1 H-exp

(6.87)

which is the probability that a trap will contain an electron. The function (1 - ffiEt)) is then the probability that the trap is empty. In Equation (6.87), we have assumed that the degeneracy factor is one, which is the usual approximation made in this analysis. However, if a degeneracy factor is included, it wil] eventually be absorbed in other constants later in the analysis.

For process 2, the rate at which electrons are emitted from filled traps back into the conduction band is proportional to the number of filled traps, so that

R,n = E,N,MEr) (6.88)

where

Ren - emission rate (#/cm-s) E - constant ffiEt) = probability that the trap is occupied

In thermal equilibrium, the rate of electron capture from the conduction band and the rate of electron emission back into the conduction band must be equal. Then

Ren = /?cn (6-89)

Process 2: The inverse of process 1-the emission of an electron that is initially occupying a trap level back into the conduction band.

Process 3: The capture of a hole from the valence band by a trap containing an electron. (Or we may consider the process to be the emission of an electron from the trap into the valence band.)

Process 4: The inverse of process 3-the emission of a hole from a neutral trap into the valence band. (Or we may consider this process to be the capture of an electron from the valence band.)

In process I, the rate at which electrons from the conduction band are captured by the traps is proportional to the density of electrons in the conduction band and proportional to the density of empty trap states. We can then write the electron capture rate as

Re = CM( fy(E,))n (6.86)

where

Ren = capture rate (#/cm -s)

= constant proportional to electron-capture cross section Nj = total concentration of trapping centers n electron concentration in the conduction band ff(Et) = Fermi function at the trap energy

The Fermi function at the trap energy is given by

so that

E.NJfoiEr) = C Af;(l - fFo(Et))no (6.90)

where ffo denotes the thermal-equilibrium Fermi function. Note that, in thermal equilibrium, the value of the electron concentration in the capture rate term is the equilibrium value о- Using the Boltzmann approximation for the Fermi funcrion, we can find in terms of C as

E - пГ (6.91)

where is defined as

-iE,-E,)

n = Nc exp

(6.92)

The parameteris equivalent to an electron concentrarion that would exist in the conduction band if the trap energy Ef coincided with the Fermi energy Ef.

In nonequilibrium, excess electrons exist, so that the net rate at which electrons are captured from the conduction band is given by

R, = R R, (6.93)

which is just the difference between the capture rate and the emission rate. Combining Equations (6.86) and (6.88) with (6.93) gives

Rn - [CnNril - ff(E,))n] - [EnNJriEr)] (6.94)

We may note that, in this equation, the electron concentration n is the total concentration, which includes the excess electron concentration. The remaining constants and terms in Equarion (6.94) are the same as defined previously and the Fermi energy in the Fermi probability function needs to be replaced by the quasi-Fermi energy for electrons. The constants E and C are related by Equation (6.91), so the net recombination rate can be written as

Rn = C Ndn(i - ffiEJ) - njAEr)] (6,95)

If we consider processes 3 and 4 in the recombination theory, the net rate at which holes are captured from the valence band is given by

Rp = CpNApffiEJ - p41 - ffiEM (6.96)

where Cp is a constant proportional to the hole capture rate, and p is given by

-{Er-E,)

(6.97)

In a semiconductor in which the trap density is not too large, the excess electron and hole concentrations are equal and the recombination rates of electrons and holes are equal. If we set Equation (6.95) equal to Equation (6.96) and solve for the Fermi function, we obtain

ME,) = , C n + C,p-

C (n + n) + Cp(p + p)

We may note that np = nj. Then, substituting Equation (6.98) back into either Equation (6.95) or (6.96) gives

R R C,CpNt{np~nj)

CAn+n)CJp p)

(6.99)

Equation (6.99) is the recombination rate of electrons and holes due to the recombination center dt E = EjAf we consider thennal equiHbrium, then np - П{\р{\ = n?, so that - Rp - 0. Equation (6.99), then, is the recombination rate of excess electrons and holes.

Since R in Equation (6.99) is the recombination rate of the excess carriers, we may write

(6Л00)

where 8nis the excess-carrier concentration and т is the lifetime of the excess carriers.

6-5.2 Limits of Extrinsic Doping and Low Injection

We simplitied the ambipolar transport equation, Equation (6.39), from a nonlinear differential equation to a linear differential equation by applying limits of extrinsic doping and low injection. We may apply these same limits to the recombination rate equation.

Consider an n-type semiconductor under low injection. Then

0 Pcb 0 8p, /70 > n\ nQ > p

where 8p is the excess minority carrier hole concentration. The assumptions of no y> rt and По p imply that the trap level energy is near midgap so that n and p are not too different from the intrinsic carrier concentration. With these assumptions, Equation (6.99) reduces to

RCpN.Sp (6.101)

The recombination rate of excess carriers in the n-type semiconductor is a function of the parameter Cp, which is related to the minority carrier hole capture cross section. The recombination rate, then, is a function of the minority carrier parameter in the same way that the ambipolar transport parameters reduced lo their minority carrier values.

The recombination rate is related to the mean carrier lifetime. Comparing Equations (6.100) and (6.101), we may write j

CpNtSp

(6.102)

where

(6.103)

(np - nj)

R = ---Ц--- (6Л05)

Xpoin -h n ) + r o(p + p)

pConsider an intrinsic semiconductor containing excess carriers. Then n = ni -\- 8n and p = ni -\-Sn. Also assume that n p = щ.

Solution

Equation (6,105) now becomes

2nibn + (Sny

(2nj -Sn){Tpu + r o) If we also assume very low injection, so that 5n <$: 2/г then we can write

ГрО + r,i{) T

where г is the excess carrier lifetime. We see that т = t> + to in the intrinsic material. Comment

The excess-carrier lifetime increases as we change from an extrinsic to an intrinsic semiconductor

Intuitively, we can see that the number of majority carriers that are available for recombining with excess minority carriers decreases as the extrinsic semiconductor

where Zpo is defined as the excess minority carrier hole lifetime. If the trap concentration increases, the probability of excess carrier recombination increases; thus ;the excess minority carrier lifetime decreases.

Similarly, if we have a strongly extrinsic p-type material under low injection, we assume that

Po > 0. Po > po Po > p le liferime then becomes that of the excess minority carrier electron lifetime, or

Again note that for the n-type material, the lifetime is a function of Cp, which is related to the capture rate of the minority carrier hole. And for the p-type material, the lifetime is a function of which is related to the capture rate of the minority carrier electron. The excess-carrier lifetime for an extrinsic material under low injection reduces to that of the minority carrier.

Objective example 6.7

To determine the excess-carrier lifetime in an intrinsic semiconductor.

If we substitute the definitions of excess-carrier fifetimes from Equations (6T03) and (6T04) into Equation (6.99), the recombination rate can be written as

becomes intrinsic. Since there are fewer carriers available for recombining in the intrinsic material, the mean lifetime of an excess carrier increases.

TEST YOUR UNDERSTANDING

£6.12 Consider silicon at Г 300 К doped at concentrations of Nj = 10 cm~ and

Na =0. Assume that n = /? = л, in the excess carrier recombinahon rate equation and assume parameter values of ro = ro = 5 x 10~ s. Calculate the recombination rate of excess carriers if Sn = Sp = 10 cm\ t-ШЭ 01 x V)

6.6 1 SURFACE EFFECTS I

In all previous discussions, we have implicitly assumed the semiconductors were infinite in extent; thus, we were not concerned with any boundary conditions at a serai-conductor surface. In any real application of semiconductors, the material is not infinitely large and therefore surfaces do exist between the semiconductor and an adjacent medium.

6.6,1 Surface States

When a semiconductor is abruptly terminated, the perfect periodic nature of the idealized single-crystal lattice ends abruptly at the surface. The disruption of the periodic-potential function results in allowed electronic energy states within the energy bandgap. In the previous section, we argued that simple defects in the semiconductor would create discrete energy states within the bandgap. The abrupt termination of the periodic potential at the surface results in a distribution of allowed energy states within the bandgap, shown schematically in Figure 6.17 along with the discrete energy states in the bulk semiconductor.

The Shockley-Read-Hall recombination theory shows that the excess minority carrier lifetime is inversely proportional to the density of trap states. We may argue

Electron energy

Surface

Figure 6.17 I Distribution of surface states within the foibidden bandgap.

6. в Surface Effects

Surface Distance x

Figure 6.181 Steady-state excess hole concentration versus distance from a semiconductor surface.

that, since the density of traps at the surface is larger than in the bulk, the excess minority carrier lifetime at the surface will be smaller than the corresponding lifetime in the bulk material. If we consider an extrinsic n-type semiconductor, for example, the recombinarion rate of excess carriers in the bulk, given by Equation (6.102), is

where Sps is the concentration of excess minority carrier holes in the bulk material. We may write a similar expression for the recombinarion rate of excess carriers at the surface as

Ks - (6.107)

where Bp is the excess minority carrier hole concentration at the surface and ros is the excess minority carrier hole lifetime at the surface.

Assume that excess carriers are being generated at a constant rate throughout the entire semiconductor material. We showed that, in steady state, the generation rate is equal to the recombination rate for the case of a homogeneous, infinite semiconductor. Using this argument, the recombination rates at the surface and in the bulk material must be equal. Since т^,oл < V, then the excess minority carrier concentration at the surface is smaller than the excess minority carrier concentrarion in the bulk region, or Sps < Spti. Figure 6JS shows an example of the excess-carrier concentration plotted as a function of distance from the semiconductor surface.

Objective example 6.s

To determine the steady-state excess-carrier concentration as a function of distance from the surface of a semiconductor.

Consider Figure 6.18, in which the surface is at x = 0. Assume that rn the n-type semiconductor Spfi = 10 cin~ and TpQ = 10~ s in the bulk, and г^ол = Ю s at the surface. Assume zero applied electric field and let D = 10 cm/s.

CHAPTER 6 Nonequilibrium Excess Carriers in Semiconductors Solution

From Equations (6.Ш6) and (6Л07), we have

SO that

TPO )

10- \

From Equation (6.56), we can write

diSp) , 8p

(6.108)

The generation rate can be determined from the steady-state conditions in the bulk, or

The solution to Equation (6.107) is of the form

(6.109)

Asx -l-oo, Spix) Spn = gXpo = 10 cm--, which implies ihat Л = 0. At ,r 0. we have

Sp{i)) hp, = lO- + В - 10- cm--

so that В -9 X 10 *. The entire solution for the minority carrier hole concentration as a function of distance from the surface is

Sp{x) lO-d -0.9/)

where

Lp ./Щ^ У(Т0)(10 - 3E6 дт

Comment

The excess carrier concentration is smaller at the surface than in the bulk.

6,6,2 Surface Recombination Velocity j

A gradient in the excess-carrier concentration exists near the surface as shown in Figure 6.18; excess carriers from the bulk region diffuse toward the surface where they recombine. This diffusion toward the surface can be described by the equarion

. dm

sSp

surf

(6.110)

surf

where each side of the equation is evaluated at the surface. The parameter n is the unit outward vector normal to the surface. Using the geometry of Figure 6.18,

6.6 Surface Effects

d{Sp)jdx is a positive quantity and h is negative, so that the parameter 5 is a positive quantity.

A dimensional analysis of Equation (6.110) shows that the parameter s has units of cm/sec, or velocity. The parameter s is called the surface recombination velocity. If the excess concentrations at the surface and in the bulk region were equal, then the gradient term would be zero and the surface recombination velocity would be zero. As the excess concentration at the surface becomes smaller, the gradient term becomes larger, and the surface recombination velocity increases. The surface recombination velocity gives some indication of the surface characteristics as compared with the bulk region.

I Equation (6.110) may be used as a boundary condition to the general solution given by Equation (6.109) in Example 6.8. Using Figure 6.18, we have that = - 1, and Equation (6.110) becomes

- s8p

surf

(6.111)

surf

We have argued that the coefficient Л is zero in Equation (6.109). Then, from Equation (6.109), we can write that

8psuri = 8p(0) = gzpo + В

(6.112a)

d(Sp) dx

surf

В

(6.112b)

Substituting Equations (6.112a) and (6.112b) into Equation (6.111) and solving for the coefficient B, we obtain

(Dp/Lp)+s

The excess minority carrier hole concentration can then be written as

(6.113)

, ( sLe-\

(6.114)

Objecrive

To determine the steady-state excess concentration versus distance from the surface of a semiconductor as a function of surface recombination velocity.

Consider, initially, the case when the surface recombination velocity is zero, or =0.

Solution

Substituting s - 0 into Equation (6.114), we obtain

Now consider the case when the surface recombination velocity is infinite, or 5 = oo.

EXAMPLE 6.9

CHAPTER6 NonequHibnum Excess Gamers in Semiconductors

Solution

Substituting s oc into Equation (6.114), we obtain

Comment

For the case when s 0, the surface has no effect and the excess minority carrier concentration at the surface is the same as in the bulk. In the other extreme when s = oo, the excess minority carrier hole concentration at the surface is zero.

An infinite surface recombination velocity implies that the excess minority carrier concentration and lifetime at the surface are zero.

EXAMPLE 6Л0

Objective

To determine the value of surface recombinaion velocity corresponding to the parameters given in Example 6.8.

From Example 6.8, we have that gro - Ю' * cm \ Dp = lOcm-/s, - 3E6 iim, and V№) = 10 cm

Solution

Writing Equation (6.114) at the surface, we have

{Dp/Lp) + s

Solving for the surface recombination velocity, we find that

Lp \6p(0)

which becomes

31.6 X 10

2.85 X 10-* cm/s

Comment

This example shows that a surtace recombitiation velocity of approximately 5 = 3 x 10 * cm/s could seriously degrade the performance of semiconductor devices, such as solar cells, since these devices tend to be fabricated close to a surface.

In the above example, the surface influences the excess-carrier concentrarion to the extent that, even at a distance of = 31.6 дт from the surface, the excess-carrier concentration is only two-thirds of the value in the bulk. We will see in later chapters that device performance is dependent in large part on the properties of excess carriers.

6.7 I SUMMARY

The processes of excess electron and hole generation and recombination were discussed. The excess carrier generation rate and recombination rate were defined.

Excess electrons and holes do not move independently of each other, but move together. This common movement is called ambipolar transport.

The ambipolar transport equation was derived and limits of low injection and extrinsic doping were applied to the coefficients. Under these conditions, the excess electrons and holes diffuse and drift together with the characteristics of the minority carrier, a result that is fundamental to the behavior of semiconductor devices.

The concept of excess carrier lifetime was developed.

Examples of excess carrier behavior as a function of time, as a function of space, and as a function of both time and space were examined.

The quasi-Fermi level for electrons and the quasi-Fermi level for holes were defined. These parameters characterize the total electron and hole concentrations in a semiconductor in nonequilibrium.

The Shockley-Read-Hall theory of recombination was considered. Expressions for the excess minority carrier lifetime were developed.

The effect of a semiconductor surface influences the behavior of excess electrons and holes. The surface recombination velocity was defined.

GLOSSARY OF IMPORTANT TERMS

ambipolar diffiision coefficient The effective diffusion coefficient of excess carriers, ambipolar mobility The effective mobility of excess carriers.

ambipolar transport The process whereby excess electrons and holes diffuse, drift, and recombine with the same effective diffusion coefficient, mobility, and lifetime,

ambipolar transport equation The equation describing the behavior of excess carriers as a function of time and space coordinates.

carrier generation The process of elevating electrons from the valence band into the conduction band, creating an electron-hole pair.

carrier recombination The process whereby an electron Tails* into an empty state in the valence band (a hole) so that an electron-hole pair are annihilated.

excess carriers The term describing both excess electrons and excess holes.

excess electrons The concentration of electrons in the conduction band over and above the thermal-equilibrium concentration.

excess holes The concentration of holes in the valence band over and above the thermal-equilibrium concentration.

excess minority carrier lifetime The average time that an excess minority carrier exists before it recombines.

generation rate The rate (#/cm* -s) at which electron-hole pairs are created.

low-Jevel injection The condition in which the excess-carrier concentration is much smaller than the thermal-equilibrium majority carrier concentration.

minority carrier diffusion length The average distance a minority carrier diffuses before recombining: a parameter equal to VDt where D and r are the minority carrier diffusion coefficient and lifetime, respectively.