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some basic differences in electrical characteristics caused by variations in band structure by considering some simplified energy bands.

There are several possible energy-band conditions to consider. Figure 3.19a shows an allowed energy band that is completely empty of electrons. If an electric field is applied, there are no panicles to move, so there will be no current. Figure 3.19b shows another allowed energy band whose energy states are completely full of electrons. We argued in the previous section that a completely full energy band will also not give rise to a current. A material that has energy bands either completely empty or completely full is an insulator. The resistivity of an insulator is very large or, conversely, the conductivity of an insulator is very small. There are essentially no charged particles that can contribute to a drift current. Figure 3.19c shows a simplified energy-band diagram of an insulator. The bandgap energy of an insulator is usually on the order of 3.5 to 6 eV or larger, so that at room temperature, there are essentially no electrons in the conducfion band and the valence band remains completely full. There are very few thermally generated electrons and holes in an insulator.

Figure 3.20a shows an energy band with relatively few electrons near the bottom of the band. Now, if an electric field is applied, the electrons can gain energy, move to

AJ lowed energy band (empty)

Allowed

energy

band

(almost

empty)

Allowed energy band (full)

Allowed

energy

band

(almost

full)

Conduction

band

(empty)

Valence

band

(full)

Figure 3.19 1 Allowed energy bands showing (a) an empty band, (b) a completely full band, and (c) the bandgap energy between the two allowed bands.


Conduction band (almost empty)

Empty electronic states

Valence band (almost full)

Figure 3.20 f Allowed energy bands showing (a) an almost empty band, (b) an altnost full band, and (c) the bandgap energy between the two allowed bands.



Partially

-----------------filled

i -ш band

1 III, mmlitf-- .....ffi Till Ш №1 if



Upper

у / band

Lower

band ) Electrons

(a) (b)

Figure 3.211 Two possible energy bands of a metal showing (a) a partially filled band and (b) overlapping allowed energy bands,

higher energy states, and move through the crystal. The net flow of charge is a current. Figure 3.20b shows an allowed energy band that is almost full of electrons, which means that we can consider the holes in this band. If an electric field is applied, the holes can move and give rise to a current. Figure 3.20c shows the simplified energy-band diagram for this case. The bandgap energy may be on the order of 1 eV. This energy-band diagram represents a semiconductor for Г > 0 К. The resistivity of a semiconductor, as we will see in the next chapter, can be controlled and varied over many orders of magnitude.

The characteristics of a metal include a very low resistivity. The energy-band diagram for a metal may be in one of two forms. Figure 3.2 la shows the case of a partially full band in which there are many electrons available for conduction, .so that the material can exhibit a large electrical conductivity. Figure 3.21b shows another possible energy-band diagram of a metal. The band splitting into allowed and forbidden energy bands is a complex phenomenon and Figure 3.21b shows a case in which the conduction and valence bands overlap at the equilibrium interatomic distance. As in the case shown in Figure 3.21a, there are large numbers of electrons as well as large numbers of empty energy states into which the electrons can move, so this material can also exhibit a very high electrical conductivity.

3.3 I EXTENSION TO THREE DIMENSIONS

The basic concept of allowed and forbidden energy bands and the basic concept of effective mass have been developed in the last sections. In this section, we will extend these concepts to three dimensions and to real crystals. We will qualitatively consider particular characteristics of the three-dimensional crystal in terms of the E versus к plots, bandgap energy, and effective mass. We must emphasize that we will only briefly touch on the basic three-dimensional concepts; therefore, many details will not be considered.

One problem encountered in extending the potential function to a three-dimensional crystal is that the distance between atoms varies as the direction through the crystal changes. Figure 3.22 shows a face-centered cubic structure with the [100] and [110] directions indicated. Electrons traveling in different directions encounter different potenual patterns and therefore different space boundaries. The E versus к diagrams are in general a function of the space direction in a crystal.



3.3 Extension to Three Dimensions

[110] direction


[lOOJ direction

Figure 3.22 I The (100) plane of a face-centered cubic crystal showing the [1001 and [110] directions.

ЗЗЛ The fc-Space Diagrams of Si and GaAs

Figure 3.23 shows an E versus к diagram of gallium arsenide and of silicon. These simplified diagrams show the basic properties considered in this text, but do not show many of the details more appropriate for advanced-level courses.

Note that in place of the usual positive and negative к axes, we now show two different crystal directions. The E versus к diagram for the one-dimensional model

>


1100]

>

Figure 3.231 Energy band structures of (a) GaAs and (b) Si. (FromSze (Щ)

Conduction

band

t i

*\ 1

Valence

\ i

band

[111] 0 iiooj

к



CHAPTERS IntroductiontotheQuantum Theory of Sol i ds

was symmetric in к so that no new information is obtained by displaying the negative axis. It is normal practice to plot the [100] direction along the normal -\-k axis and to plot the 1111] portion of the diagram so the +/: points to the left. In the case of diamond or zincblende lattices, the maxima in the valence band energy and minima in the conduction band energy occur at /: = 0 or along one of these two directions.

Figure 3.23a shows the E versus к diagram for GaAs. The valence band maximum and the conduction band minimum both occur eX к - 0. The electrons in the conduction band tend to settle at the minimum conduction band energy which is at it - 0. Similarly, holes in the valence band tend to congregate at the uppermost valence band energy. In GaAs, the minimum conduction band energy and maximum valence band energy occur at the same к value. A semiconductor with this property is said to be a direct bandgap semiconductor; transitions between the two allowed bands can take place with no change in crystal momentum. This direct nature has significant effect on the optical properties of the material. GaAs and other direct bandgap materials are ideally suited for use in semiconductor lasers and other optical devices.

The E versus к diagram for silicon is shown in Figure 3.23b. The maximum in the valence band energy occurs гхк - 0 as before. The minimum in the conduction band energy occurs not at - 0, but along the [100] direction. The difference between the minimum conduction band energy and the maximum valence band energy is still defined as the bandgap energy Eg. A semiconductor whose maximum valence band energy and minimum conduction band energy do not occur at the same к value is called an indirect bandgap semiconductor. When electrons make a transition between the conduction and valence bands, we must invoke the law of conservation of momentum. A transition in an indirect bandgap material must necessarily include an interaction with the crystal so that crystal momentum is conserved.

Germanium is also an indirect bandgap material, whose valence band maximum occurs -dtk = 0 and whose conduction band minimum occurs along the [111] direction. GaAs is a direct bandgap semiconductor, but other compound semiconductors,j such as GaP and AI As, have indirect bandgaps.

3.3.2 Additional Effective IVIass Concepts

The curvature of the E versus k diagrams near the minimum of the conduction Ъш energy is related to the effective mass of the electron. We may note from Figure 3.Z that the curvature of the conduction band at its minimum value for GaAs is largg than that of silicon, so the effective mass of an electron in the conduction band о GaAs will be smaller than that in silicon.

For the one-dimensional E versus к diagram, the effective mass was defined by Equation (3.41) as l/m* = IjT?- -dE/dk. A complication occurs in the effective mass concept in a real crystal. A three-dimensional crystal can be described by к vectors. The curvature of the E versus к diagram at the conduction band minimum not be the same in the three к directions. We will not consider the details of the varii effective mass parameters here. In later sections and chapters, the effective mass parameters used in calculations will be a kind of statistical average that is adequate for most device calculations.




3.4 I DENSITY OF STATES FUNCTION

As we have stated, we eventually wish to describe the current-voltage characteristics of semiconductor devices. Since current is due to the flow of charge, an important step in the process is to determine the number of electrons and holes in the semiconductor that will be available for conduction. The number of carriers that can contribute to the conduction process is a function of the number of available energy or quantum states since, by the Pauli exclusion principle, only one electron can occupy a given quantum state. When we discussed the splitting of energy levels into bands of allowed and forbidden energies, we indicated that the band of allowed energies was actually made up of discrete energy levels. We must determine the density of these allowed energy states as a function of energy in order to calculate the electron and hole concentrations.

ЗАЛ Mathematical Derivation

To determine the density of allowed quantum states as a function of energy, we need to consider an appropriate mathematical model. Electrons are allowed to move relatively freely in the conduction band of a semiconductor, but are confined to the crys tal. As a first step, we will consider a free electron confined to a three-dimensional infinite potential well, where the potential well represents the crystal. The potential of the infinite potential well is defined as

V(jt,y,)0 forO<x<a (3.59)

0 < у < a

0 < z < a V(x, y, г) - oo elsewhere

where the crystal is assumed to be a cube with length a. Schrodingers wave equation in three dimensions can be solved using the separation of variables technique. Extrapolating the results from the one-dimensional infinite potential well, we can show (see Problem 3.21) that

2m E

(3.60)

where n, Пу, and л^-; are positive integers, (Negative values of л^, /j , and n- yield the same wave function, except for the sign, as the positive integer values, resulting in the same probability function and energy, so the negative integers do not represent a different quantum state.)

We can schematically plot the allowed quantum states in к space. Figure 3.24a shows a two-dimensional plot as a function of and ky. Each point represents an allowed quantum state corresponding to various integral values of Пд and Пу, Positive and negative values of kj, ky, or k- have the same energy and represent the same




1


Figure 3.24 ) (a) A two-dimensional array of allowed quantum states in к space, (b) The positive one-eighth of the spherical к space.

energy state. Since negative values of kx, ky, or k- do not represent additional quantum states, the density of quantum states will be determined by considering only the positive one-eighth of the spherical к space as shown in Figure 3.24b.

The distance between two quantum states in the A;, direction, for example, is given by

fx-hl -.v = ( t + 1)

(3.6Г

Generalizing this resuh to three dimensions, the volume Vk of a single quantum state is

(3.62)

We can now determine the density of quantum states in к space. A differential volume in к space is shown in Figure 3.24b and is given by 4jrA: dk, so the differential density of quantum states in к space can be written as .

gr{k)dk = 2[~

The first factor, 2, takes into account the two spin states allowed for each quanti state; the next factor, , takes into account that we are considering only the quantt states for positive values of k, ky, and k. The factor Алк^ dk is again the differei tial volume and the factor (n/a) is the volume of one quantum state. Equation (3.63] may be simplified to

gT(k)dk =

nkdk



k = л/2т£ (3.65b)

The differential dk is

= у i <f £ (3.66)

Then, substituting the expressions for and dk into Equation (3.64), the number of energy states between E and E dE is given by

7та {2mE\ 1 / m

Since h - h/27t. Equation (3.67) becomes

gAE) dE = -- . (2m)- . dE (3.68)

Equation (3.68) gives the total number of quantum states between the energy E and £ + in the crystal space volume of a?. If we divide by the volume a, then we will obtain the density of quantum states per unit volume of the crystal. Equation (3.68) then becomes

4jr(2m)- r-g{E)-j (3.69)

The density of quantum states is a function of energy E, As the energy of this free electron becomes small, the number of available quantum states decreases. This density function is really a double density, in that the units are given in terms of states per unit energy per unit volume.

Objective example з.з

To calculate the density of states per unit volume over a particular energy range.

Consider the density of states for a free electron given by Equation (3.69). Calculate the density of states per unit volume with energies between 0 and 1 eV.

Equation (3.64) gives the density of quantum states as a function of momentum, through the parameter L We can now determine the density of quantum states as a function of energy E, For a free electron, the parameters E and k are related by

2m E

(3.65a)



CHAPTER 3 I ntroduction to the Quantum Theory of Solids Solution

The volume density of quantum states, from Equation (3.69), is

= j g{E)dE = ----j s/EdE

The density of states is now

47г12(9.11 X 10 )] 2 in-iv. ,n7 -i

;V = -----* - * (L6 X 10 )- = 4.5 X 10 m

(6.625 X 10-4) 3

= 4.5 X 10 states/cm

Comment

The density of quantum states is typically a large number. An effective density of states in a semiconductor, as we will see in the following sections and in the next chapter, is also a lare number, but is usually less than the density of atoms in the semiconductor crystal.

3.4.2 Extension to Semiconductors

in the last section, we derived a general expression for the density of allowed eb tron quantum states using the model of a free electron with mass m bounded in a three-dimensional infinite potential well. We can extend this same general model to a semiconductor to determine the density of quantum states in the conduction band and the density of quantum states in the valence band. Electrons and holes are confined within the semiconductor crystal so we will again use the basic model of the infinite potential well.

The parabolic relationship between energy and momentum of a free electron was given in Equation (3.28) as Я - 12m = frk/2m. Figure 3.16a showed the conduction energy band in the reduced к space. The E versus к curve near = 0 at the bottom of the conduction band can be approximated as a parabola, so we may write

E = E, + -- (3.7

where E,. is the bottom edge of the conduction band and m* is the electron effectii mass. Equation (3.70) may be rewritten to give

E - E, -- (3.71



The general form of the £ versus к relation for an electron in the bottom of a conduction band is the same as the free electron, except the mass is replaced by the effective mass. We can then think of the electron in the bottom of the conduction band as being a *free electron with its own particular mass. The right side of Equation (3.71) is of the same form as the right side of Equation (3.28), which was used in the derivation of the density of states function. Because of this similarity, which yields the free conduction electron model, we may generalize the free electron results of Equation (3.69) and write the density of allowed electronic energy states in the conduction band as

gc(E) ------ VВ - E,

(3.72)

Equation (3.72) is valid for E > E As the energy of the electron in the conduction band decreases, the number of available quantum states also decreases.

The density of quantum states in the valence band can be obtained by using the same infinite potential well model, since the hole is also confined in the semiconductor crystal and can be treated as a free particle. The effective mass of the hole is m*. Figure 3.16b showed the valence energy band in the reduced к space. We may also approximate the E versus к curve near : = 0 by a parabola for a free hole, so diat

(3.73)

Equafion (3.73) may be rewritten to give

E.,-E =

(3.74)

Again, the right side of Equation (3.74) is of the same form used in the general derivation of the density of states function. We may then generalize the density of states function from Equation (3.69) to apply to the valence band, so that

(3.75)

Equation (375) is valid for £ < E .

We have argued that quantum states do not exist within the forbidden energy band, so g(E) - 0 for < E < E Figure 3.25 shows the plot of the density of quantum states as a function of energy. If the electron and hole effective masses were equal, then the functions gc{E) and gv(E) would be symmetrical about the energy midway between Ec and F or the midgap energy, midgap-





8viE)

Figure 3.25 I The density of energy states in the conduction band and the density of energy states in the valence band as a function of energy.

TEST YOUR UNDERSTANDING

E3.2 Determine the total number of energy states in silicon between E. and E- -\- кТ at

Г = 300K. (t-ыOI ZVZ -suy) E3.3 Determine the total number of energy states in silicon between E,. and - кТ at

T = 300 K. siOl X Z6L suy)

3.5 I STATISTICAL MECHANICS

In dealing with large numbers of particles, we are interested only in the statistical behavior of the group as a whole rather than in the behavior of each individual particle. For example, gas within a container will exert an average pressure on the walls of the vessel. The pressure is actually due to the collisions of the individual gas molecules with the walls, but we do not follow each individual molecule as it collides with the wall. Likewise in a crystal, the electrical characterisrics will be determined by the statistical behavior of a large number of electrons.

3.5.1 Statistical Laws

In determining the statistical behavior of particles, we must consider the laws that the i particles obey. There are three distribution laws determining the distribution of particles among available energy states. !




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