Chen C.J. Physics of Solar Energy

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“ChenSolarEnergy” — 2011/5/17 — 17:56 — page 183 — #210

9.2 The Shockley–Queisser Limit  183

After some simpliﬁcation, maximum power is

= I

= V



1 −

1 + ln ln(I

)

ln(I

)



. (9.16)

The ﬁll factor η

deﬁned as

≡

, (9.17)

=1−

1 + ln ln(I

)

ln(I

)

. (9.18)

Typically, η

=0.8 − 0.85.

9.2 The Shockley–Queisser Limit

In 1961, William Shockley and Hans Queisser made a thorough analysis of pn-junction

solar cell, and established an upper limit for the eﬃciency of single-junction photovoltaic

cells as a consequence of the principle of detailed balance[77]. The eﬃciency is deﬁned

as the ratio of power delivered to a matching load versus the incident radiation power

to the solar cell. Three parameters are involved: The temperature of the Sun, T



;the

temperature of the cell, T

; and the energy gap of the semiconductor, E

. Actually,

eﬃciency only depends on two dimensionless ratios:



, (9.19)

. (9.20)

Typically, k



=0.5eV,k

− 0.025 eV, and E

is on the order of 1–2 eV. Therefore,

the typical order of magnitude is x

≈ 2 −−4, and x

≈ 40 −−80.

The analysis of Shockley and Queisser was based on the following assumptions:

1. A single p–n-junction

2. One electron–hole pair excited per incoming photon

3. Thermal relaxation of the electron–hole pair energy in excess of the band gap

4. Illumination with unconcentrated sunlight

The above assumptions are well satisﬁed by the great majority of conventional solar

cells, and the limit is well veriﬁed by experiments, unless one or more assumptions

are explicitly removed, for example, using concentrated sunlight or using tandem solar

cells. In these cases, the arguments of the Shockley–Queisser theory are still valid.

In the literature, the notation FF is often used to represent the ﬁll factor, a factor of eﬃciency in

the treatment of solar cells by Shockley and Queisser. To maintain consistency of notations, we use

instead.

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184  Semiconductor Solar Cells

Figure 9.6 Generation of an electron–hole pair. A photon of energy greater than the band

gap of the semiconductor can excite an electron from the valence band to the conduction band. The

electron–hole pair energy in excess of the band gap dissipated into thermal energy of the electrons

quickly (with a time scale of 10

−11

s). The useful part of the photon energy for converting to electrical

energy equals the band gap.

9.2.1 Ultimate Eﬃciency

Shockley and Queisser ﬁrst considered the eﬀect of band gap [77]. Assuming that

solar radiation is proportional to blackbody radiation of temperature T



,thepower

spectrum is (see Eq. 2.96),

u(, T



) d =

2πq



exp(/k



) − 1

d. (9.21)

Equation 9.21 represents the radiation power density at the surface of the Sun. At

the location of Earth, the spectral power density is diluted by a factor f deﬁned by

Eq. 2.103,

f =









6.96 × 10



[1.5 × 10

]

=2.15 × 10

−5

. (9.22)

Shockley and Queisser evaluated the ratio of the power of electron–hole pairs and

the incident radiation power by assuming that the absorptivity of the semiconductor

for photons with energy less than E

is zero and those greater than E

is unity. This

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9.2 The Shockley–Queisser Limit  185

simpliﬁcation is also used to evaluate the rate of radiative combination of the electron–

hole pair thus generated; see section 9.2.3. For photons with >E

, the electron-pair

energy is quickly thermalized to E

; see Fig. 9.6. By replacing one of variables  in Eq.

9.21 is by E

, the power of thus generated electron–hole pairs is

2πq



∞



d

exp(/k



) − 1

2πq

f(k



)



∞

− 1

. (9.23)

On the other hand, the incident radiation power is

2πq



∞



exp(/k



) − 1

d =

2πq



)

. (9.24)

The eﬃciency as a function of the dimensionless variable x



∞

− 1

. (9.25)

The integral in Eq. 9.25 can be evaluated using a simple numerical integration program

or the quickly converging expansion in Problem 9.1. The result is shown in Fig. 9.7. A

Figure 9.7 Ultimate eﬃciency of solar cells. Only photons with energy greater than the band

gap can be absorbed, and the electron–hole pair energy in excess of the band gap is dissipated into

thermal energy of the electrons, which make an ultimate limit to the eﬃciency of solar cells. The

maximum eﬃciency is 0.44, at E

=2.2 k



.ForT



= 5800 K, k



≈ 0.5 eV, the optimum band

gap is 1.1 eV.

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186  Semiconductor Solar Cells

qualitative explanation of the result is as follows. If the band gap is small, the range

of photon absorption is large. However, most of the photon energy is dissipated as

heat; see Fig. 9.6. For a large band gap, the range of spectral absorption is reduced.

Therefore, it should have a maximum somewhere. The position and the value of the

maximum eﬃciency can be obtained using a numerical program, and the results are

g max

=2.2 k



; η

u max

=0.44. (9.26)

Taking T



= 5800 K, k



≈ 0.5 eV. The optimum band gap is 1.1 eV. Shockley and

Queisser called it the ultimate eﬃciency.

9.2.2 Role of Recombination Time

The ultimate eﬃciency determines the maximum open-circuit current of a solar cell.

If the solar radiation power received by a solar cell is P

, the power of the electron–

hole pairs generated by the solar radiation is η

. It corresponds to the maximum

short-circuit current of the solar cell,

. (9.27)

The open-circuit voltage at the terminals of the solar cell is determined by the diode

equation 9.4. Combining Eq. 9.4 with Eq. 9.27, we have the nominal power, deﬁned as

the product of the short-circuit current and the open-circuit voltage,

= I

)



− 1



. (9.28)

Clearly, the reverse saturation current of a pn-junction, I

, is the limiting factor, de-

termined by Eq 8.48,

= qn









. (9.29)

The reverse saturation current, Eq. 9.29, can be estimated using actual data from

semiconductors. A general observation is as follows. The higher the reverse saturation

current, the smaller the open-circuit voltage. By looking at Eq. 9.29, it is clear that

the determining factor is the recombination time, τ

and τ

. Once an electron–hole

pair is generated by absorbing a photon, the pair has a tendency to recombine by

generating radiation or giving up the energy to the lattice. Shockley and Queisser

found a fundamental limit due to radiative recombination of electron–hole pairs based

on a detailed-balance argument.

9.2.3 Detailed-Balance Treatment

In the steady state, the electron–hole pairs in a solar cell are undergoing two major

processes: the generation of the pairs by solar radiation with a rate F

and various

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9.2 The Shockley–Queisser Limit  187

recombination processes. The generation rate F

is calculated by assuming that the

spectral density of sunlight is a blackbody radiation with temperature T



but diluted

because of the distance between the Sun and Earth by a factor f =2.15×10

−5

,deﬁned

by Eq. 9.22. The number of electron–hole pairs generated per unit area per unit time

on a surface perpendicular to the sunlight is expressed by the power of the electron–hole

pair (Eq. 9.23),

2πq



∞



exp(/k



) − 1

d. (9.30)

Many factors that contribute to recombination, such as those related to defects or

surfaces, can be reduced or avoided. The radiative recombination F

, however, is a

process which sets a fundamental limit on the eﬃciency of solar cells. To evaluate

the rate of radiative recombination of the electron–hole pairs, Shockley and Queisser

considered the equilibrium of the solar cell with the environment at the cell temperature

, typically 300 K, without sunlight. The principle of detailed balance requires that

the generation rate must equal the recombination rate. This principle, manifested as

Kirchhoﬀ’s law, states that emissivity equals absorptivity at any given wavelength.

With respect to the simpliﬁed model of Shockley and Queisser [77], for photons with

energy greater than the energy gap of the semiconductor, the emissivity is 1; otherwise

it is 0 (see Fig. 9.8). The rate of radiative electron–hole recombination can be calculated

using an integral similar to Eq. 9.30 but at the environment temperature T

2πq



∞



exp(/k

) − 1

d. (9.31)

Figure 9.8 A simpliﬁed optical model of semiconductors. To evaluate the eﬀect of radiative

recombination, Shockley and Queisser used a simpliﬁed optical model of semiconductors. For photons

with energy greater than the band gap of the semiconductor, the absorptivity is 1, and hence the

emissivity is also 1. Otherwise the emissivity is 0.

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188  Semiconductor Solar Cells

Typically, the parameter x

in Eq. 9.20 equals E

≈ 40 – 80, and thus the term

1 in the denominator of the integrand of Eq. 9.31 can be neglected. To high accuracy,

the result is

2πq

)



∞

−x

dx =

2πq

)

−x



+2x



. (9.32)

The above result is valid when the semiconductor is at equilibrium. With sunlight,

excess minority carriers are generated. For example, if the bulk of a solar cell is made

of p–type semiconductor, according to Eq. 8.34, with an external voltage V , the electron

concentration changes from its equilibrium value n

= n

exp





. (9.33)

Thus, the radiative recombination rate is increased to

(V )=F

exp





. (9.34)

At a steady state, the rate of electron–hole pair generation must equal the rate of

radiative recombination plus the rate of electron consumption due to the current I

drawn by the external circuit,

= F

(V )+

. (9.35)

By shorting the terminals together, the voltage V is zero, and the short-circuit current

= q (F

− F

) . (9.36)

Using Eq. 9.34, the current on the external load, I,isgivenas

I = I

+ qF



1 − e

qV/k



. (9.37)

Deﬁning the reverse saturation current I

= qF

, (9.38)

Eq. 9.37 becomes

I = I

+ I



1 − e

qV/k



. (9.39)

The open-circuit voltage can be obtained by setting I =0,



− 1



. (9.40)

Because F

 F

, Eq. 9.40 can be simpliﬁed to



− 1



. (9.41)

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9.2 The Shockley–Queisser Limit  189

9.2.4 Nominal Eﬃciency

Shockley and Queisser deﬁned the nominal eﬃciency as

. (9.42)

Using Eqs. 9.41, 9.27, and 9.20, after some reduction, we ﬁnd the expression

= η

, (9.43)

where the ultimate eﬃciency η

is deﬁned by Eq. 9.25, and the detailed balance

eﬃciency η

is deﬁned as



− 1



, (9.44)

Using Eqs. 9.30 and 9.31, we obtain







∞

− 1

−x



+2x



. (9.45)

As shown in Eq. 9.44, the detailed-balance eﬃciency η

is the fundamental upper

limit of the ratio of the open-circuit voltage and the band gap of the semiconductor in

volts. It depends on the temperature of the cell. If the temperature of the solar cell is

very low, x

→∞, the detailed-balance eﬃciency becomes

→

ln (Ce

) →

+lnC

→ 1, (9.46)

because the expression C varies much less than the exponential. Therefore, at very low

cell temperatures, if there is no other recombination mechanism other than radiative

recombination, the open-circuit voltage approaches the band gap energy in volts and

the nominal eﬃciency approaches the ultimate eﬃciency.

The reverse saturation current in Eq. 9.38 is the absolute limit of the observed

reverse saturation current (Eq. 9.29). In addition to radiative recombination, there are

other types of recombination processes that increase the reverse saturation current and

then reduce the eﬃciency of solar cells; see Section 9.3.

9.2.5 Shockley–Queisser Eﬃciency Limit

With a load of matched impedance, the output power of a solar cell can be maximized.

As shown in Section 9.1.3, the maximum power is related to the nominal power by a

form factor η

approximately

= η

≈ η



1 −

1 + ln ln(I

)

ln(I

)



. (9.47)

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190  Semiconductor Solar Cells

Figure 9.9 Eﬃciency limit of solar cells. Eﬃciency limit as determined by detailed balance.

The solar radiation is approximated by blackbody radiation at 5800 K. The temperature of the solar

cell and its surroundings is assumed as 300 K. The abscissa is the band gap of the semiconductor

in electron volts. The values of several important solar cell materials are marked. The ultimate

eﬃciency is determined by the absorption edge and thermalization of excited electrons. The nominal

eﬃciency, which is always lower than the ultimate eﬃciency, is a result of radiative recombination of

electrons and holes. While driving an external load at maximum power, the radiative recombination

is further increased, and the eﬃciency is further reduced. The detailed-balance limit of eﬃciency is a

fundamental limit for the conditions listed by Shockley and Queisser. After Ref. [77].

The detailed-balance eﬃciency limit, together with the ultimate eﬃciency and the

nominal eﬃciency, for T



= 5800 K and T

= 300 K, is shown in Fig. 9.9. The abscissa

is the band gap of the semiconductor in electron volts. Values of several important solar

cell materials are indicated. The ultimate eﬃciency is determined by the absorption

edge and thermalization of excited electrons. The nominal eﬃciency, which is always

lower than the ultimate eﬃciency, is a result of radiative recombination of electrons and

holes. While driving an external load at maximum power, the radiative recombination

is further increased, and the eﬃciency is further reduced.

In the original paper [77], Shockley and Queisser also discussed several factors that

aﬀect the eﬃciency limit, such as both sides of a solar cell can radiate which only one

side receives sunlight, the non-blackbody behavior of the cell, nonradiative electron–

hole pair recombination processes, and the diﬀerence between the solar constant and

the AM-1.5 solar radiation. The eﬃciency is further reduced by a few percentage points.

9.2.6 Eﬃciency Limit for AM1.5 Radiation

Shockley and Queisser assumed that solar radiation is a blackbody radiation [77]. The

actually solar radiation spectrum received on Earth is inﬂuenced by scattering and the

absorption of water vapor, carbon dioxide, and so on; thus the spectrum is diﬀerent

from blackbody radiation; see Section 5.2. How is the eﬃciency limit for actual solar

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9.3 Nonradiative Recombination Processes  191

Figure 9.10 Eﬃciency limit of solar cells for AM1.5 solar radiation. Eﬃciency limit as

determined by detailed balance for the AM1.5 solar radiation. The abscissa is the band gap of the

semiconductor in electron volts. The values of several important solar cell materials are marked.

Similar to Fig. 9.9, the ultimate eﬃciency, the nominal eﬃciency, and the detailed-balance limit of

eﬃciency, the fundamental limit for the conditions listed by Shockley and Queisser, are shown.

radiation spectrum changed from that of blackbody radiation?

The problem can be resolved using the same method of Shockley and Queisser.

Instead of using the blackbody radiation spectrum, the measured solar radiation, the

standard AM1.5 spectrum (see Plate 1) is used to compute the integrals in Eqs. 9.25 and

9.45. Because the strongest absorption and scattering are in the mid- to far-infrared and

ultraviolet regions, visible light is basically unchanged. Because the infrared radiation

with photon energy lower than the energy gap of the semiconductor does not participate

in the generation of the electron–hole pair, and ultraviolet photons would lose more

energy in the thermalization process, eﬃciency with regard to total radiation power

is slightly higher. The strong absorption peaks and valleys are smoothed out by the

integration process. Therefore, the ﬁnal result is qualitatively identical to that from a

blackbody radiation approximation. Thus, the blackbody approximation is appropriate

and useful.

9.3 Nonradiative Recombination Processes

The Shockley–Queisser limit is solely based on the thermodynamics of radiative recom-

bination of electron–hole pairs. There are several other recombination mechanisms and

factors that could limit the solar-cell eﬃciency. Some of the factors are intrinsic and

others can be mitigated or avoided by better cell design and manufacturing.

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192  Semiconductor Solar Cells

As shown in Section 9.2.2, the most serious limiting factor to solar cell eﬃciency is

the recombination rate of electrons and holes. According to the basic equation of solar

cells (Eq. 9.5), the open-circuit voltage is determined as

. (9.48)

The dark current I

is determined by Eq. 8.48. The solar cells are always made of a

thicker, lightly doped substrate and a thinner, highly doped ﬁlm. For example, the

typical silicon solar cell is made of a p–n

junction. Therefore, only one of the two

terms dominate. Thus, Eq. 8.48 can be simpliﬁed to



. (9.49)

The open-circuit voltage is

= const +

ln τ. (9.50)

Obviously, the greater the recombination time τ, the greater the open-circuit voltage

, and the eﬃciency will be greater.

If there are several recombination processes, the rate is additive, and thus the inverse

of the recombination time is additive,

+ ... +

. (9.51)

Each additional recombination process causes a reduction of the total recombination

time, and thus a reduction of eﬃciency.

Figure 9.11 The Auger recombination process. The electron–hole pair can recombine and

transfer the energy E

into either a free electron near the conduction band edge E

, (a), or free hole

near the valance band edge, E

, (b). Then the excited electron quickly loses its excess energy to the

lattice as phonons.