SUMMARY

transparent conducting material arranged in a certain pattern. When a voltage is applied between the plates, the molecular alignment is altered and the polarization of incident light is no longer rotated by 908. By using orthogonally oriented polarizing sheets in front of and behind the cell, and a mirror at the back, we can arrange for incident light to be reflected when there is no applied voltage, but for no light to be reflected from those areas where there is an applied voltage.

Then we see the familiar black-and-white alphanumeric display patterns.

Liquid-crystal displays used in flat-screen televisions and laptop computers, for instance, are transmissive, or “backlit,” rather than reflective, but the principle of operation is the same. The backlighting is done with very small fluorescent tubes or LEDs, together with a white panel behind the LCD that scatters light from the tubes to produce a uniform illumination of the

LCD. The light from the screen is strongly polarized. †

The light-emitting material in organic light-emitting diodes (OLEDs) consists of large organic molecules or polymers in a very thin layer, typically only a few hundred nanometers thick. The emitting layer is sandwiched between a cathode array and an anode array, with additional conductive layers serving to facilitate the injection of electrons and holes into the emitting layer. The color and brightness of the light pro- duced by electron – hole recombination in the emitting layer depend on the type of organic molecules used and on the strength of the applied current. The electrode layers are anode and cathode strips, and the emitting pixels (picture elements) are at the intersections of these strips. The application of different current levels to different pixels determines which pixels are on or off for display or video. The emitting layers for OLEDs can be produced in large and flexible sheets, suggesting applications such as foldable electronic “newspapers” that can be updated minute by minute.

Transistors, consisting basically of two adjacentpnjunctions (pnpornpn), are the most important application of semiconductor junctions, and their operation may be understood within the electron – hole framework we have used to discuss LEDs.

In much of laser physics it is sufficient to regard any atom as simply a “black box” for electrons, with the special property that the electrons inside can only be in certain energy slots. We can adopt a similar view for the electronic structure of molecules.

Of course, we cannot ignore the fact that molecules can also vibrate and rotate. However, the most important vibrational and rotational characteristics of molecules are, for us, very similar to their electronic characteristics. First and foremost among these similarities, of course, is the restriction of the vibrational and rotational energies to a fixed set of allowed values. Just as an electron can jump to a higher (or lower) energy level with the absorption (or emission) of a photon, so too can the molecule as a whole “jump” to a different vibrational or rotational state with the simultaneous absorption (or emission) of a photon. In fact, the electronic, vibrational, and rotational states of a molecule canallchange as the molecule absorbs or emits a photon. Molecular spectra are more complicated than atomic spectra, but this simply means that the black box we call a molecule is more complicated on the inside than an atom.

The properties of solids are determined to a large extent by the outermost occupied electron orbitals of its constituent atoms or molecules. In crystalline solids the allowed electron energies are spread into energy bands as a consequence of the tight packing and periodic arrangement of atoms in a crystal lattice. The concept of energy bands provides a satisfactory interpretation of insulators, conductors, and semiconductors.

Semiconductor junctions are an especially interesting and important application of the quantum mechanics (band theory) of solids. In particular, the concept of a missing electron, or hole, as a sort of particle in its own right, greatly facilitates our understanding of semiconductor junctions. The basicpnjunction acts as a diode, passing a current when it is forward biased but not when it is reverse biased. Light-emitting diodes are important not only in lighting and alphanumeric displays but also as the gain media of diode lasers.

The existence of atomic and subatomic particles as the basic building blocks of matter in all its forms is arguably the most basic and significant discovery of post-Newtonian science. A strong argument can also be made that the most far-reaching technological developments since the mid-20th century have involved the controlled manipulation of quantum states of these particles. Among these developments are nuclear power sources and transistor-based computer technology. The laser is another example. In this case populations of excited atomic and molecular states are created and controlled to generate light.

APPENDIX: ENERGY BANDS IN SOLIDS

In Section 2.6 we used the quantum mechanical result that the allowed electron energies in crystalline solids occur in bands, with forbidden energy gaps between these bands.

We will now use a simple one-dimensional model of a solid to show how this band structure arises in quantum mechanics. This will also serve as an example of a full solution to the one-dimensional Schro¨dinger equation

d²c dx²þ2m

h²[EV(x)]c¼0: (2:A:1)

For the case of an electron in a periodic potential,

V(xþd)¼V(x), (2:A:2)

where the distance d is the lattice spacing in our one-dimensional model. Note that (2.A.2) implies that V(xþnd)¼V(x), wheren is any integer, so in our model the solid is infinitely long.

IfV(x) were identically zero, the reader could easily show by substitution that the solution of (2.A.1) is the free-particle plane-wave

c(x)¼ue^ikx, (2:A:3) withusome constant (complex) amplitude andksuch that

E¼h²k²

2m : (2:A:4)

For a potential that is not identically zero, and that satisfies (2.A.2), it is natural to try to satisfy the Schro¨dinger equation (2.A.1) with a wave function of the form

c(x)¼u(x)e^ikx, (2:A:5a) withka real number andu(x) now not a constant, but a function with the periodicity of the potential:

u(xþd)¼u(x): (2:A:5b) Indeed, it may be shown that a solution of the Schro¨dinger equation, with a potential satisfying (2.A.2), must be of the form (2.A.5). This statement is Floquet’s theorem, and in solid-state physics it is called Bloch’s theorem. We will use it in our treatment of the one-dimensional solid.

Different models of a one-dimensional solid are characterized by different choices of the potentialV(x) satisfying (2.A.2), but the most important results are insensitive to the specific V(x) chosen. A particularly simple choice is the series of “square wells” shown in Fig. 2.31. This will serve as a crude idealization of the sort of potential encountered by an electron in a crystal lattice, each square well representing the effect of an atom at a lattice site. The lattice spacing in Fig. 2.31 isaþb, withathe width of each potential well.

V(x) V₀

–(a+b) –b 0 a a+b x

Figure 2.31 A model of the potentialV(x) encountered by an electron in a one-dimensional crystal lattice.

APPENDIX: ENERGY BANDS IN SOLIDS 57

We consider first the solution in the “unit cell” 0,x,aþb. For 0,x,a, V(x)¼0, and

c(x)¼Ae^iaxþBe^iax, (2:A:6)

whereAandBare constants and

a¼ 2mE h² ₁₌₂

: (2:A:7)

This solution is just a sum of free-particle, plane-wave solutions, one wave propagating to the right and the other to the left. This is the most general possible solution of (2.A.1) with V¼0. Similarly, for a,x,aþb, where V(x)¼V0, the general solution of (2.A.1) is⁸

c(x)¼Ce^bxþDe^bx, (2:A:8) whereCandDare constants and

b_¼ ^2m

h²(V0E)

₁₌₂

: (2:A:9)

According to Bloch’s theorem the wave function must have the form (2.A.5). We there- fore use the solutions (2.A.6) and (2.A.8) to identify the functionu¼ce^2ikx:

u(x)¼Ae^i(ak)xþBe^i(aþk)x, 0,x,a, (2:A:10a) u(x)¼Ce^(bik)xþDe^(bþik)x, a<x<aþb: (2:A:10b) The wave equation (2.A.1) is a second-order differential equation. As such it demands that bothcanddc_/dxbe continuous functions ofxbecause a function that is differentiable must be continuous. This means thatuanddu/dxmust also be continu- ous functions ofx. In particular, continuity ofuanddu/dxatx¼0 requires the follow- ing relations amongA,B,C, andDin (2.A.10):

AþB¼CþD, (2:A:11a)

i(ak)Ai(aþk)B¼(bik)C(bþik)D: (2:A:11b) Nowu(x) must, according to Bloch’s theorem, have the periodicity of the potential.

Thus,uanddu/dxmust have the same values atx¼aas atx¼2b. This condition of periodicity requires that

Ae^i(ak)aþBe^i(aþk)a¼Ce^(bik)bþDe^(bþik)b, (2:A:11c) i(ak)Ae^i(ak)ai(aþk)Be^i(aþk)a¼(bik)Ce^(bik)b(bþik)De^(bþik)b: (2:A:11d)

8Since we will be interested in the caseV₀.E, in whichbis a real number, we have written (2.A.8) in terms of real exponentials. IfV₀,E, thenbis purely imaginary and (2.A.8) is a sum of two plane waves, as in (2.A.6) for the caseV₀¼0.

The conditions (2.A.11) are four linear, homogeneous, algebraic equations for the four “unknowns”A,B,C,D. A trivial, uninteresting solution isA¼B¼C¼D¼0.

In order for a nontrivial solution to exist, the 44 determinant of the coefficients must vanish. After some algebra we find that this condition for a nontrivial solution takes the form

b²a²

2ab ^sinhbbsinaaþcoshbbcosaa¼cosk(aþb): (2:A:12) Sinceaandbare fixed in our model of a one-dimensional solid, this equation imposes a relation amonga,b, andk. As we now show, this relation gives rise to allowed energy bands separated by forbidden energy gaps.

It is convenient to consider a special case of (2.A.12) in whichV0is very large andbis very small (Fig. 2.32). Specifically, we takeV0!1andb!0 in such a way thatV0b remains a finite number. Sinceb²V0for largeV0, this limit is such thatb²bhas a finite limit asb!1andb!0. For convenience we denote this limiting value 2P/a, which is the same as defining

P¼ lim

b!1lim

b!0 1 2b²ab

: (2:A:13)

Sincebb¼(1/b)(b²b), it follows thatbb!0 in this limit. Thus,

b!1limlim

b!0coshbb¼lim

x!0coshx¼1, (2:A:14a) and similarly

b!1limlim

b!0

b²a²

2ab ^sinhbb¼ 1 aalim

b!1lim

b!0

b²ab 2

sinhbb bb ¼ 1

aalim

b!1lim

b!0

b²ab

2 ¼ P

aa, (2:A:14b) since

limx!0

sinhx

x ¼1: (2:A:15)

V(x)

–a 0 a 2a x

Figure 2.32 The Kronig – Penney model for the potential energyV(x) for an electron in a one- dimensional crystal lattice. This is the limit of the potential of Fig. 2.31 for the case in whichV₀is very large andbis very small, such thatV0bis a finite number. In this limit the lattice spacing isa.

APPENDIX: ENERGY BANDS IN SOLIDS 59

This limit ofb²_!1andb!0, such thatb²bstays finite, is called theKronig – Penney model. It is useful as a simplification of (2.A.12). With (2.A.14), the condition (2.A.12) in this limit reduces to

Psinaa

aa þcosaa¼coska: (2:A:16) IfPis very small, the first term on the left may be neglected, and (2.A.16) becomes cosaa¼coska, ora_¼k(except possibly for a trivial shift of 2p_/a). Using the defi- nition (2.A.7) ofa, we see thata_¼kgives the free-particleE2krelation (2.A.4).

If P is very large, on the other hand, then (2.A.16) can only make sense when (sinaa)/aais very small. In the limitP!1, then, we must have

aa¼np, n¼+1,+2,+3,. . .: (2:A:17) From the expression (2.A.7) fora, this condition is seen to restrict the electron energy to one of the values

E_n¼n²p²h²

2ma² , n¼1, 2, 3,. . .: (2:A:18) These allowed energies are those for an electron in a single, infinitely deep (V0!1) square well of widtha(Problem 2.7). They may be regarded heuristically as the allowed levels of an electron in an isolated “atom” in the present model.

From the discussion in Section 2.6 we expect these discrete energy levels to broaden into bands when the atoms are brought together to form a crystal lattice. In Fig. 2.33 we plot the left-hand side of Eq. (2.A.16) for a case in whichP¼1 has been arbitrarily chosen. Obviously, those values of aa for which this function exceeds unity do not allow (2.A.16) to be satisfied, since jcoskaj 1 for all (real) values of ka. Those values ofafor whichj(Psinaa)/aaþcosa_aj.1 define theforbiddenvalues ofE via the relation (2.A.4):

E¼h²a²

2m ¼p²h² 2ma²

aa p

2: (2:A:19)

2 sinaaaa + cosaa

1

0

0 4

a

8 12

–1

α

Figure 2.33 Plot of the left side of Eq. (2.A.16) for the Kronig – Penney model whenP¼1.0.

On the other hand, those values ofaafor whichjP(sina_a)/aaþcosaaj 1 permit (2.A.16) to be satisfied for real values ofk, as required by Bloch’s theorem, and the corresponding energies (2.A.19) are theallowedelectron energies.

In Fig. 2.34 we plot the allowed energies, in units ofp²h²=2ma², for the example P¼1 of Fig. 2.33. These are the results for a particularly simple one-dimensional model crystal. The theory for a three-dimensional crystal lattice leads similarly from the period- icity of the potential to a band structure for the allowed electron energies. This result is brought out by the one-dimensional model we have considered, and so we will not pursue a more complicated, albeit more realistic, three-dimensional model.

† Using certain physical assumptions, we can prove Bloch’s theorem as follows. First, the periodicity of the potential in (2.A.1) suggests immediately that the probability distribution jc(x)j²for the electron should also be periodic:

jc(xþd)j²¼ jc(x)j², (2:A:20) which means that

c(xþd)¼Cc(x), (2:A:21) with

jCj²¼1: (2:A:22) Note that (2.A.21) implies that

c(xþnd)¼Cⁿc(x): (2:A:23) Our one-dimensional crystal is assumed to be infinitely long; this is implicit in the assumption of the periodicity of the potential. The underlying assumption, of course, is that there are enough atoms in a real crystal to make the model of an infinite lattice a reasonable one. That is, “edge effects” in a real crystal are assumed to be very small. In this vein it is also reasonable to suppose

Allowed energies

Allowed energies Energy gaps

E

1 –1

Allowed energies

Figure 2.34 Allowed energies given by the Kronig – Penney model forP¼1.0. These energies are given by (2.A.19) for those values ofaafor which Eq. (2.A.19) allowsjcoskaj 1. The allowed energies appear in bands.

APPENDIX: ENERGY BANDS IN SOLIDS 61

there is some integerN, perhaps very large, such that

c(xþNd)¼C^Nc(x)¼c(x): (2:A:24) It can be assumed that the distanceNd, after which the wave function repeats itself according to (2.A.24), is large enough that the assumption (2.A.24) will not affect any physical predictions of the model. In other words, edge effects associated with the artificial periodic boundary condition (2.A.24) do not have any real physical consequences.

Equation (2.A.24) implies thatC^N¼1, which means thatCmust be one of theNth roots of unity:

C¼e^2piM=N, M¼0, 1, 2,. . .,N1: (2:A:25) It then follows from (2.A.21) thatc(x) must have the form

c(x)¼e^2piMx=Ndu(x)¼e^ikxu(x), (2:A:26a) withk¼2pM/Ndand

u(xþd)¼u(x): (2:A:26b) That is, Eqs. (2.A.21) and (2.A.25) are satisfied when c(x) has the form (2.A.26). Bloch’s theorem is easily extended to the case of a three-dimensional lattice. † It is instructive to plotEvs.k, as shown in Fig. 2.35 for the exampleP¼1. We also show for comparison the free-particle E2k relation (2.A.4). In the E2k curve the energy gaps occur at those values ofk for which the right-hand side of (2.A.16) is þ1, that is, for

k¼np

a, n¼+1,+2,+3,. . .: (2:A:27)

60

40

20

0 2

Energy gap

Energy gap

ka

4 6

( a)² = 2ma²E h² α

Figure 2.35 Plot ofEvs.kfor the Kronig – Penney model withP¼1.0.

This may be understood physically as follows. The wave function

C(x,t)¼c(x)e^iEt=h ¼u(x)e^i(kxEt=h), (2:A:28) for an electron propagating down the lattice has wavelengthl_¼2p_/jkjassociated with the plane-wave factor

e^i(kxvt)¼e^i(pxEt=h): (2:A:29)

If 2a¼nl, wheren¼1, 2, 3,. . ., the spacing between the potential barriers in Fig. 2.35 is an integral number of half wavelengths. This means that the waves reflected from these barriers are all in phase and interfere constructively. In other words, when

jkj ¼2p l ^¼

np

a , n¼1, 2, 3,. . ., (2:A:30) the wave (2.A.28) is strongly reflected and forbidden from propagating unhindered down the lattice. This is why the energies associated with thekvalues (2.A.30), or equivalently (2.A.27), are forbidden.

† In Section 2.7 we invoked the concept of aneffective mass,m, of an electron in a crystal lattice. To see how this concept arises, suppose that a forceFacts on the electron. The rate of change of the electron energy as a result of this force is

dE

dt ¼Fv, (2:A:31)

wherevis the electron velocity. Now the force equals the rate of change of the momentum p¼hk:

F¼dp dt ¼h dk

dt: (2:A:32)

Thus

dE dt ¼hvdk

dt, (2:A:33)

or

v¼1 h dE=dt

dk=dt ¼1 h dE

dk, (2:A:34)

which is the clear quantum analog of the well-known expression for the group velocity of a light pulse,v^g¼(dk/dv)²¹(Section 8.3). The acceleration of the electron is therefore

a¼dv dt¼1

h d dt

dE dk

¼1 h dk

dt d dk

dE dk ¼F

h²d²E

dk²: (2:A:35)

APPENDIX: ENERGY BANDS IN SOLIDS 63

This equation has the formF¼ma, which defines the effective mass as

m¼h² d²E dk² 1

: (2:A:36)

The physical basis for effective mass can be understood as follows. The total force acting on the electron is the external forceF_extplus the forceF_crysdue to the atoms of the crystal lattice. This total force equalsma. In our derivation above, however, the forceFis onlyFext;Fcrysis accounted for only indirectly via theE2krelation for the electron in the crystal. Thus,marises from the proportionality of the electron acceleration to theexternalforce. †

PROBLEMS

2.1. (a) Equation (2.2.18) withn⁰¼2 and n¼3, 4, 5,. . .gives theBalmer seriesof the hydrogen spectrum. In what region of the electromagnetic spectrum (e.g., infrared, visible, ultraviolet) are the wavelengths of the Balmer series?

(b) Equation (2.2.18) withn⁰¼1 andn¼2, 3, 4,. . .gives theLyman series of hydrogen. In what region of the spectrum are the wavelengths of the Lyman series?

(c) Equation (2.2.18) withn⁰¼3 andn¼4, 5, 6,. . .gives thePaschen seriesof hydrogen. In what region of the spectrum are these wavelengths?

2.2. Verify Eq. (2.3.6).

2.3. Given the fact that the molecular weight of water is 18, estimate the average distance between two water molecules in ice.

2.4. Show that the magnetic force acting on the charge carriers in the Hall-effect exper- iment of Fig. 2.20 is upward, regardless of whether the charges are positive or negative.

2.5. Assuming for GaAs a dielectric constante _¼13.0e0, and an effective massm¼ 0.07m, estimate the energy required to ionize donor impurities.

2.6. (a) Consider emission into air of an LED employing a pnjunction without the dome indicated in Fig. 2.29. Show that total internal reflection at the interface with air reduces the emission efficiency by the factor 12cosu_c, whereu_cis the critical angle for total internal reflection. (Note: The solid angle of a cone with apex angleuis 2p[12cos(u_/2)].)

(b) Show that the reduction factor calculated in part (a) is approximately equal to 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

11=n²

p , where n is the refractive index of the LED, and estimate this factor for GaAs. What is the expected increase in efficiency for a GaAs LED when it is designed with the plastic dome as in Fig. 2.29?

2.7. Consider a particle of massmin an infinitely deep, one-dimensional square well of widtha. Between the walls the particle is free (V¼0), but because it cannot penet- rate the walls the wave function must vanish atx¼0 andx¼aand for allxoutside those limits. Using the Schro¨dinger equation (2.A.1), show that the normalized

stationary-state wave functions are given by c_n(x)¼ 2

a

1=2

sin npx a

, n¼1, 2, 3,. . ., with corresponding allowed energies

En¼n²p²h² 2ma² : 2.8. The binding energy of the ion H2þ

(the energy required to separate to infinity the two protons and the electron) is216.3 eV at the equilibrium separation 0.106 nm.

(a) What is the contribution to the energy from the Coulomb repulsion of the nuclei?

(b) What is the contribution to the energy from the Coulomb attraction of the electron to the nuclei?

(c) The Hellman – Feynman theorem says, in effect, that the force between the nuclei in a molecule can be calculated from the electrostatic repulsion between the nuclei and the electrostatic attraction of the nuclei to the electron distribution. According to this theorem, where must the squared modulus of the electron wave function in H2þ

have its maximum value?

(d) Estimate the rotational constantBefor H2þ

, and compare your result with the value 29.8 cm²¹tabulated in Herzberg’sSpectra of Diatomic Molecules.⁴

PROBLEMS 65

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