentrations are independent of time, the rate at which the electrons and holes are generated and the rate at which they recombine must be equal. For direction band-toband transition (In thermal equilibrium) Gno G po Rno R po Gno G po Rno R po Direct bandgap semiconductor

Excess Carrier Generation and Recombination When external force (electric, optical, thermal) is applied, excess electrons and holes are create in pairs g n gp With generation of excess carriers, concentration of electrons and holes are increase above their thermal equilibrium n n 0 n, p p 0 p excess electron concentrations Electrons and holes are recombined at the same time of generation are equal ' n R R ' p in a nonequilibrium condition np n 0 p 0=n i 2 excess hole concentrations

Excess Carrier Generation and Recombination

Excess Carrier Generation and Recombination Excess carriers recombination rate Ø The rate at which electrons recombine must be proportional to the electron concentration and must also be proportional to the hole concentration. Ø The net rate change in electron concentration

Excess Carrier Generation and Recombination Low-Level Injection and High-Level Injection Can be easily solve by impose the condition of low-level injection Low-level injection: excess carrier concentration is much less than thermal equilibrium majority carrier concentrations n-type material: no>>po, p-type material: po>>no, δn(t) << no δn(t)<<po High-level injection: excess carrier concentration is comparable to or greater than thermal equilibrium majority carrier concentrations n-type material: no>> po, p-type material: po>>no, δn(t) >= no δn(t)>= po

Excess Carrier Generation and Recombination Carrier Recombination Rate Under Low-level Injection Carrier recombination rate Under low-level injection: p-type material: po>> no and po >> n(t) Ø The solution to the equation is an exponential decay from the initial excess concentration, or where,

Excess Carrier Generation and Recombination (low - levelinjection) ( p-type, low level injection) n-type material, no>>po ( n-type, low level injection) Carrier Generation and Recombination

Continuity Equations Characteristics of Excess Carriers Ø For the x component of the particle current density shown and from the calculus, the Taylor expansion gives Ø The net increase in the number of holes per unit time in the differential volume element is given by

Continuity Equations Characteristics of Excess Carriers The recombination rate holes including thermal-equilibrium recombination and excess recombination The recombination lifetime which includes thermal-equilibrium carrier lifetime and excess carrier lifetime Continuity equation for holes Continuity equation for electrons

Time-Dependent Diffusion Equation Characteristics of Excess Carriers The current density in material is By dividing current density the charge of each individual particle, we obtain particle flux Thus the continuity equations can be rewritten as

Time-Dependent Diffusion Equation Characteristics of Excess Carriers Time-dependent diffusion equations for holes and electrons, respectively.

Time-Dependent Diffusion Equation Characteristics of Excess Carriers The thermal equilibrium concentrations, no and po, are not function of time. For homogeneous semiconductor, no and po are also independent of space coordinates Homogeneous semiconductor Ø involving the total concentrations, p and n, and terms involving only the excess concentrations, δp and δn.

Quasi-Fermi Energy Levels Ø Quasi-Fermi Energy Levels At thermal equilibrium, Ø the electron and hole concentrations are functions of the Fermi-level. Ø The Fermi level remains constant throughout the entire material Ø The carrier concentrations is exponentially determined by the Fermi-level

Quasi-Fermi Energy Levels Ø Quasi-Fermi Energy Levels At non-thermal equilibrium Ø If excess carriers are created, thermal equilibrium no longer exists and Fermi energy is strictly no longer defined Ø However, we may define a quasi-Fermi level for electrons and a quasi-Fermi level for holes that apply for nonequilibrium. Ø In such a way, the quasi-Fermi levels for electrons and holes specified for non- thermal equilibrium conditions do not hold constants over the entire material Ø If δn and δp are the excess electron and hole concentrations, respectively, we may write Ø where Efn, and Efp, are the quasi-Fermi energy levels for electrons and holes, respectively.

Quasi-Fermi Energy Levels

Thermionic Emission Process THERMIONIC EMISSION PROCESS– condition where the carriers have sufficient energy to ‘thermionically’ emitted into the vacuum. Thus, the electrons are emitted across a barrier. This occurs because thermal energy given to the carrier overcomes the binding potential, also known as work function.

Thermionic Emission Process q - electron affinity is the energy difference between the condition band edge & the vacuum level in the s/cond. q s – work function (energy between Fermi level & vacuum level in the s/cond). If energy > q - electron can be thermionically emitted into the vacuum. (a) (12) • ELECTRON DENSITY WITH ENERGY> q may be written as : (b) NC – effective density of states in cond. band. Vn – is the difference between bottom of cond. band & Fermi level. (a) The band diagram of an isolated ntype semi-conductor. (b) The thermionic emission process.

Tunneling Process QUANTUM TUNNELING PHENOMENA: • The energy diagram when two isolated semiconductor samples are brought close together. The distance between them is d, the potential barrier height q. Vo is equal to electron affinity q. • If d<<<, electron at left-side semiconductor may transport across the barrier & and move to the other side (even if electron energy is less than barrier height. ) (a) The band diagram of two isolated semiconductors with a distance d. (b) One-dimensional potential barrier. (c) Schematic representation of the wave function across the potential barrier.

High Field Effects At low electric field, drift velocity is proportional to the applied field, and assume that time interval between collision c is independent of the applied field. (Assumption for drift velocity < thermal velocity of carrier, 107 cm/s for Si at room temperature). As the drift velocity reach thermal velocity, its field dependence on the electric field will depart from linear relationship. Drift velocity increases less rapidly when electric field increased. At large field, the drift velocity approaches a saturation velocity. Drift velocity: Where vs – saturation velocity (107 cm/s for Si at 300 K). E 0 – constant field which is 7 x 103 V/cm for electrons and E 0 = 2 x 104 V/cm for holes in highpurity Si materials. - 2 for electrons and 1 for holes.

High Field Effects -shows the measured drift velocity of electrons and holes in Si as a function of the electric field. - Drift velocity increases less rapidly when electric field increased. - At large field, the drift velocity approaches a saturation velocity. Drift velocity versus electric field in Si.

High Field Effects Drift velocity versus electric field in Si and Ga. As. Note that for n-type Ga. As, there is a region of negative differential mobility. shows the differentiation between high-field transport in n-type Ga. As and Si. For n-type Ga. As – vs reached maximum level, then decreases when the field increases. This phenomenon is due to energy bands structure of Ga. As that allows the transfer of conduction electrons from high mobility energy minimum (called valley) to low mobility. Means that, electron transfer from the central valley to the satellite valleys along [111] direction.