Electrons and Holes

The only empty electronic states in the silicon crystal are in the CB (Figure 5.1c). An electron placed in the CB is free to move around the crystal and also respond to an applied electric field because there are plenty of neighboring empty energy levels. An electron in the CB can easily gain energy from the field and move to higher energy lev­els because these states are empty. Generally we can treat an electron in the CB as if it were free within the crystal with certain modifications to its mass, as explained later in Section 5.1.3.

Since the only empty states are in the CB, the excitation of an electron from the VB requires a minimum energy of Eg. Figure 5.3a shows what happens when a pho­ton of energy hv > Eg is incident on an electron in the VB. This electron absorbs the incident photon and gains sufficient energy to surmount the energy gap Eg and reach the CB. Consequently, a free electron and a “hole,” corresponding to a missing elec­tron in the VB, are created. In some semiconductors such as Si and Ge, the photon ab­sorption process also involves lattice vibrations (vibrations of the Si atoms), which we have not shown in Figure 5.3b.

Electrons and Holes

Electron energy

подпись: electron energy

Figure 5.3

A photon with an energy greater than Eg can excite an electron from the VB to the CB.

When a photon breaks a Si-Si bond, a free electron and a hole in the Si—Si bond are created.

подпись: figure 5.3
a photon with an energy greater than eg can excite an electron from the vb to the cb.
when a photon breaks a si-si bond, a free electron and a hole in the si—si bond are created.

(b)

подпись: (b)Although in this specific example a photon of energy hv > Eg creates an electron – hole pair, this is not necessary. In fact, in the absence of radiation, there is an electron – hole generation process going on in the sample as a result of thermal generation. Due to thermal energy, the atoms in the crystal are constantly vibrating, which corresponds to the bonds between the Si atoms being periodically deformed. In a certain region, the atoms, at some instant, may be moving in such a way that a bond becomes over­stretched, as pictorially depicted in Figure 5.4. This will result in the overstretched bond rupturing and hence releasing an electron into the CB (the electron effectively

Figure 5.4 Thermal vibrations of atoms can break bonds and thereby create electron-hole pairs.

Electrons and HolesBecomes “free”). The empty electronic state of the missing electron in the bond is what we call a hole in the valence band. The free electron, which is in the CB, can wander around the crystal and contribute to the electrical conduction when an electric field is applied. The region remaining around the hole in the VB is positively charged because a charge of — e has been removed from an otherwise neutral region of the crystal. This hole, denoted as /i+, can also wander around the crystal as if it were free. This is be­cause an electron in a neighboring bond can “jump,” that is, tunnel, into the hole to fill the vacant electronic state at this site and thereby create a hole at its original position. This is effectively equivalent to the hole being displaced in the opposite direction, as illustrated in Figure 5.5a. This single step can reoccur, causing the hole to be further displaced. As a result, the hole moves around the crystal as if it were a free positively charged entity, as pictured in Figure 5.5a to d. Its motion is quite independent from that of the original electron. When an electric field is applied, the hole will drift in the di­rection of the field and hence contribute to electrical conduction. It is now apparent that there are essentially two types of charge carriers in semiconductors: electrons and holes. A hole is effectively an empty electronic state in the VB that behaves as if it were a positively charged “particle” free to respond to an applied electric field.

When a wandering electron in the CB meets a hole in the VB, the electron has found an empty state of lower energy and therefore occupies the hole. The electron falls from the CB to the VB to fill the hole, as depicted in Figure 5.5e and f. This is called recombination and results in the annihilation of an electron in the CB and a hole in the VB. The excess energy of the electron falling from CB to VB in certain semiconductors such as GaAs and InP is emitted as a photon. In Si and Ge the excess energy is lost as lattice vibrations (heat).

It must be emphasized that the illustrations in Figure 5.5 are pedagogical pictorial visualizations of hole motion based on classical notions and cannot be taken too seriously, as discussed in more advanced texts (see also Section 5.11). We should remember that the electron has a wavefunction in the crystal that is extended and not localized, as the pictures in Figure 5.5 imply. Further, the hole is a concept that corre­sponds to an empty valence band wavefunction that normally has an electron. Again, we cannot localize the hole to a particular site, as the pictures in Figure 5.5 imply.

Tiiilffili

(d)

H*

Warn

ih

IimwiS

CB

A

VB

H+

Electrons and Holes
Electrons and Holes
Electrons and Holes

N

SlHiMii

 

If)

 

Electrons and Holes Electrons and Holes Electrons and Holes

Figure 5.5 A pictorial illustration of a hole in the valence band wandering around the crystal due to the tunneling of electrons from neighboring bonds.

Conduction in Semiconductors

When an electric field is applied across a semiconductor as shown in Figure 5.6, the energy bands bend. The total electron energy E is KE + PE, but now there is an addi­tional electrostatic PE contribution that is not constant in an applied electric field. A uniform electric field ‘Ex implies a linearly decreasing potential V (x), by virtue of (dV/dx) = —’Ex, that is, V = —Ax + B. This means that the PE, —eV(x), of the electron is now eAx — eB, which increases linearly across the sample. All the energy levels and hence the energy bands must therefore tilt up in the x direction, as shown in Figure 5.6, in the presence of an applied field.

Under the action of “Ex, the electron in the CB moves to the left and immediately starts gaining energy from the field. When the electron collides with a thermal vibra­tion of a Si atom, it loses some of this energy and thus “falls” down in energy in the CB. After the collision, the electron starts to accelerate again, until the next collision, and so on. We recognize this process as the drift of the electron in an applied field, as illustrated in Figure 5.6. The drift velocity v* of the electron is ixeT, x where ixe is the drift mobility of the electron. In a similar fashion, the holes in the VB also drift in an applied field, but here the drift is along the field. Notice that when a hole gains energy, it moves “down” in the VB because the potential energy of the hole is of opposite sign to that of the electron.

Electrostatic PE(x)

Electrons and Holes

Electrons and Holes

(b)

подпись: (b)

Figure 5.6 When an electric field is applied, electrons in the CB and holes in the VB can drift and contribute to the conductivity.

A simplified illustration of drift in Јx.

Applied field bends the energy bands since the electrostatic PE of the electron is

-eV(x) and V(x) decreases in the direction of ЈX/ whereas PE increases.

подпись: figure 5.6 when an electric field is applied, electrons in the cb and holes in the vb can drift and contribute to the conductivity.
a simplified illustration of drift in јx.
applied field bends the energy bands since the electrostatic pe of the electron is
-ev(x) and v(x) decreases in the direction of јx/ whereas pe increases.
»

W

I

W

ЈP

D

C

<D

‘o

(a)

подпись: (a)X

Since both electrons and holes contribute to electrical conduction, we may write the current density 7, from its definition, as

J = envde + epvdh

Where n is the electron concentration in the CB, p is the hole concentration in the VB, and Vde and Vdh are the drift velocities of electrons and holes in response to an applied electric field *EX> Thus,

[5.1]

Where iie and fXh are the electron and hole drift mobilities. In Chapter 2 we derived the drift mobility fie of the electrons in a conductor as

Vde — fte’E’x

Lie =

Exf

Where re is the mean free time between scattering events and me is the electronic mass. The ideas on electron motion in metals can also be applied to the electron motion in the CB of a semiconductor to rederive Equation 5.3. We must, however, use an effective mass m* for the electron in the crystal rather than the mass me in free space. A “free” electron in a crystal is not entirely free because as it moves it interacts with the potential energy (PE) of the ions in the solid and therefore experiences various internal forces. The effective mass m * accounts for these internal forces in such a way that we can relate the acceleration a of the electron in the CB to an external force Fext (e. g., —e*Ex) by Fcxt = m*a just as we do for the electron in vacuum by Fex, = mea. In applying the

M,

And

[5.2]

Vdh = HhZ;

[5.3]

Electrons and Holes

Electron and hole drift velocities

 

Electrons and Holes

Fex t = m* a type of description to the motion of the electron, we are assuming, of course, that the effective mass of the electron can be calculated or measured experimentally. It is important to remark that the true behavior is governed by the solution of the Schrцdinger equation in a periodic lattice (crystal) from which it can be shown that we can indeed describe the inertial resistance of the electron to acceleration in terms of an effective mass m*. The effective mass depends on the interaction of the electron with its environment within the crystal.

We can now speculate on whether the hole can also have a mass. As long as we view mass as resistance to acceleration, that is, inertia, there is no reason why the hole should not have a mass. Accelerating the hole means accelerating electrons tunneling from bond to bond in the opposite direction. Therefore it is apparent that the hole will have a nonzero finite inertial mass because otherwise the smallest external force will impart an infinite acceleration to it. If we represent the effective mass of the hole in the VB by mЈ, then the hole drift mobility will be

[5.4]

подпись: [5.4]Exh

№h — * ml

Where rh is the mean free time between scattering events for holes.

Conductivity of a

Semiconductor

подпись: conductivity of a
semiconductor

[5.5]

подпись: [5.5]Taking Equation 5.1 for the current density further, we can write the conductivity of a semiconductor as

A = en/ie + epfih

Where n andp are the electron and hole concentrations in the CB and VB, respectively. This is a general equation valid for all semiconductors.

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