Книги+1 / 2013 [Chandan_Kumar_Sarkar]_Technology_CAD

points x = –l and x = l are considered as in [1], which are one mean free path away from x = 0. The flux due to the semiconductor carriers at x = 0 due to carriers moving from x = –l is given by

The “1/2” term is because only half of the carriers move to the left while the other half moves to the right. The flux due to the semiconductor carriers at x = 0 due to carriers moving from x = +l is given by

The total flux of carriers at x = 0 may be obtained by subtracting the flux due to carriers moving from right to left from the flux of carriers moving from left to right:

Considering small mean free path, the carrier density derivative may be obtained as

The diffusion current density for electrons may be obtained by multiplying the flux with the charge of an electron:

Let the diffusion constant, Dn , be equal to the product of thermal velocity, vth, and the mean free path, l:

For holes the diffusion current density is given as

2.5.3  Total Drift-Diffusion Current

Based on the concepts derived in the previous sections, we can now establish the drift-diffusion equations [3,20]. The total hole current density in a semiconductor is composed of the sum of the drift and the diffusion components of current. Similarly, the total electron current density in a semiconductor is composed of the sum of the drift and the diffusion components of current. For electrons the total current density is given as

The total current is the sum of the electron and hole current densities multiplied by the area, A, perpendicular to the current direction:

2.5.4  Einstein Relation

In a non-uniformly doped semiconductor the doping concentration decreases with increase in x. As a result, diffusion of majority carriers takes place from a high concentration to a low concentration region along the +x direction. The flow of electrons leaves behind positively charged donor ions. The separation of positive and negative charges creates an electric field in the direction opposite to the diffusion process. At equilibrium the induced electric field prevents further diffusion.

When taking the electric field along the x direction, the energy bands are as shown in Figure 2.11. The potential energy increases in the direction of the electric field. The electrostatic potential, which varies in the opposite direction as it is defined in terms of positive charges, is given by

Electric field

Conduction band

Ei

Valence band

x

FIGURE 2.11

Energy band diagram of a semiconductor in the presence of an electric field E(x).

Considering Ei as a convenient reference, the electron potential energy may be related to E(x) as

Since the band diagram indicates electron energies, we know that the slope of this band must be such that electrons drift downhill in the field. Therefore E points uphill in the band diagram. At equilibrium no current flows. Putting

parameter (μ) and diffusion parameter (D) is given by the Einstein relationship.

2.6  Carrier Recombination and Generation

Recombination is a process by which the electrons occupy the empty states associated with holes. The carriers as a result disappear. The energy released is the difference between the initial state energy and the final state energy of the electrons. The energy is emitted in the form of a photon for radiative recombination. For non-radiative recombination, it is transmitted to one or more phonons, and for Auger recombination it is transferred as kinetic energy to another electron [1,2,21]. The different processes are shown in Figure 2.12.

Ec

E

Et

Ev

FIGURE 2.12

Carrier recombination mechanisms in semiconductors.

In band-to-band recombination an electron moves from the conduction band into the valence band state associated with the hole. In case of direct band-gap semiconductors, it is a case of radiative transition. In trap-assisted recombination, the electrons fall into the band gaps caused by some defects or some foreign atoms. The electrons occupying those band gaps move into an empty valence band state, thus completing the recombination process. So this is a two-step transition of an electron from the conduction to the valence band called Shockley-Read-Hall (SRH) recombination. Auger recombination is a process of recombination of an electron and a hole with the resultant energy given to another electron or hole. Auger recombination is different from band-to-band recombination due to this third electron or hole. All the recombination processes when reversed cause carrier generation instead of recombination. A single expression can be used to describe both generation and recombination processes. Generation of carriers by light absorption is a process that does not have recombination associated with it. This process is referred to as ionization. Impact ionization also belongs to this category. The different generation mechanisms are shown in Figure 2.13.

If the photon energy is large, an electron from the valence band may move into the conduction band generating an electron-hole pair as a result. This photon energy must be larger than the band-gap energy for electron-hole pair generation. Kinetic energy (Eph – Eg) is added to the electron and the hole due to absorption of the photon.

Carrier generation due to high-energy charged particles is similar except that the available energy may be far greater than the band-gap energy causing multiple electron-hole pairs generation. The high-energy particle gradually loses its energy and eventually stops [5,22].

Impact ionization is the counterpart of Auger recombination. It is caused by an electron/hole with energy, much greater/smaller than the conduction/ valence band edge. The process is shown in Figure 2.14.

The excess energy is transmitted to generate an electron-hole pair in band transition [23]. Avalanche multiplication is caused in semiconductor diodes

Conduction

band

Valence

band

FIGURE 2.13

Carrier generation due to light absorption and ionization due to high-energy particles.

Small volume in which generation and recombination occurs

(a)

FIGURE 2.15

(a) A conceptual description of the continuity equation. (b) Geometry used to develop the current continuity equation.

A solution to these equations can be obtained by substituting the expression for the electron and hole current. This then yields two partial differential equations as a function of the electron density, the hole density, and the electric field. The electric field is obtained from Gauss’s law as in Figure 2.15(b).

Let us now collect the various terms in this continuity equation. If δn is the excess carrier density in the region, the recombination rate R in the volume A x may be written approximately as

where τn is the electron recombination time per excess particle due to both the radiative and the non-radiative components. The particle fllow rate into the same volume due to the current density Jn(x) is given by the difference of particle current coming into the region and the particle current leaving the region:

70 Technology Computer Aided Design: Simulation for VLSI MOSFET

If G is the generation rate per unit volume, the generation rate in the volume A · x is GA x. δn/τn is the net recombination rate of electrons and U = G − R—that is, Rate of electron buildup (U) = Increase in electron concentration in xA per unit time (G) – recombination rate (R). The rates of electron

As x approaches zero, we can write equations in the derivative form for electrons and holes as

Using these expressions, the diffusion currents are

Jn(diff ) = eDn ∂δn

∂x

Jp (diff ) = −eDp ∂δp

(2.45)

∂x

Thetime-dependentcontinuityequationforelectronsandholes,validseparately:

These equations are used to study the steady-state charge profiile in p-n diodes and bipolar transistors. In steady state,

∂2 δn = δn = δn

∂x2 Dnτn L2n

(2.47)

∂2 δp = δp = δp

∂x2 Dpτp L2p

Here is the diffusion length for electrons, and Lp is the diffusion length for holes. Considering the case where an excess electron density δn(0) is maintained at x = 0, at point L in the semiconductor, the excess carrier density is maintained at δ(L). The general solution of the above second-order differential equation is

When L >> Ln and δn(L) = 0 , the semiconductor is much longer than Ln , for example in the case of the long p-n diode. A1 and A2 can be found from the boundary conditions. For a large value of x, δn = 0 at x = ∞ and so A1 = 0. Similarly δn = 0 = δn(0) at x = 0 giving A2 = δn(0) . The solution of the equation is given by

It is seen from the above equation that the carrier density decays exponentially in the semiconductor.

However, when L << Ln , the carrier density is linear from one boundary value to the other because over a short distance exponential can be approximated as linear. When excess carriers are injected into a thick semiconductor sample, both diffusion and recombination take place. Ln represents the distance over which the injected carrier density falls to 1/e of its original value. It also represents the average distance an electron diffuses before recombination.

The probability that an electron survives up to a distance x without recombination is given by

δnp (x) = e− x Ln

δnp (0)

The steady-state distribution of excess holes causes diffusion and a hole current in the direction of decreasing concentration.

It will be useful for the current calculation of the p-n junction where the injection of minority carriers across a junction will lead to exponential distribution.