Chen C.J. Physics of Solar Energy

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7.2 Many-Electron Systems  153

Figure 7.4 Experimental observation of HOMO and LUMO. Using a scanning tunneling

microscope, the energy levels and the probability density distributions of those states can be determined

by direct experimental measurements. (a) Tunneling current and tunneling conductivity curves. It

shows a peak at -2.4 eV below the vacuum level, corresponding to HOMO, and a peak at +1.5 eV

above the vacuum level, corresponding to LUMO. (b) Experimental observations of the charge density

contours of pentacene at -2.4 and +1.5 eV, corresponding to the HOMO and LUMO, respectively. The

theoretical results from numerical computations are also shown. Adapted from Ref. [18].

conductivity curves, where the peak at -2.4 eV below the Fermi level corresponding

to HOMO, and the peak at +1.5 eV above the Fermi level corresponding to LUMO.

Figure 7.4(b) shows experimental observations of the charge density contours of pen-

tacene at -2.4 and +1.5 eV, corresponding to the HOMO and LUMO, respectively. The

theoretical results from numerical computations are also shown.

7.2.4 Quantum States of a Nanocrystal

The scanning tunneling microscope is capable of not only mapping the electronic states,

but also constructing artiﬁcial atomic structures not found in nature. Figure 7.5 shows

the electronic states observed on a nanocrystal made of a linear chain of 15 copper atoms

with spacing 225 pm assembled by STM. Electron density distributions at various bias

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154  Quantum Transitions

Figure 7.5 Quantum states of a nanocrystal. (a) A nanocrystal made of 9 copper atoms

and the electron density distributions observed by STM at diﬀerent bias voltages. Each bias voltage

corresponds to the energy level of an electronic state of the nanocrystal. (b) Wavefunctions of the

electronic states, which give rise to the observed electron density distributions. Each one has a diﬀerent

wavevector. (c) Relation between the energy levels and the wavevectors of the electronic states, showing

a quadratic relation. Adapted from Ref. [18].

voltages are observed.

As shown, the wavefunction, exhibited as charge density distributions, as a function

of the energy level, controlled by the bias voltage, can be directly observed. As shown

in Section 7.1.3, the momemtum of the electron wavefunction is related to the number

of nodes in the charge density distribution. Experimentally, a quadratic dependence

on wavevector k is observed,

E = E

+const. ×k

. (7.43)

Similar to the theoretical argument in Eq. 7.27, an eﬀective mass ca be obtained by

Eq. 7.28,



∂

∂k

. (7.44)

7.3 The Golden Rule

In previous sections, the stationary states of atoms, molecules, and solid systems were

described. For the utilization of solar energy, an understanding of the interaction of

radiation with the quantum systems that cause transitions among various quantum

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7.3 The Golden Rule  155

states is of paramount importance. A quantum system can absorb a solar photon and

be excited from a state with a lower energy value to a state with a higher energy value

or decay from a state with a higher energy value to a state with a lower energy value

by emitting a photon or giving up the energy to the surrounding atoms.

A simple, sketchy theory was provided by Einstein in 1916, as shown in Section 2.5.

However, Einstein did not provide an explicit expression of those coeﬃcients for a

concrete system. That problem was resolved by Dirac in 1927 in a beautiful and com-

prehensive treatment of time-dependent quantum mechanical perturbation theory [22].

In that paper, a ﬁrst-order time-dependent perturbation theory provided an explicit

expression for the absorption and emission coeﬃcients of radiation, thus also the rate of

transition among quantum states. The resulting formula is so important that in 1950,

Enrico Fermi coined a term Golden Rule for it in his book Nuclear Physics [29].

Dirac’s 1927 paper deals with a number of issues concerning the interaction of

radiation with atomic systems [22]. It is profound and diﬃcult to read. Here, we

provide an easy-to-follow derivation of the Golden Rule. As we will see, the derivation

process provides a good understanding of the process of the interaction of radiation

with atomic systems.

The derivation process also provides the quantum-mechanical foundation of the

principle of detailed balance: a symmetry between absorption and emission.

7.3.1 Time-Dependent Perturbation by Periodic Disturbance

Consider a quantum system with a time-independent Hamiltonian

.Itcouldbe

a molecule, or a piece of semiconductor. In the absence of external disturbance, the

equation of motion of a state |Ψ is

i

∂

∂t

|Ψ =

|Ψ, (7.45)

where the state |Ψ is understood to be in general time dependent. Because the Hamil-

tonian

is time independent, the system has a series of stationary states, labeled by

an integer n,

|n = E

|n. (7.46)

A general solution of Eq. 7.45 can be written as

|Ψ =

∞



n=0

−iE

t/

|n, (7.47)

where c

are constant coeﬃcients. That solution can be veriﬁed by direct substitution

into Eq. 7.45.

Assuming that for t ≤ 0 the system is in an initial state with index i,

|Ψ = e

−iE

t/

|i,t=0, (7.48)

where the normalization condition implies c

=1.

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156  Quantum Transitions

The interaction of the system with radiation is described by a periodic disturbance

rad

(t)=

iωt

†

−iωt

, (7.49)

which is turned on for t>0. It is easy show that the disturbance

rad

(t) is Hermitian.

The ﬁrst term represents an outgoing wave, and the second term represents an incoming

wave. Consider the incoming term ﬁrst. The total Hamiltonian becomes

H =

†

−iωt

. (7.50)

The equation of motion becomes

i

∂

∂t

|Ψ =



†

−iωt



|Ψ. (7.51)

We look for a trial solution of Eq. 7.51 of the form

|Ψ = e

−iE

t/

|i +

∞



n=i

(t)e

−iE

t/

|n, (7.52)

where the coeﬃcients c

(t) are small but time dependent and are to be determined

by the equation of motion 7.51. At t = 0, the disturbance is not yet eﬀective, and

therefore, c

(0) = 0. Substituting Eq. 7.52 into Eq. 7.51, we have three types of

terms. Because the disturbance

†

is small, the ﬁrst nonvanishing terms with

†

are

∞



n=i

(t)

−iE

t/

|n =

†

|ie

−iE

t/−iωt

. (7.53)

Now we look for the probability of transition to a ﬁnal state, |f. Multiplying both

sides with a bra of that ﬁnal state, f|, using the orthogonality condition f |n = δ

one obtains

i

(t)

= f|

†

|ie

−i(E

−E

)t/−iωt

. (7.54)

The coeﬃcient c

(t) can be obtained by direct integration using the initial condition

(0) = 0,

(t)=f|

†

|i

−E

−ω)t/

− 1

− E

− ω

. (7.55)

The probability of ﬁnding a ﬁnal state |f at time t is

≡|c

(t)|

2π



t |f|

†

|i|



sin

[(E

− E

− ω)(t/2)]

π [E

− E

− ω]

(t/2)



. (7.56)

Denote E

− E

− ω = u and t/2 = a; the function in square brackets has a sharp

peak near u = 0, as shown in Fig. 7.6. The area under it is 1:



∞

−∞

sin

πau

du =1. (7.57)

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7.3 The Golden Rule  157

Figure 7.6 Condition of energy conservation. The integrand in Eq. 7.58 approaches a delta

function when a approaches inﬁnity. If the time of the experiment is not too short, the condition of

energy conservation is valid.

As a →∞, it approaches a delta function,

lim

a→∞

sin

πau

= δ(u). (7.58)

Therefore, when t →∞, the function in square brackets in Eq. 7.56 approaches a delta-

function δ(E

− E

− ω). Also, from Eq. 7.56, the probability of |f is proportional

to t, and therefore, the transition rate is

≡

2π



|f|

†

|i|

δ(E

− E

− ω). (7.59)

Equation 7.59 is the Golden Rule for atomic systems with discrete energy levels. The

Bohr frequency condition,

ω = E

− E

, (7.60)

comes naturally. The radiation ﬁeld can only transfer an energy quantum of ω to the

atomic system, which is the essence of Einstein’s theory of photons and provides an

explanation of the line spectra, especially the Ritz combination principle.

7.3.2 Golden Rule for Continuous Spectrum

For systems with a continuous energy spectrum, the parameter f becomes a continuous

variable. The number of states ΔN in an energy interval ΔE

is given by the density of

states of the ﬁnal state, ΔN = ρ(E

)ΔE

. The transition rate R

should be integrated

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158  Quantum Transitions

over a range of states,

2π



lim

t→∞



∞

−∞

|f|

†

|i|



sin

[(E

− E

− ω)(t/2)]

π [E

− E

− ω]

(t/2)



ρ(E

) dE

2π





∞

−∞

|f|

†

|i|

δ(E

− E

− ω) ρ(E

)dE

2π



|f|

†

|i|

ρ(E

+ ω).

(7.61)

Similarly, the Bohr frequency condition E

= E

+ω comes naturally. Equation 7.61 is

the Golden Rule for the continuous energy spectrum for the case of electronic excitation

of an atomic system.

7.3.3 Principle of Detailed Balance

The above treatment is for the absorption of a photon from the radiation ﬁeld. By

changing the sign of exponential in Eq. 7.50,

H =

+ λ

iωt

, (7.62)

and repeating the treatment, Eq. 7.59 becomes

2π



|f|

V |i|

δ(E

− E

+ ω), (7.63)

where the Bohr frequency condition becomes

ω = E

− E

, (7.64)

which means that the atomic system decays from a state of higher energy to a state of

lower energy by emitting a photon to the radiation ﬁeld. For a continuous spectrum,

the corresponding Golden Rule is

2π



|f|

V |i|

ρ(E

− ω). (7.65)

By exchanging the initial state and the ﬁnal state, it is clear that the matrix element

is identical in both absorption and emission processes,

|i|

V |f |

= |i|

†

|f|

. (7.66)

The symmetry of absorption and emission is the foundation of the principle of detailed

balance, which plays a crucial role in the understanding of the eﬃciency limit of solar

cells.

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7.4 Interactions with Photons  159

7.4 Interactions with Photons

The interaction of an atomic system with photons can be treated as a special case of

the Golden Rule by using an explicit form of the interaction term

V . In Section 2.1.6,

the classical Hamiltonian of an electron in electric and magnetic ﬁelds is Eq. 2.32,

H =

(p − qA)

+ qφ, (7.67)

where A is the vector potential of the electromagnetic ﬁeld and φ is the electric (scalar)

potential. The Hamiltonian in quantum mechanics has exactly the same form, but the

dynamic variables become operators,

H =

(

p − qA)

+ qφ. (7.68)

To treat the problems of the interaction between the solar radiation ﬁeld and atomic

systems, two simpliﬁcations can be made: First, the spatial variation of the radiation

ﬁeld, represented by the vector potential A, is usually over 1 μm, much slower than

the variation of the wavefunctions, which is less than 1 nm. Thus it can be treated as

a constant over coordinates and consequently commutes with p. Second, the square

of the vector potential, A

, is much smaller than the other terms and thus can be

neglected. Equation 7.68 becomes

H = −



∇

+ qφ −

(A • ˆp). (7.69)

Here the ﬁrst two terms represent the Hamiltonian for the atomic system,

,andthe

last term is the perturbation potential

V .

A radiation ﬁeld, a stream of sunlight, can be well represented by a plane wave.

Take z as the direction of travel. The disturbance term in the total Hamiltonian is

V = −

•

p e

i(k

z −ωt)

. (7.70)

Equation 7.70 can be written in a more convenience form. Using Eqs. 2.33 and 2.8

and recalling that the term eA is a small perturbation, the spatial variation (on the

order of 1 μm) is much slower than the variation of the atomic wavefunction, thus can

be considered as a constant, and the temporal variation of the disturbance potential

follows the e

−iωt

factor,

V = −iωqA

•

r e

−iωt

= E

• qr e

−iωt

(7.71)

which has an intuitive explanation. Because the spatial and temporal variations of

the electric ﬁeld of the radiation are much slower than that of the atomic states, the

radiation ﬁeld can be treated as uniform and static: The interaction of the radiation

ﬁeld is the electric ﬁeld intensity of the light, E, acting on the dipole moment of the

atomic system, qr. Hence, it is often called the dipole approximation.

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160  Quantum Transitions

Problems

7.1. Using the deﬁnition of Hermitian operators (Eq. 7.8), show that for an arbitrary

matrix

A, both

†

A and i(

†

−

A) are Hermitian.

7.2. A running wave in the x-direction with wavevector k

can be represented by a

function

ψ(x)=Ce

x−ωt

. (7.72)

Using de Broglie’s relation

= k

, (7.73)

prove that the momentum operator is

ˆp

ψ(x)=−i

∂

∂x

ψ(x). (7.74)

7.3. Using the coordinate representation of the state in one dimension, the wavefunc-

tion |ψ = ψ(x), and the deﬁnition of momentum

ˆp

|ψ = −i

∂

∂x

|ψ, (7.75)

prove the commutation relation, Eq. 7.9,

ˆxˆp

− ˆp

ˆx = i. (7.76)

7.4. The deﬁnition of angular momentum in quantum mechanics is

m =

r ×

p. (7.77)

Prove that the angular momentum operator is Hermitian.

7.5. For the case of a string of atoms, the equation is the same as for an electron in a

one-dimensional box (Eq. 7.17). At the two ends of the atom chain, the wavefunction

should reach a maximum. The boundary conditions are



(0) = 0, (7.78)



(L)=0. (7.79)

Write the wavefunctions satisfying those boundary conditions.

7.6. Using the same conditions in Problem 7.5, obtain the energy levels of that system.

7.7. Using the same conditions in Problem 7.5, obtain the normalized wavefunctions

of that system.

7.8. Using the same conditions in Problem 7.5, obtain the wavefunctions with time

variation.

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Chapter 8

pn-Junctions

To date, most solar cells are made of semiconductors. A semiconductor is character-

ized by a relatively narrow energy gap, typically a fraction of an electron volt to a

few electron volts. Electrons can be excited by a photon from the valence band to

the conduction band and form an electron–hole pair. The electron–hole pair stores a

substantial portion of the photon energy. After formation of the electron–hole pairs,

the pn-junction separates the electrons and holes to generate external electric current.

In this chapter, we will learn the basic physics of semiconductors and pn-junctions.

8.1 Semiconductors

8.1.1 Conductor, Semiconductor, and Insulator

When a large number of atoms come together to form a solid, the wavefunctions of

atoms interact to form extended states which are similar to the one-dimensional crystal

in Section 7.2.4. A number of energy bands are formed. In general, the number of states

is inﬁnite. According to the Pauli exclusion principle, each state can only be occupied

Figure 8.1 Conductor, semiconductor, and insulator. (a) For conductors, the highest occupied

energy level is in the middle of an energy band. (b) For semiconductors, the highest occupied energy

level matches the top of the valance band but the energy gap to the conduction band is small. (c) If

the energy gap is big, the solid is an insulator.

161

Physics of Solar Energy C. Julian Chen

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162  pn-Junctions

Figure 8.2 Band gaps of a number of semiconductors. The wavelength of light corresponding

to the energy gap is also shown. Most of the semiconductors used in solar cells have an energy gap

corresponding to the photons of near-infrared or visible light.

by one electron. Imagine electrons arte added to a system one by one, stating from the

state with lowest energy. At one point, the number of electrons equals the number of

protons in the system and the system becomes neutral.

According to the relative position of the energy bands and the highest occupied

energy level, there are three diﬀerent cases, as shown in Fig. 8.1.

In Fig. 8.1(a), the highest occupied energy level is in the middle of an energy band.

The electrons can move to the unoccupied parts of the energy band. This type of

material is called a conductor or a metal.

If the highest occupied energy level matches the top of an energy band, which is

called the valence band, marked as E

, and the distance to the next energy band is

large, the electrons are not easily excited to the higher band. This type of materials is

called an insulator; see Fig. 8.1 (c).

An important case between those two is the semiconductor, where the gap between

the top of the valance band E

and the bottom of the next energy band E

is small

such that when the temperature is not too low electrons can be excited to the next

energy band, the conduction band. Typically the energy gap is less than a few electron

volts. Once the electrons are excited to the conduction band, some conduction can

take place. See Fig. 8.1(b).

Figure 8.2 shows the band gaps of a number of important semiconductors related

to solar cells.

8.1.2 Electrons and Holes

At low temperature, pure semiconductors have almost no mobile electrons, and the

conductivity is very low. By raising the temperature, electrons in the valance band

can be excited to the conduction band; see Fig. 8.3. Therefore, a semiconductor has

an important property: Conductivity depends critically on temperature — the higher

the temperature, the higher the conductivity.