Chen C.J. Physics of Solar Energy

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“ChenSolarEnergy” — 2011/5/17 — 17:56 — page 173 — #200

8.3 Analysis of pn-Junctions  173

It is equivalent to two ﬁrst-order diﬀerential equations in the same sense that x

is equivalent to x = 2 and x = −2,

dδn

(x)

√

δn

(x) (8.40)

dδn

(x)

= −

√

δn

(x), (8.41)

which represent decays to the +x and −x directions, respectively. To account for

junction current, for each side of the junction, only one of the two is needed. The

diﬀusion current of electrons is

= qD

dδn

(x)

= q



δn

(x). (8.42)

At x = 0, using Eq. 8.35, the junction current of electrons is

=0)=−q





exp





− 1



. (8.43)

Similarly, for holes,

=0)=q





exp





− 1



. (8.44)

The total junction current is

I = q











exp





− 1



. (8.45)

Furthermore, using the approximate relations

, (8.46)

Eq. 8.45 can be reduced to

I = qn











exp





− 1



. (8.47)

8.3.4 Shockley Equation

Denoting a constant

≡ qn









, (8.48)

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174  pn-Junctions

Eq. 8.47 is simpliﬁed to the well-known form of the diode equation, also known as the

Shockley equation,

I = I



qV/k

− 1



. (8.49)

By applying a large reversed bias voltage on the diode, the exponential term van-

ishes. The current I = −I

. Therefore, the constant I

is the reverse saturation current

density. A question that remains to be answered is whether in the above derivation

of the diode equation only the current at the border of the space-charge region (the

transition region) and the neutral region is counted. What about the current deep in

the neutral regions? The simple answer is, the electric current must be continuous. As

the minority carriers are recombined with the majority carriers, there is a continuous

current of majority carriers to replenish the lost ones. Because the concentration of

majority carriers is orders of magnitude greater than that of the minority carriers, the

disturbance caused by such a current is very minor.

Figure 8.9 shows the current–voltage behavior of a pn-junction. According to the

diode equation 8.49, in the ﬁrst quadrant with a forward bias at room temperature,

the current is growing exponentially, about one order of magnitude per 60 mV. In the

third quadrant, as the applied voltage exceeds 100 mV the current reaches a saturated

value, which is dominated by the drift current. The two sides are highly asymmetric,

which makes the pn-junction the most widely used rectiﬁer.

The quality of a rectiﬁer critically depends on the magnitude of the reverse satura-

tion current density I

. As shown in Eq. 8.48, the decisive factor is the minority-carrier

lifetimes τ

and τ

. The longer the lifetime, the smaller the reverse saturation current

density, and more improved the quality of the rectiﬁer. This fact is equally important

Figure 8.9 Current-voltage behavior of a pn-junction. With a forward bias, at room tem-

perature, the current is growing exponentially, about one order of magnitude per 60 millivolts. In the

third quadrant, as the applied reverse voltage exceeds 100 millivolts, the current reaches a saturate

value, which is dominated by the drift current.

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8.3 Analysis of pn-Junctions  175

for solar cells. As we will show in Chapter 9, the most signiﬁcant limiting factor of

solar cell eﬃciency is minority-carrier lifetime. Therefore, in the design and manufac-

ture of solar cells, eﬀorts should be devoted to reducing the rate for minority-carrier

recombination and thus prolong the minority-carrier lifetime.

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176  pn-Junctions

Problems

8.1. A typical silicon solar cell has the following parameters: The p-type material has

acceptor concentration N

=1× 10

−3

, hole diﬀusion coeﬃcient D

=40cm

/s,

and lifetime τ

=5μs. The n-type material has donor concentration N

=10

−3

electron diﬀusion coeﬃcient D

=40cm

/s, and lifetime τ

=1μs. The permittivity of

free space is 

=8.85×10

−14

F/cm, and the relative permittivity of silicon is 

=11.8.

The intrinsic carrier concentration in silicon is n

=1.5 × 10

−3

. Assuming the

built-in potential is qV

=0.75 V, calculate the following:

1. Width of transition layer, W

Hint: The donor concentration is too high, thus the n-layer is very thin. The width

of transition layer is eﬀectively that of the p-layer. The dielectric constant (permittivity)

of silicon is the product of the permittivity of free space and the relative permittivity

of silicon.

2. Diﬀusion lengths for holes and electrons

3. Saturation current density I

4. The pn-junction capacitance of the 10-cm × 10-cm solar cell.

8.2. Show that excess minority carriers decay with x by

δp

(x)=δp

(0)e

−x/L

(8.50)

and

δn

(x)=δn

(0)e

−x/L

, (8.51)

where the diﬀusion lengths are deﬁned as



(8.52)



. (8.53)

8.3. Using diﬀusion length as a parameter, show that the junction current is

I = q







qV/k

− 1



. (8.54)

8.4. The mobility of holes in silicon is μ

=480 cm

/V·s at room temperature. What

is its diﬀusion coeﬃcient D

Hint: use Einstein’s relation.

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Chapter 9

Semiconductor Solar Cells

The photovoltaic eﬀect, the direct generation of electric power by light in a solid mate-

rial, was discovered by British scientists William Grylls Adams and his student Richard

Evens Day in the 1870s using selenium. A few years later, Charles Fritt of New York

constructed the ﬁrst photovoltaic module for generating power from sunlight. However,

the eﬃciency of the selenium solar cells was less than 0.5%, which meant it would not

generate suﬃcient energy economically.

An important breakthrough was made in the 1950s by Gerald Pearson, Darryl

Chapin, and Calvin Fuller at Bell Labs. Using silicon, they demonstrated a solar cell of

eﬃciency 5.7%, ten times greater than that of the selenium solar cell; see Section 1.4.

Solar cells ﬁrst found applications in space. The eﬃciency of silicon cells has been

improved to about 24% in the early 2000s, very close to the theoretical limit of 28%.

To date, semiconductor solar cells account for roughly 90% of the market share.

Especially, silicon solar cells account for more than 80% of the solar cell market. Thin-

ﬁlm solar cells, especially those based on CIGS (copper–indium–gallium–selenide) and

CdTe-CdS, are second to silicon solar cells in market share. Organic solar cells, which

will be presented in Chapter 10, is a promising emerging technology.

9.1 Basic Concepts

The solar cell is a solid-state device which converts sunlight, as a stream of photons,

into electrical power. Figure 1.22(b) shows the structure of a typical silicon solar cell.

Thebaseisapieceofp–type silicon, lightly doped with boron, a fraction of a millimeter

thick. A highly doped n–type silicon, with a thickness of a fraction of one micrometer

was generated by doping with phosphorus of much higher concentration. Because of

the built-in potential of the pn-junction, electrons migrate to the n–type region, and

generate electric power similar to an electrochemical battery.

According to the theory of quantum transitions presented in Chapter 7, radiation,

as a stream of photons, interacts with a semiconductor in two ways; see Fig. 9.1. A

photon with energy greater than the gap energy of the semiconductor material can

be absorbed and create an electron–hole pair. An electron–hole pair can recombine

and emit a photon of energy roughly equal to the energy gap of the semiconductor.

According to the principle of detailed balance (see section 7.3.3), the probabilities of

177

Physics of Solar Energy C. Julian Chen

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178  Semiconductor Solar Cells

Figure 9.1 Interaction of radiation with semiconductors. According to the theory of quantum

transitions, a photon with energy greater than the gap energy of the semiconductor material can be

absorbed and create an electron–hole pair. An electron–hole pair can recombine and emit a photon of

energy roughly equal to the energy gap of the semiconductor.

the two processes should equal. This fact has a signiﬁcant consequence to the eﬃciency

of solar cells; see Section 9.2.3.

Because the potential energy of the electron–hole pair equals the value of the en-

ergy band, the best material should have a band gap close to the center of the solar

spectrum. Another factor that aﬀects the eﬃciency of solar cells is the type of the

energy gap. Depending on the relative positions of the top of the valence band and

Figure 9.2 Direct and indirect semiconductors. Depending on the relative positions of the top

of the valence band and the bottom of the conduction band in the wavevector space, the energy gap

of a semiconductor can be direct or indirect. Direct semiconductors have a much higher absorption

coeﬃcient than that of indirect semiconductors. As shown, Si is an indirect semiconductor and GaAs

is a direct semiconductor.

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9.1 Basic Concepts  179

Figure 9.3 Absorption spectra of

semiconductors commonly used

for solar cells. The most commonly

used material for solar cells, silicon,

is an indirect semiconductor, and has

a relatively low absorption coeﬃcient,

typically 10

−1

. A thickness of

0.01 cm is required for eﬃcient absorp-

tion. Direct semiconductors, such as

GaAs, CuInSe

and CdTe, have ab-

sorption coeﬃcients ranging from 10

to 10

−1

. A thickness of a few mi-

crometers is suﬃcient for an almost

complete absorption.

the bottom of the conduction band in the wavevector space, the energy gap of a semi-

conductor can be direct or indirect; see Fig. 9.2. For semiconductors with a direct

gap, such as GaAs, CuInSe

, and CdTe, a photon can directly excite an electron from

the valence band to the conduction band; the absorption coeﬃcient is high, typically

greater than 1 ×10

−1

. For semiconductors with an indirect gap, such as Ge and Si,

the top of valence band and the bottom of the conduction band are not aligned in the

wavevector space, and the excitation must be mediated by a phonon, in other words,

by lattice vibration. Therefore, the absorption coeﬃcient is low, typically smaller than

1 × 10

−1

. A thicker substrate is required. Figure 9.3 shows the absorption spectra

of commonly used semiconductor materials for solar cells. Table 9.1 gives the properties

of frequently used solar cell materials.

Table 9.1: Properties of Common Solar-Cell Materials.

Material Ge CuInSe

Si GaAs CdTe

Type Indirect Direct Indirect Direct Direct

Band gap (eV) 0.67 1.04 1.11 1.43 1.49

Absorption edge (μm) 1.85 1.19 1.12 0.87 0.83

Absorption coef. (cm

−1

)5.0×10

1.0×10

1.5×10

3.0×10

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180  Semiconductor Solar Cells

9.1.1 Generation of Electric Power

Figure 9.4 shows a quantitative explanation of how the electric power is generated by

a solar cell. As shown in Fig. 9.4(a), a photon generates a electron–hole pair in the

p–type region. Because of the built-in electric ﬁeld, which points towards the p–type

region, the electrons drift into the n–type region, whereas the holes stay in the p–type

region. By connecting the two terminals in Fig. 9.4 together, almost all the electrons

generated by the photons can migrate to the n–type region, and complete the circuit.

The short-circuit current is the current of the electrons generated by sunlight. As shown

in Fig. 9.4(b), if the two regions are not connected externally, the charges accumulated

in the two regions generate a potential across the junction capacitance. The potential

becomes a forward voltage for the diode. The thickness of the transition region is

reduced, and a forward diode current is created. When the forward diode current

equals the drift current of the electrons generated by the photons, an equilibrium is

established. The voltage on the two terminals is the open-circuit voltage of the solar

cell under illumination.

Figure 9.4 Separation of holes and electrons in a solar cell. (a) The incident photons

generate electron–hole pairs. The built-in electrical ﬁeld in the transition region is pointing toward

the p–type region. The electrons, negatively charged, is dragged by the ﬁeld into the n–type region.

By connecting the two terminals, an electrical current is generated. The short-circuit current is

determined by the rate of electron–hole pair generation from the radiation. (b) If the terminals are

not connected, the electrons migrate to the n–type region are accumulated and builds a voltage across

the junction capacitance. The direction of the voltage is the same as the forward bias voltage of the

diode, which generates a current to compensate the electron current. At equilibrium, an open-circuit

voltage is established.

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9.1 Basic Concepts  181

9.1.2 Solar Cell Equation

A solar cell can be represented by a current source connected in parallel with a pn-

junction diode; see Fig. 9.5. The current source is the photocurrent generated by

incoming sunlight, deﬁned by Eq. 9.27. the diode equation is changed to

I = I



qV/k

− 1



− I

. (9.1)

which is the fundamental equation of solar cells, in a format consistent with the diode

equation. However, in Eq. 9.1, while the voltage is always positive, the current is

always negative. This is understandable because the diode is a passive device which

consumes energy. By considering the solar cell as a battery, the direction of current

should be reversed. Therefore, a better form of the solar cell equation is

I = I

− I



qV/k

− 1



, (9.2)

where both voltage and current are always positive; see Fig. 9.5.

The open-circuit voltage is the voltage when the current is zero, deﬁned by the

condition



− 1



= I

. (9.3)

Consequently,



− 1



. (9.4)

Because I

is always much bigger than I

, Eq. 9.3 can be simpliﬁed to

. (9.5)

Figure 9.5 Equivalent circuit of solar cell. A solar cell can be represented by a current source

connected in parallel with a pn-junction diode. The current source is the photocurrent generated by

incoming sunlight.

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182  Semiconductor Solar Cells

9.1.3 Maximum Power and Fill Factor

The output power of a solar cell is determined by the product of voltage and current,

P = IV . As discussed in Section 1.4.2, it is always smaller than the product of the

short-circuit current I

and the open-circuit voltage V

, see Fig. 1.25. The rated power

of a solar cell is the maximum power output with an inﬂux of photons of one sun, or 1

kW/m

, under favorable impedance-matching conditions. In general, the condition of

maximum power is

dP = IdV + VdI =0, (9.6)

in other words,

= −

. (9.7)

According to the solar cell equation 9.2, the output power as a function of the output

voltage V is

P = IV =



− I



qV/k

− 1



V. (9.8)

From Fig. 1.25, we observe that the voltage of maximum power is only slightly

smaller than the open-circuit voltage. Introduce a voltage oﬀset v,wewrite

V = V

− v. (9.9)

Using Eq. 9.5, Eq. 9.8 can be simpliﬁed to

P ≈ I

− v)



1 − e

−qv/k



. (9.10)

Taking the derivative of P with respect to v, the condition of maximum power is

qv/k

=1+

. (9.11)

Again, using Eq. 9.5, Eq. 9.10 becomes

qv/k

=1+ln

. (9.12)

Because I

 I

, we ﬁnd

v =

ln ln

. (9.13)

Therefore, the voltage at maximum power is

= V

− v = V



1 −

ln ln(I

)

ln(I

)



, (9.14)

andthecurrentatmaximumpoweris

= I



1 − e

−qv/k



= I



1 −

ln(I

)



. (9.15)