Marder M.P. Condensed Matter Physics

Подождите немного. Документ загружается.

Excitons

643

than a lattice spacing. In this limit, the exciton is called a Mott-Wannier exci-

ton.

The same arguments used in Section 18.3.1 to discuss interaction of electrons

with weak impurity potentials apply again. Assume that the characteristic distance

between electron and hole is much larger than a lattice spacing, so that the wave

functions for both electron and hole can be constructed as products of Bloch waves

with wave vector k near 0. Then the interactions of electron and hole with the pe-

riodic ionic potential can be eliminated from the Hamiltonian, provided that their

masses are replaced by the appropriate effective masses. That is, the electron and

hole obey

-h

-rt^i

V- + V-

2m*

" 2m*

e°\7n-r

-£

Wn,

(21.24)

Equation (21.24) is exactly the equation obeyed by the electron and proton in

forming the hydrogen atom. It can be separated into two equations, one describing

the center-of-mass motion of the exciton and the other describing the binding by

defining

m*r„ +

r = r,

to give

2(m*

+ m*)

-Vl-

*cm(fl)

-w-

2ß

e°r

*b(?),

cm= center of mass.

b=binding.

with the reduced mass /i given by

m*m*

+ m*

(21.25)

(21.26)

(21.27)

(21.28)

(21.29)

There is no need to repeat the solution of the hydrogen atom any further. The bound

states of Eq. (21.28) have energies

\xe

e°

me°

and are characterized by a radius

* _ e°h

° ~ e

13.6 eV /=i...°c.

•

0.529 Â.

(21.30)

(21.31)

Because \x is typically half an electron mass or less, while e for a semiconductor

may be around 10, the scale on which the bound state lives is 20 times larger than

the usual hydrogen atom; therefore, the picture of the electron and hole sitting in a

classical background dielectric is consistent. Binding energies are on the order of a

644 Chapter 21. Optical Properties of Semiconductors

2.12 2.13 2.14 2.15 2.16

Energy (eV)

Figure 21.8. Absorption measured in

CU2O.

Cuprous oxide is an indirect gap semiconduc-

tor, and absorption begins to increase as a square root, at an energy of

2.045

eV.

This square

root form is predicted by Elliott (1957) for phonon-assisted transitions to the lowest-lying

exciton state; direct transitions are forbidden by symmetry. At about 0.1 eV higher in en-

ergy a series of peaks corresponding to electron transitions between the top of the valence

band and excited exciton states begins. The first (/ = 2), second, and third excited states

are clearly visible. [Experiments performed at 77 K by Baumeister (1961), p 361.]

hundredth of a Rydberg, or 0.1 eV. The dispersion relations of excitons have been

measured with great accuracy, as reported by Ueta et al. (1986).

According to Eq. (21.30), the presence of excitons produces a sequence of

discrete energy states sitting near the conduction band edge. Usually these peaks

are not resolved individually, but merge into a bump centered around hu = £,

However, at low temperatures in CU2O, a sequence of distinct absorption peaks can

be measured, as shown in Figure 21.8. In these measurements, an electron makes

a transition from the valence band to various excited states of the excitons. In

addition, just like the hydrogen atom, Eq. (21.28) must have a continuous collection

of unbound solutions, which enhance the absorption for

Hco

> E

. Why are such

effects observed in GaAs (Figure 21.5) but not InSb (Figure 21.4)? The static

dielectric constant of InSb, recorded in Table 22.2, is 17.88, and the effective mass

is 0.01m, reducing the energy scale of the excitons to 3

■

10~

eV and making

them invisible even at temperatures as low as 5 K.

21.4.2 Frenkel Excitons

The Frenkel exciton consists of a bound state of electrons and holes in insulators,

where the spatial extent of the exciton is on the scale of angstroms, and the con-

tinuum approximation that served the Mott-Wannier exciton fails. These excitons

Optoelectronics 645

have excited

states,

just like more conventional molecules, and transitions between

these excited states allow optical absorption in insulators. In addition, these exci-

tons are mobile and can travel through the lattice as particles. A calculation of their

properties, using the Hartree-Fock approximation, appears as Problems 4 and 5.

21.4.3 Electron-Hole Liquid

When a sufficiently large density of excitons is created in a semiconductor, by

strong illumination, the excitons can condense into a liquid-like state that manifests

itself as an electron-hole drop, first predicted by Keldysh and Kozlov (1967) and

reviewed by Hensel et al. (1977) and Rice (1977).

21.5 Optoelectronics

Semiconductor electronics can be made to interact with light in numerous different

ways,

ranging from the conversion of photons into electrical energy to the con-

version of electrical energy to laser beams. These various applications are largely

implicit in the Kubo formula, Eq. (20.70), but great ingenuity continues to be ex-

pended in obtaining efficient performance.

21.5.1 Solar Cells

A solar cell is essentially a diode that has been fabricated so as to absorb as much

light as possible in the neighborhood of

the

junction.

Every photon that is absorbed

generates an electron-hole pair. The pairs that are generated inside the depletion

region are immediately ripped apart by the enormous electric fields and sent to

opposite sides. Because the depletion region is very narrow, the formation of pairs

in the quasi-neutral regions is more important. Minority carriers formed within a

distance L

or L

of the depletion zone can diffuse there and are then swept across

to the other side of the junction. This picture predicts that shining light of intensity

/ on a diode will produce a current density

j = — 75 (L„ +L„) S is some constant of proportionality taking (21.32)

into account the efficiency of absorption, and

making the units come out right.

Because the sizes of the quasi-neutral regions are independent of the voltage

across the diode, the main effect of light is simply to lower the current-voltage

curve of the diode downwards, as shown in Figure 21.9. When the solar cell is

under illumination but not connected to a circuit, the separation of electron-hole

pairs causes the voltage across it to rise until the current-voltage curve intersects

the point where current vanishes, at a voltage on the order of kßT/e, or around

0.025 V. Connected to a load, the voltage drops slightly and current begins to flow,

generating power. There is an ideal load at which power generation is maximum,

while if the cell is connected across an open circuit the voltage across it drops to

zero,

and it again generates no power. Solar cells can convert sunlight to power

646 Chapter

21.

Optical Properties of Semiconductors

Figure 21.9. The current-voltage characteristic for a solar cell is essentially the ideal

diode equation (19.78) shifted vertically downwards. In the lower right quadrant the diode

is forward biased, but current flows in the opposite direction, generating power.

with an efficiency of over 25%, but are not yet an economical source of energy

under most conditions.

21.5.2 Lasers

The laser is based upon the idea of stimulated emission, first conceived by Ein-

stein (1916) and first measured by Ladenburg (1928). Three decades followed the

first experimental observations before a coherent beam of radiation was actually

produced, and several more decades followed before this "solution in search of a

problem" became important commercially. Possible applications of the laser were,

however, evident right from the outset.

Hecht (1992) describes the complicated history of the race to complete the

laser. G. Gould eventually won extensive patent rights to the laser based upon per-

sonal notes written in 1957, but it was an influential paper of Schawlow and Townes

(1958) that started the international scramble to build a working device, won by

Maiman (1960). The basic idea is fairly simple. If an excited electronic state that

should be nearly empty in thermal equilibrium is somehow given a macroscopic

occupation of electrons, then as soon as the electrons begin to travel to the ground

state the process can proceed explosively. The light emitted by electrons as they

jump down the energy levels triggers other electrons

jump down simultaneously,

and light emission grows exponentially. To see why, go back to Einstein's original

argument.

Stimulated and Spontaneous Emission. Einstein (1916) deduced stimulated

emission from deceptively simple arguments about thermal equilibrium. Consider

an isolated cavity filled with material whose index of refraction is n, and focus

upon two electronic states

and 2, of energies £i and £i +

£12-

Einstein first ob-

Optoelectronics 647

served that all systems prepared in an excited state decay at some rate, and he took

the spontaneous

transition rate

from state 2 to

to be

f (\ f \ ^

transition rate >

proportional to the proba-

_ __

«sp

A2lJ2\

—

J\)

bility that state 2 be occupied and that state 1 be

{21.33)

empty,

and

are Fermi factors for the respec-

tive states.

Next suppose the cavity to be so well isolated that the material inside is in

complete equilibrium with a bath of photons at temperature T. Whenever a photon

of the right frequency collides with an electron in state 1, it has the chance to be

absorbed and bring the electron to state 2. The rate Rn at which electrons make

transitions from the lower to the upper state is proportional to the probability

that the lower state is occupied, to the probability (1

—

fi)

that the upper state is

unoccupied, and to the number of photons Nß

of energy £12 in the cavity.

noted in Eq. (20.72), transitions between electronic states are never completely

restricted to a single energy, but can be produced by photons in some small range

d£. Accordingly, the number of photons producing the transition is proportional

to A

Dph(£i2), where D

h is the energy density of photon states. Because the

density of photon states will depend upon the index of refraction of the medium,

complicated electron-electron interactions are subtly being included with almost

no increase in the complexity of the theory. Thus

= B\2f\(l

—

fl)N^

D„\

{^\2)

is a

temperature-independent coefficient (21.34)

to be determined later.

The reverse rate,

R21,

at which an electron jumps down from an excited state,

producing photons, must at least be as large as R

. Einstein guessed that the rate

would also rise in proportion to the number of photons in the cavity. Choosing

coefficient

#21

defined to correspond closely to the coefficient in Eq. (21.34) gives

Ä21=ß2l/2(l-/l)^e

Dph(£l2)+A2l/2(l-/l). (21.35)

The transition rate is the sum of the spontaneous transition rate, independent of

the photons, and a term proportional to the photon number.

The contribution proportional to Afe

is the stimulated emission

rate.

Why photons

should cause such transitions is not immediately clear, but considering how thermal

equilibrium is maintained shows that these transitions must be present.

In thermal equilibrium the population of electrons is stationary, and Rn and

/?2i must be equal. Because

ri-, r \

Just put in Fermi functions to verify. Equivalent

j1\\ J\)

— ßE\2 to the statement that the probability of occupying

,«, ~s\

1-1

r \ ^ >

any quantum state in equilibrium is proportional \L\..5\})

/l(,l-j2) toe-'

9£

one has

*12 =

*21

(21.37)

^ßi2^V

£l2

h(£i2)

/3£l2

[ß2iAr

£|2

(£

)+A

] (21.38)

648

Chapter

21.

Optical Properties of Semiconductors

(£

)ß

-A

/3£

(£

)fi

,-A

i] "

£l2

= (21.39)

l/(exp[/3£i

]- 1)

(£i

)ß

i and 5

= (2|,40)

If Eq. (21.39) is to hold independent of temperature.

Equation (21.40) can also be written as

Ä21=fi2l/2(l-/l)(Afe

+ l)0ph(£l2). (21-41)

In this form, change in electron energies by emission and absorption of photons

reveals a perfect parallel with the change in neutron energies by emission and

absorption of phonons described in Eq. (13.134). The probability of transitions

where electron energy increases is proportional to Nß

, while the probability of

transitions where it decreases is proportional to N^

+ 1, with the trailing factor of

1 responsible for spontaneous emission.

Einstein's argument is completed by asserting that in nonequilibrium situa-

tions,

as when a stream of light energy hits electrons and is ultimately dispersed as

heat, the occupation probabilities f\ and /

need not have the values used to obtain

Eq. (21.36), but the transition rates R\

and R

\ will continue to depend upon f\

and f

in the ways indicated by Eqs. (21.34) and (21.41). Thus, away from equi-

librium, the absorption of energy by electrons from photons will be proportional

Rn-Ri\ =Ä2i [(/i -H)

El2

(1 -/i)]Z)

(£

). (21.42)

The absorption of energy is given by the difference between absorption and

emission, as given by Eqs. (21.34) and (21.41).

The coefficient B\

is the only remaining unknown of the theory, and it can

be determined by comparison with the Kubo-Greenwood formula. When the

photon density A^£

is very large, spontaneous emission becomes negligible, and

Eq. (21.42) must coincide with the results of the semiclassical theory of absorp-

tion. Therefore, turn to Eq. (20.71), which predicts that if one is considering only

the the two quantum states

and 2, then the conductivity is

Re[a

aß

(uj)} = ^^-(/i -f2)F

(cü)(l\P

\2)(2\P

\l). (21.43)

F\2(ÜJ)

is the lineshape, a function with unit area, and peaked at

—

cj|

■

In practical situations, there will not simply be two levels

and 2, but instead a

large number N of atoms or impurities that can be excited from one level to another.

Summing over these many atoms will have the effect of averaging over all possible

spatial orientations of the excited state |2) and replacing the matrix elements by

their spatial average Y,ß lOI^I

- Using Eqs. (20.14) and (20.21) to relate

conductivity to the attenuation constant a = — g shows that radiation intensity /

grows as exp[gx] where the gain is

*M = y— [te) *i2M(/2-/i)

3m2c2

■ (21.44)

Optoelectronics

649

The net rate

R21 —R12

at which electrons travel between states 2 and

equals

the rate at which photons are produced per time. To find the total number of pho-

tons,

multiply g in Eq. (21.44) by c/n to find the time rate of change of light in-

tensity /. Next recall that light intensity is proportional to photon number through

/ = N£

hui, and finally that in a medium with macroscopic index of refraction n,

the density of photon states per volume is

-3 2

n ui

h(uj)

= . This is density of states per volume; a fac- (21.45)

7T^C^

tor of V is needed to find total numbers of

photons.

Multiplying Eq. (21.44) by Eq. (21.45), multiplying by Vc/n, and integrating over

gives

«,,-«„

= !"«„ =

-^ g)

4(/,

-A)

£ KM

W£ii(21

.46)

N is the number of host atoms or impurity sites, while N^

is the number of

photons. Bi2 can be determined from Eqs. (21.42) and (21.45) if desired, but

this rate equation for photons is more useful than B\i. The lineshape F\2(ui)

integrates to 1.

Equation (21.44) contains a theoretical motivation for building a laser. If it is

possible to construct an initial state in which fa > fa, then the attenuation will be

negative, and light will grow exponentially as it passes through the medium until

state 2 is depleted. Under these circumstances, — a = g as computed in Eq. (21.44)

is renamed the gain, and the symbol a is reserved for sources of attenuation and

loss that compete with the growth from population inversion.

In order to obtain a laser, it is not enough for the intensity of a beam of light

passed into the system to grow. It must be possible for external light sources to be

turned off, while light emission from the system continues to grow spontaneously.

This condition can be achieved by placing material in a cavity with reflecting ends,

as shown in Figure 21.10. If

the

reflection coefficient of the two ends is % then the

condition for amplification in the laser cavity is

Ji exp[x(e — C()] > 1. Light has to be able to make a round trip in (21.47)

the cavity and end up stronger than at the be-

ginning. The gain g due to stimulated emis-

sion competes with loss processes summarized

in the coefficient a.

Pumping. The discussion has hinged upon the assumption of population inver-

sion:

that an excited state has been populated so that fa > fa. Occupying an excited

state preferentially at the expense of one lying lower in energy is easy to describe,

but violates conditions of thermal equilibrium, so it is not immediately obvious

how to achieve it in practice.

Lasers can be created in numerous different ways and in many different types

of materials, including liquids and gases. The following discussion will focus upon

solid-state

lasers,

which does not refer exclusively to lasers formed of semiconduc-

tors,

but includes any case in which the laser material is solid. The most impor-

tant ideas are metastable states and pumping. For the purposes of laser physics,

650 Chapter

21.

Optical Properties of Semiconductors

Figure 21.10. Light in a laser cavity reflects several times back and forth from mirrored

ends of reflection coefficient

so as to stimulate more light emission before exiting.

a metastable state is an excited energy level with a lifetime on the order of 10~

to 10~

s. Characteristic non-radiative decay times of excited states in solids are

usually on the order 10~

to 10~

s, because this is the time that phonons typi-

cally require in order to transfer energy from an excited level. A first requirement

of a solid-state laser is therefore to find an excited state that cannot be depleted by

single phonons, either because available transitions exceed the maximum phonon

frequency or because symmetry causes matrix elements [analogous to the ones

in Eqs. (13.99) or (20.70)] between initial and final states to vanish. Transitions

driven by multiple phonons are still possible, but sufficiently unlikely that emis-

sion of photons has a chance to win.

A metastable state has a long lifetime, and by the energy uncertainty principle

it can only be excited from the ground state by energy lying in a very narrow

range. Because only a laser can produce light with such a narrow linewidth, it

would seem that a laser is required to populate the levels upon which laser action

is to rely. This difficulty is overcome by pumping, illustrated in Figure

21.11.

Rather than occupying the metastable state directly, one pushes electrons into one

or more states with short lifetimes that lie above the desired metastable state, and

that rapidly decay to it in non-radiative fashion.

Solid-State Ionic Lasers. One way to make a laser is by employing the discrete

energy levels provided by color centers in insulating crystals (Section 22.4). The

first laser, built from a crystal of ruby by Maiman (1960), was a three-state laser in-

volving pump states and a metastable state decaying to the ground state, as shown

in Figure 21.12(A). In order to occupy the

E state upon which the laser relied, the

crystal was illuminated with a powerful flashlamp that succeeded in placing more

electrons into

E than were left in the ground state. Numerous researchers real-

ized that a four-state laser would have considerable advantages because if the state

at the bottom of the laser transition was normally unoccupied, constraints upon

the occupation of the upper level become less severe. Three-state lasers typically

operate in pulsed fashion, with bursts of laser radiation following flashes of pump-

ing light, while four-state lasers can operate continuously. The energy levels of

Nd in yttrium aluminum garnet (YAG), the impurity in one of the most important

solid-state lasers appear in Figure 21.12(B).

Optoelectronics

651

Figure

21.11.

(A) The problem of occupying a metastable atomic state is a bit like pump-

ing water from a trough into a funnel. The pumping action can be sloppy and erratic, but

the water nevertheless exits the funnel in a narrow steady beam. (B) For a semiconductor

laser, the pumping scheme is a bit different. As electrons travel along, they pass into a

region where the doping has changed, and the Fermi level drops precipitously from con-

duction to valence band, leading to a laser transition somewhat like water falling through

a drain. The double heterostructure laser ensures that the optical transitions take place in

a confined spatial region by trapping the electrons with an additional slight drop in Fermi

level of the conduction band that resembles a basin.

Figure 21.12. (A) Energy levels of Cr

+ in AI2O3 (ruby). Electrons are pumped to high

levels,

and they quickly decay to metastable state

E. The laser transition then occurs to

(B) Energy levels of Nd in Y

(Nd:YAG). The metastable state is

3/2

and the

laser transition occurs to state

/f,

Transitions to other states also occur, but the transition

/n/2 is the most likely, at 60% probability. Energy levels are being described in LS

coupling, as described by Schiff (1968), p. 435, or Landau and Lifshitz (1977), p. 250,

where the superscript gives

+ 1, the subscript gives J, and the capital letter indicates L.

652 Chapter

21.

Optical Properties of Semiconductors

Figure 21.13. A double heterojunction structure makes use of two junctions to trap elec-

trons and force them into radiative transitions

in a

small region. Electrons arriving from

the n-doped region on the left find themselves trapped in

p-type

region and hemmed in

on both sides. The central layer also has a higher index of refraction than the surrounding

regions, confining light in a cavity.

Light-Emitting Diodes

and

Semiconductor Lasers. The light-emitting diode

and semiconductor laser exploit the fact that electrons

«-type material traveling

in the conduction band must rapidly fall

the valence band after entering

p-type

material. In the process, they emit photons whose energy equals the band gap. The

light-emitting diode simply extracts this radiation. To obtain

laser,

is necessary

to obtain

high spatial density of recombining electrons and holes, and to trap the

light they emit

in a

cavity. Practical devices are based upon the double hetero-

junction structure shown in Figure

21.13.

The central region acts like

trap where

electrons collect in large numbers and recombine with high efficiency.

If an indirect gap semiconductor such as silicon were used

this application,

electrons would collect

the lowest point

the conduction band but then find

themselves unable

make the transition

the valence band without assistance

from phonons. The efficiency

the device would be unacceptable. That

why

direct gap semiconductors such as GaAs are needed for optical applications.

Problems

Optical measurements:

(a) Use the locations

the peaks for light and heavy holes to estimate the fre-

quency to at which the experiment in Figure 21.3 was conducted.

(b) The data

Figure 21.6 allow one to estimate uj

h(ök). Using the data

K, what are o;

h and

Ski

Lifetime of hydrogen excited state: Many

the quantities appearing in the

theory of lasers can be evaluated explicitly for hydrogen. Consider transitions

between the 2P and

states, using the wave functions

eXp[

ao]

and

eXP

f (21.48)

2V-

"Kür,