Notes on Nelson’s The Physics of Solar Cells · 2019-07-27 · Notes on Nelson’s The Physics of Solar Cells Jimmy Qin Summer 2019 And God said, Let there be PV devices: and there

Notes on Nelson’s The Physics of Solar Cells

Jimmy Qin

Summer 2019

And God said, Let there be PV devices: and there were PV devices.And God saw the PV device, and it was good; and God divided the electrons from the holes.(Gen. 1.3 VNIV)1

These are notes on Jenny Nelson’s awesome book The Physics of Solar Cells. In fact, the bookalso has very thorough and rigorous treatment of classic semiconductor phenomena, so that stuffis good too. Hopefully I will add more stuff from other sources later.

Contents

1 Introduction 3

1.1 PV cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Ideal characteristics of PV device . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Thermodynamics and basic ideas of PV 6

2.1 Currents from thermodynamics. Detailed balance . . . . . . . . . . . . . . . . . . 6

2.2 Limiting efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Electrons and holes in semiconductors 9

3.1 Direct and indirect bandgaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Carrier and current densities under bias . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Drift-diffusion equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Generation and recombination 12

4.1 Basic ideas of generation and recombination . . . . . . . . . . . . . . . . . . . . . 13

4.2 Rates from Fermi Golden Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1Very New International Version

1

Jimmy Qin Notes on photovoltaics

4.3 Photogeneration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.4 Recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4.1 Radiative recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.4.2 Auger recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4.3 Shockley-Read-Hall recombination . . . . . . . . . . . . . . . . . . . . . . . 16

4.5 Transport equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5 Junctions 18

5.1 Types of PV action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.2 Metal-semiconductor junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.1 Schottky contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.2.2 Ohmic contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 Surface and interface states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3.1 Neutrality level, φ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3.2 Fermi-level pinning and other effects . . . . . . . . . . . . . . . . . . . . . 21

6 Analysis of the pn-junction 22

6.1 A very complicated solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6.1.1 Why write it this way? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2 Approximate form in the dark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.3 Approximate form in the light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.3.1 Form of PV device J(V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.3.2 Effects of external parameters . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.3.3 Ideas for design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6.4 Simple derivation of the J(V ) characteristic . . . . . . . . . . . . . . . . . . . . . 26

7 Monocrystalline solar cells 26

7.1 Characteristics of an ideal solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.2 Silicon: material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.3 Silicon: solar cell design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.3.1 Improving the Si solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.3.2 Problems with Si solar cells . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.4 GaAs: material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.5 GaAs: solar cell design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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7.5.1 Improving the GaAs solar cell . . . . . . . . . . . . . . . . . . . . . . . . . 29

8 Thin film solar cells 29

8.1 Amorphous silicon (a-Si) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8.1.1 Grain boundaries in polycrystalline materials . . . . . . . . . . . . . . . . . 30

9 Managing light 32

9.1 Minimizing reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

9.2 Increasing photon concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

9.3 Photon trapping (photon confinement) . . . . . . . . . . . . . . . . . . . . . . . . 33

9.4 Photon recycling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

10 Into the future: strategies for high efficiency 34

10.1 Multiple band gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

10.2 Intermediate band and multiple band cells . . . . . . . . . . . . . . . . . . . . . . 36

10.3 Hot carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1 Introduction

Solar photovoltaic, or PV, energy conversion is a one-step conversion process, turning light energyinto electrical energy. The very basic idea is that photons hit a semiconductor, which makeselectrons jump out. The electrons then swept away by the electric field in the depletion region of apn-junction, and we have a current. Because there is also a voltage difference over the pn-junction,we also have a voltage. Because we have both a current and a voltage in the same place, we havepower, by P = IV . Power is energy.

The very simple description above, in fact, suggests one of the most important concepts in PVdevices:

Photovoltaic operation hinges on the idea that electrons are created (by energy transfer due tophotons) in an asymmetric potential.

1.1 PV cells

The photovoltaic cell is the building block of a photovoltaic device. It can be thought of as atwo-terminal device which conducts like a diode in the dark, and generates a photovoltage whencharged by the sun. It is usually around 100 cm2 in area, and looks blue or black because thesurface is treated to reflect as little light as possible. A pattern of metal contacts is imprinted onthe surface to make electrical contact.

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Each cell creates a dc photovoltage of around 0.5− 1 V. Cells are connected in series (perhaps 30cells in series) to create modules. This increases the dc ouput voltage to perhaps 12 V. Modulesare further connected in series or parallel to create arrays or generators. The array is connectedto a battery which stores energy. This ensures that the operation does not depend too heavily onwhether the sun is shining.

1.2 Some definitions

The solar cell, in operation, can be thought of as a current source. The characteristics of thebattery depend on the illumination and the load. In fact, solar cells are one of the most importantexamples where the source is of (roughtly) constant current, rather than constant voltage. Thecurrent density is roughly proportional to the area of the cell and also depends on the light intensityand frequency spectrum.

Define the open circuit voltage Voc to be the voltage across the terminals of the solar cell whennothing connects the terminals. As usual, we define the current-voltage characteristic to bethe function I(V ), where I is the current through the PV device and V is the voltage across theterminals. Finally, because current is proportional to the PV area, we define the short circuitcurrent density Jsc to be the limiting value

limA→∞

I

Awhen the PV device is shorted.

This is not divergent because the device has some internal resistance.

1.3 Ideal characteristics of PV device

Besides inefficiencies due to the non-ideal nature of any real device, there are also theoreticallimitations on the power generated by an ideal PV device. We will study those here.

Let us define the quantum efficiency QE(E) to be the probability that an incident photon ofenergy E will deliver one electron to the external circuit. Then the short circuit current densitymust be an integral over all possible photon energies:

Jsc = q

∫ ∞0

dEbs(E)QE(E).

Here, bs(E) is the incident spectral photon flux density, or the number of photons of energyE which are incident on unit area in unit time. QE(E) is typically peaked in the visible range. Itmight depend on the semiconductor material, how good the coating is, etc.

Now let’s study the simplest model of the I(V ) characteristic. Recall that the PV device actsjust like a pn-junction (i.e. a diode) under no illumination. Also recall that the PV device underillumination is kind of like a current source, where the current generated is voltage-independent.If we put these two pieces of information together, it’s reasonable that approximately

J(V ) = Jsc − Jdark(V ), or J(V ) = Jsc − J0(eqV/kT − 1).

Here, J(V ) is the total current under illumination and V is the bias voltage. J0 describes the(small) amount of current that passes through the diode when V < 0. We see that here, wedefined the sign of J(V ) to be opposite to the polarity of V , as shown in the following figure.

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To get some intuition, the J(V ) graph looks like this:

Therefore, the cell generates power for 0 < V < Voc. For V < 0, the device consumes power andacts as a photodetector. For V > Voc, the device again consumes power like an LED and emitslight.

When the contacts are isolated, J(V ) = 0. This gives an expression for the open-circuit voltageVoc:

Voc =kT

qln(

Jsc

J0

+ 1).

Unsurprisingly, the open-circuit voltage increases with increasing light intensity.

The power P reaches a maximum at the cell’s maximum power point (Vm, Jm). From the J(V )characteristic, you can see that Vm is not much smaller than Voc. Therefore, the fill factor

FF :=JmVmJscVoc

describes the “squareness” of the J(V ) curve. The efficiency of the cell is unsurprisingly

η =PmPinc

=JmVmPinc

=JscVocFF

Pinc

,

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where Pinc is the incident power density. Materials with higher open-circuit voltage tend to havelower short-circuit current density, so there is an inherent tradeoff between the two. This is relatedto the semiconductor bandgap.

In real cells, there is some intrinsic resistance, both due to the contacts and due to the device.We can model these by parasitic resistances Rs and Rsh, called series and shunt resistances,respectively.

The new formula is

J(V ) = Jsc − J0(eq(V+JARs)/kT − 1)− V + JARs

Rsh

.

2 Thermodynamics and basic ideas of PV

Let us study the basic thermodynamics of PV and solar energy from the sun. The spectral photonflux can be approximated as the radiation from a blackbody at T ≈ 5760 K:

bs(E) =2Fsh3c2

E2

eE/kT − 1,

where Fs = π sin2 θsun and θsun ≈ 0.26◦ is the half-angle subtended by the sun, as seen by somebodyon the Earth.

The spectral photon flux is a number count. To turn this into an energy per area per time, orirradiance L, we must multiply by the energy per photon:

L(E) = Ebs(E).

2.1 Currents from thermodynamics. Detailed balance

There is a limit on PV performance due to the principle of detailed balance. Basically, the idea isthat if the PV device absorbs solar radiation, it must also radiate away energy to its surroundingenvironment (i.e. to the heat bath).

Suppose a PV device is in the dark. It absorbs photons and also spontaneously emits them. Ifwe define the absorption a(E) and the emissivity ε(E) to be the probabilities of absorbing oremitting a photon of energy E, and also define the probability of photon reflection to be R(E),

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then we have

jabs(E) = q(1−R(E))a(E)ba(E) and jrad(E) = q(1−R(E))ε(E)ba(E) =⇒ a(E) = ε(E).

In thermal equilibrium, the absorption is equal to the emissivity. The reason we can use thesame ba(E) in both equations is that we assume both the environment and the PV device areblackbodies at the same temperature, so they have the same spectrum.

Now, suppose the PV device is under illumination by a solar photon flux bs(E). Approximately,the absorption current is

jabs(E) = q(1−R(E))a(E)(bs(E) + ba(E)).

The radiation current (i.e. heat loss) is a little different because the chemical potential is nolonger zero. That is because as a result of illumination, part of the electron population has raisedelectrochemical potential energy. We can think about it this way: after an electron is created, it is“immediately” swept into a high-voltage region with a different µ, and this process is faster thanthe lifetime of radiative recombination. Let’s denote it by µ. Then

jrad(E) = q(1−R(E))ε(E)be(E, µ).

We know from before that a(E) = ε(E), which holds even out of equilibrium because it is anintrinsic property of the device. We can thus find the net current density,

jnet(E) = jabs(E)− jrad(E) = q(1−R(E))a(E)

(bs(E) + ba(E)− be(E, µ)

).

Here, be(E, µ) is the photon flux emitted normal to the surface, which is taken to be the usualblackbody formula

be(E, µ) =2Fen

2s

h3c2

E2

e(E−µ)/kT − 1,

and Fe is a geometrical factor depending on indices of refraction and so on. We will use thisblackbody radiation formula to derive the I(V ) characteristic of a diode in the next section (!)

We can think of the current as two contributions: one from net absorption,

jabsnet (E) = q(1−R(E))a(E)bs(E)

and one from net emission, or radiative recombination:

jradnet (E) = q(1−R(E))a(E)

(be(E, µ)− be(E, µ = 0)

).

The radiative recombination corresponds to a loss of energy, similar to the sink of a heat engine.The absorbed solar radiant energy can never be fully utilized.

2.2 Limiting efficiency

Consider a semiconductor with perfectly full valence band and perfectly empty conduction band;they are separated by a bandgap Eg. Let the chemical potential of band i be µi and the ambienttemperature be T .

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It will turn out that jabsnet (E) gives the photocurrent contribution Jsc and jrad

net (E) gives the dioderectifying contribution, Jdark(V ). We will flesh these ideas out here.

Claim: The ideal photocurrent is a function of only the bandgap Eg and the incident spectrum.Reasoning: Note the word “ideal.” We will make lots of assumptions here. From the above, wecan obtain the photocurrent density at short circuit by integrating jabs

net (E) over all energies E:

Jsc = q

∫ ∞0

dE(1−R(E))a(E)bs(E)⇐⇒ Jsc = q

∫ ∞0

dEQE(E)bs(E),

from a previous definition of quantum efficiency QE(E). We could guess, similar to Einstein’sidea behind the photoelectric effect, that in the ideal case R(E) = 0, we have

QE(E) = a(E) = 1 if E ≥ Eg and 0 if E < Eg.

Therefore,

J idealsc = q

∫ ∞Eg

bs(E)dE.

This implies that ideal photocurrent depends only on the bandgap and the incident spectrum.

Important point: we must give the spectrum bs(E) before making any statement about PV deviceefficiency.

Claim: The diode term −J0(eqV/kT − 1) in the current-voltage characteristic J(V ) can be thoughof as due to the radiative recombination term jrad

net (E).Reasoning: The main question is “what is µ?” In an ideal material with lossless carrier transport,µ = qV . This is important because the electrons are swept away before they can emit their photons.According to the formula

be(E, µ) =2Fen

2s

h3c2

E2

e(E−µ)/kT − 1,

there is a large exponential dependence on µ, or equivalently on qV . This is the same, in fact, asthe diode formula

Jdark ∼ (eqV/kT − 1).

Therefore, the simple step-like absorption function yields

J(V ) = q

∫ ∞Eg

(bs(E)− be(E, qV ) + be(E, 0))dE.

Therefore, the net electron current is dependent on V and can even switch sign. For V > Voc,the radiative recombination current is higher than the absorption current. This means that morelight is being sent out than is coming in, like an LED.

The limiting efficiency follows from

η ≤ VmJ(Vm)

Pswhere Ps =

∫ ∞0

Ebs(Es)dE.

We see that all the energy for E < Eg has been wasted. However, it is not true that lowering Egwill always make the PV device more efficient! That is because Vm < Voc < Eg, so lowering thebandgap also lowers the operating voltage of the device.

Conclusions for the ideal solar cell:

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• All the light energy for E < Eg is wasted.

• If Eg is too small, the operating voltage Vm will be too small. If Eg is too large, the operatingcurrent Jm will be too small. (The optimal bandgap seems to be around 1-1.5 eV and thelimiting efficiency is around η = 30 − 40%. The preferred materials are therefore III-Vsemiconductors like GaAs (Eg = 1.42 eV) and InP(Eg = 1.35 eV). Silicon, which is cheaper,has Eg = 1.1 eV which is not as good but still okay.)

• The current characteristic J(V ), including the diode rectifying contribution, can be thoughtof as a result of different processes of blackbody radiation.

Reasons why real solar cells do not achieve ideal performance:

• Incomplete absorption, or nonzero reflection R(E) > 0.

• Non-radiative recombination of photogenerated carriers. We know that radiative recombi-nation is unavoidable, but defects in the semiconductor bulk can make excited charges gettrapped (and “recombine” with the positively charged defect) before being collected on theother side of the device, so the current is lower than it should be.

• Parasitic resistance in the device.

3 Electrons and holes in semiconductors

Clearly I’ve learned this stuff like twice already, but it is good to get a fresh perspective. There isquite a bit of material here that is done more rigorously than in Razeghi, for example, so is niceto read.

3.1 Direct and indirect bandgaps

Here is an example of an indirect bandgap, which is when the maximum energy of the valenceband occurs at different wavevector compared to the minimum energy of the conduction band:

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Because a photon of energy Eg has very small momentum, an Eg-photon-absorption process byitself is not sufficient to surmount the energy barrier: the momentum needs to change as well! Thisis typically supplied by a phonon. Conclusion: in indirect bandgap semiconductors, photon ab-sorption can only happen if there are enough phonons available with the right value of momentum~k.

3.2 Carrier and current densities under bias

Consider the behavior and distribution functions of the electrons and holes under a bias. Let usdefine the intrinsic potential energy Ei to be that number, close to the center of the bandgap,which gives the correct populations when we talk about the Fermi energy:

n = nie−β(Ei−EF ) and p = nie

−β(EF−Ei).

As you can see, the higher the Fermi energy, the more electron carriers. Just as we expect. (Above,ni is the intrinsic carrier density.)

Nonequilibrium carrier densities: The above was in equilibrium. What is the semiconductordevice is not in equilibrium, for example, if there is light illumination and hence nonzero netcurrent? Then the equilibrium relation

np = n2i is no longer true!

However, we can make a great simplification: if the nonequilibrium disturbance is not too great,we can assume the populations of electrons and holes relax to achieve a state of quasi-thermalequilibrium, but with different quasi-thermal parameters. Thus, we will have to have to introducetwo different Fermi energies EFn and EFp , and we say the Fermi-levels have split. Generally, EFnand EFp can be functions of position and of time. This means there is a local quasi-thermalequilibrium distribution. In the Boltzmann approximation,

n = nie−β(Ei−EFn ) and p = nie

−β(EFp−Ei).

In fact, the temperatures could be different, Tn and Tp. We will assume they are the same andequal to the ambient temperature, T , which is a good approximation in the case that the electricfield is not too strong, so the different in effective mass between electrons and holes is not so greatthat one species could gain a lot of kinetic energy (due to the ~E-field) and hence be at a differenttemperature.

In fact, the difference in quasi-Fermi levels,

µ = EFn − EFp ,

is the chemical potential from the previous chapter. The population product is

np = nieµ/kT .

Nonequilibrium current densities: In fact, the spatiotemporal dependence of the quasi-thermal distribution function allows us to derive a Boltzmann transport equation for f(~k, ~r, t)and hence a current density. Let’s see how this works.

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Define fn(~k, ~r, t) to be the quasi-thermal distribution function for electrons. We have

dfndt

= ∂tfn + ∂t~r · ∇~rfn + ∂t~k · ∇~kfn.

Let us put ∂t~r = ~v and ∂t~k = ~F , and also assume Boltzmann statistics with form of the distributionfunction f0 (i.e. that the equilibrium situation is fn = f0), which means the derivatives of fn areclose to the derivatives of f0, so

∇~rfn = − f0

kT∇~r(Ec − EFn) and ∇~kfn = −f0~v

kT,

where ~v = ∇~kE is the group velocity. We take it to be the same velocity as ∂t~r, which is reasonableif you analogize with classical mechanics. Because the force on the electron is actually related tothe spatial variation of the conduction band energy (equivalently, related to the spatial variationof the Fermi energy)

∇~rEc = −~F ,we have

dfndf

=f0

kT~v · (~∇~rEFn) + ∂tfn,

and due to Liouville’s theorem, etc. we have dfndt

= 0 in the steady-state. If we use the relaxationtime approximation

∂tfn = −1

τ(fn − f0),

and therefore

fn = f0(1− τ~v

kT· (∇~rEFn)).

We can use this distribution to determine the current density,

Jn(~r, t) = − q~m∗n

∫~k

~kgc(~k)f(~k, ~r, t).

Here, gc(~k) is the density of states. The result is

Jn(~r, t) = µnn∇~rEFn(~r, t)

where µnn = qkT

∫~kτ( ~~k

m∗n)2gc(~k)f0(E(~k)). Important point: µn ∝ τn. We see that the current

density is greater if the electron fluid takes longer to relax! That makes sense if we think of thecurrent as something propagating only before the fluid can reach equilibrium.

3.3 Drift-diffusion equations

We saw in the previous section that J(~r) = Jn(~r) + Jp(~r), where

Jn(~r) = µnn∇EFn(~r) and Jp(~r) = µpp∇EFp(~r).

Drift-diffusion equations: We claim that we can recast the derivative ∇EFn(~r), and hence theabove formulas for the currents, as

Jn(~r) = qDn∇n+ qµnFn and Jp(~r) = −qDp∇p+ qµpFp.

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Here, F is the electrostatic field. The first contribution is the diffusive contribution and the secondcontribution is the drift part (i.e. due to the electric field).Reasoning: Starting with n = nie

−β(Ei−EFn ), we can solve for EFn to get

EFn = Ei + kT lnn = Ec − kT lnNc + kT lnn and EFp = Ev + kT lnNv − kT lnn.

Here, we have used the formula Ei = 12(Ec + Ev)− kT

2ln(Nc

Nv) and split the electron contributions

and one side and the hole contribution on the other side because actually the electrons and holesare thought of as “independent.”

We will assume the electron and hole effective masses are constant throughout the bulk, so ∇Nc =∇Nv = 0. We have

∇Ec = qF −∇χ and ∇Ev = qF −∇χ−∇Eg,

where F is the electrostatic field and χ is the electron affinity. Ignoring ∇Eg and ∇χ gives thedesired result, where we used the Einstein mobility-diffusion relations

µn =qDn

kTand µp =

qDp

kT.

4 Generation and recombination

This is a very dense and technical chapter. We will study the detailed physics behind the Poissonequation, carrier transport, and carrier generation and recombination. All of these things arelinked.

The basic equation of device physics are based on two principles:

• The number of carriers of each type is conserved. If G,U are volume rates of generation andrecombination, then

∂tn =1

q∇ · ~Jn +Gn − Un and ∂tn =

−1

q∇ · ~Jn +Gn − Un

• The electrostatic potential obeys Poisson’s equation:

∇2φ =q

εs(−ρfixed + n− p).

Here, εs = ε0ε is the permittivity and ρfixed is fixed charge, such as interface charge fromimpurities.

We already know how to find J and n from last chapter, at least in quasi-thermal distributions.(For example, the drift-diffusion relation, etc.) In this chapter, we will study how to find thevarious important contributions to G and U .

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4.1 Basic ideas of generation and recombination

Generation and recombination are opposite processes which create and annihilate charge carriers,respectively. For every generation process, there is a corresponding recombinative process, due tothe microscopic reversibility.

If the process is band-to-band, an electron and hole are created or annihilated, so two chargecarriers per process. If the process is from a trap state to a band, only one carrier is createdor annihilated in the process. The most important generation process for PV devices is opticalgeneration (i.e. absorption of a photon). The most important recombination processes are radia-tive recombination (release of a photon), non-radiative recombination (release of a phonon), orAuger recombination (kinetic energy transferred to another free carrier). Even though there is acorresponding generation process for all of these, they are not as prominent due to the differencesin occupation levels. (So although the square of matrix element is the same, the strength of theprocess depends on whether the initial starting point of the process is prominent in the materialor not.)

There is actually thermal generation and recombination, Gth = Uth, which exactly cancel eachother out. We will ignore this effect and consider G and U to be made of other non-thermal effectson top of the thermal effect, which always exists in quasi-thermal equilibrium.

4.2 Rates from Fermi Golden Rule

The transition rate Γ from |i〉 to |f〉 due to the interaction Hint and emission or absorption ofenergy E is

Γ =2π

~|〈i|Hint|f〉|2δ(Ef − Ei ∓ E).

Here, the −E is for processes with Ei > Ef and the +E is for Ei < Ef , which correspond tophoton emission and photon absorption, respectively.

Claim: The photon distribution function f(E) can be derived from occupations of the conductionand valence bands, fc and fv.Reasoning: Let

fv =1

eβ(Ev−EFp ) − 1and fc =

1

eβ(EFn−Ec) − 1

be the occupation probabilities of states in the valence and conduction bands, respectively. Thenthe rates of net absorption and emission are

Rabs =2π

~|〈c|Hint|v〉|2 × f(fv − fc) and Rem =

2π

~|〈v|Hint|c〉|2 × fc(1− fv).

The quasi-equilibrium occupation function for photons is established by setting the net rate equalto zero, 0 = Rabs −Rem. The result is

f =1

e(E−µ)/kT − 1where µ = EFn − EFp .

This gives another interpretation of µ: the chemical potential of radiation. A more intenselight field will cause a greater separation in quasi-Fermi levels. In this interpretation, the illumi-nation from light necessitates a change in the chemical potential, µ, which makes the populationsreadjust until equilibrium is established.

13


4.3 Photogeneration

Photogeneration is the generation of mobile electrons and holes via the absorption of light in thesemiconductor. Because a fraction α(E)dx of the photons of energy E entering a slab of thicknessdx will be absorbed, the light intensity will go like

I(x) = I0e−αx

where x is the depth into the bulk of the absorbing semiconductor. This is known as the Beer-Lambert law. Therefore, the carrier generation rate g(E, x) a depth x into the material is

g(E, x) = (1−R(E))α(E)bs(E)e−αx,

where we allowed for the possibility of reflection and I0 → bs(E) is the incident photon flux.

Here is an important point...Claim: If E > Eg, then E is irrelevant to the generation of photovoltaic energy.Reasoning: Why? The point is that the excited electron is thermalized very quickly, in thesense that in the process

Ev → Ec + ε→ Ec → Ev,

the second arrow is really fast and the last arrow is relatively slow. So there is time for the electronto get swept along to the other side of the pn-junction, but only after it has thermalized to thebottom of the conduction band. Conclusion: the number of excitation events is the importantquantity, not the total energy absorbed. This is the essential difference between PV devices andsimply heating something up by putting it in the sun.

Now, let us describe the absorption of photons with AMO physics. In the dipole approximation,

Hint =iqE0

2mω~d · ~p,

where ~d is the dipole moment and ~p is the momentum operator. Therefore, we can write

α(E) =Aαm

∫~k~k′|Mcv|2δ(Ec − Ev − E)gc(~k

′)gv(~k),

where Aα = 4π2q2hnscmE

and Mcv = 〈v,~k|~d · ~p|c,~k′〉. This allows for the generality of indirect bandgaps.

We can use this machinery to study the absorption in both direct gap and indirect gap materials.We claim that in the Boltzmann approximation,

α(E) ∼ (E − Eg)1/2 in direct-gap materials

andα(E) ∼ (E − Eg)2 in indirect-gap materials.

Actually, the mathematics is complicated and uninteresting. I am content just with these depen-dences. In real life, α(E) ∼ (E − Eg)1/2 in direct-gap materials is seldom seen, because the DoSfunctions are not parabolic even moderately far away from the extrema of the band.

14


4.4 Recombination

Now let’s study recombination. We distinguish between two categories: unavoidable recombina-tive processes are due to essential physical processes (spontaneous emission/radiative recombina-tion, Auger recombination, stimulated emission, etc.) and avoidable ones are due to imperfectionsin the material, or defects (trap states, etc.).

4.4.1 Radiative recombination

Let us derive the rate of spontaneous emission, or radiative recombination. This is easy becausewe already have some information from the detailed balance argument from before.

We found before that the equilibrium distribution of photons, feq, satisfies

feq(fv − fc) = fc(1− fv).

Away from equilibrium, the new rate is

Rabs

Reqabs

=feq

f.

Under an incident field (the nonequilibrium scenario), the distribution function is defined to be

f(E) =nph(E)

gph(E),

where gph(E) = 8πn2s

h3c3E2 (n is the index of refraction) is the photon density of states and nph(E) =

U(E)E

is the number of photons around energy E. Substituting everything in gives

Rabs =Req

absE

U(E)× gphfeq =

8πn2

h3c2

α(E)E2

eβ(E−µ) − 1.

The upshot is that the nonequilibrium scenario is described by a nonzero chemical potential µ,and that the net recombination rate (subtracting away the rate at thermal equilibrium) is

Urad =

∫ ∞0

α(E)(be(E, µ)− b(E, 0))dE.

The µ came from the equilibrium distribution function feq(E) that we derived earlier. Note thatthis is a rate and not a total energy, because we are integrating photon numbers rather thanintegrating photon number times energy.

We can simplify this expression. For a nonequilibrium semiconductor, np = n2i eµ/kT . Therefore

Urad = Brad(np− n2i ),

where Brad is an integral of some things. This simplifies further in doped material. For example,if the semiconductor is n-type with doping density Nd, then the carrier densities in equilibriumare n0 = Nd, p0 = n2

i /Nd, thus

np− n2i = (Nd + ∆n)(n2

i /Nd + ∆n)− n2i ≈ Nd∆n

where ∆n is the density of both photogenerated electrons and holes. Thus

Urad =p− p0

τp,rad

, where τp,rad =1

BradNd

.

15


4.4.2 Auger recombination

In this kind of recombination, a collision between two similar carriers results in the excitation ofone carrier to a higher kinetic energy, and the other carrier drops down to the lower-energy bandand recombines with a carrier of opposite polarity. The excited carrier will lose the extra energyas heat and relax to the band edge. A picture for when the two initial carriers are electrons isshown below.

In this band-to-band Auger recombination, either an electron and two holes, or two electronsand a hole, are involved. Therefore the law of mass action says the net rate is

U1Aug = Ap(n

2p− n20p0) and U2

Aug = An(np2 − n0p20).

Unsurprisingly, Auger processes are important when carrier densities are high: in low bandgap,heavily doped materials, or at high-T . We can argue from the above rates (similarly to how weargued for the rate of spontaneous emission in doped semiconductors) that

τn,Aug =1

AnN2a

and τp,Aug =1

ApN2d

in p- and n-type materials, respectively.

Unlike pure radiative recombination, Auger recombination can occur in indirect bandgap materials.

4.4.3 Shockley-Read-Hall recombination

The most important recombination processes in real semiconductors (which contain impurities)and those which involve defects or trap states in the band gap. These are important because ifyou have a state inside the bandgap, the rates are exponentially increased compared to processesthat have to traverse the entire bandgap at once.

Perhaps the main physical difference between a trap and a band is that a band is delocalized,while a trap is spatially localized - literally, a defect in the semiconductor. We can think of thetrap as “capturing” otherwise free carriers. Traps can actually perform different kinds of processes.Localized states which mainly capture and release only one type of carrier are referred to as traps,and those which capture both types of carrier (so an electron and hole could recombine at thedefect site) are called recombination centers. Canonical traps may be closer to the bandgapedges, and recombination centers typically lie further into the bandgap.

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Let us derive the SRH recombination rate. This is a nice exercise in physical modeling.Consider a semiconductor with a single kind of trap of density Nt at energy Et in the bandgap,occupied with probability ft. If the trap is empty, it can capture electrons from the conductionband; if filled, it can capture holes from the valence band.

The rate at which electrons are trapped is therefore

Untrap = BnnNt(1− ft), where Bn = vnσn.

Here, vn is the thermal velocity of the electron, σn is the cross section or “scattering length” of asingle trap for electrons. The rate at which electrons are freed from the trap is

Gntrap =

Ntftτnesc

.

Here, τnesc is a characteristic release lifetime. We could estimate it based on energy barriers andamount of thermal vibration in the trap, etc. In quasi-equilibrium we demand that Un

trap = Gntrap,

which implies thatGn

trap = BnntNtft, where nt = nieβ(Et−Ei)

is the value of the electron density when the electron Fermi level is equal to the trap level. Notethat Gn

trap is independent of the density of conduction electrons, which makes sense because it isrelevant for electrons coming out of the trap state. A similar idea works for the holes, and we set

USRH = Untrap −Gn

trap = Uptrap −G

ptrap.

This fixes the value of ft, and we find that

USRH =np− n2

ip+ptBnNt

+ n+ntBpNt

.

For doped semiconductors, we find as before that

USRH =n− n0

(BnNt)−1for p-type, and USRH =

p− p0

(BpNt)−1for n-type.

In real materials, such traps are most likely to occur at surfaces and interfaces, due to brokenbonds and impurities, etc. We call an impurity an electron trap if the time for electron releaseis much longer than the time for capture of a hole, and vice versa for a hole trap. Electron trapsare close to the conduction band and hole traps are close to the valence band.

4.5 Transport equations

The most general form of Boltzmann transport equation is the drift diffusion equation modifiedby generation and recombination, i.e.

Dn∂xxn+ µnF∂xn+ µnn∂xF − U +G = 0,

Dp∂xxp+ µpF∂xp+ µpp∂xF − U +G = 0.

Here, F is the electric field. This is the equation being solved, along with the Poisson equation,in simulations of device physics such as the program DEVSIM that I used before.

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5 Junctions

How a solar cell works is this: even if we apply a bias voltage V across the device, the current canstill flow the “wrong” way and hence generate power rather than consuming it. This is because ofa built-in voltage - the voltage of the pn-junction. Once our bias voltage overpowers the built-involtage, the electrons will no longer have wind sweeping them to the “wrong” side and there is nomore power generated; instead, power is consumed.

There are several equivalent ways to think of the driving force. We can think of it as a light-induced gradient in the quasi-Fermi levels for electrons and holes; because the Fermi level is notspatially uniform, there will be a current. Alternatively, we can think of it as two paths: onewhich lets holes flow but blocks electrons, and one which lets electrons flow but blocks holes. Wewill think about these interpretations now.

5.1 Types of PV action

Recall the formula we derived in Ch. 3,

Jn(~r, t) = µnn∇~rEFn(~r, t).

Hence, the quasi-Fermi level must vary with position for there to be a current. We also saw thatthe above equation can be rephrased as the drift-diffusion equation, which in its most general formis

Jn = qDn∇(n− n0) + µn(n− n0)(qF −∇χ− kT∇ lnNc)

Jp = −qDp∇(p− p0) + µp(p− p0)(qF −∇χ+ kT∇ lnNv).

The first term represents diffusion and the second represents drift, which is due to the net electricfield, which is a composite of the electrostatic field as well as certain gradients. We may summarizethe possible effects as follows. The various conditions which can give rise to charge separation ina semiconductor are

• Gradient in the vacuum level or work function (equivalently, a change in doping)

• Gradient in the electron affinity

• Gradient in the bandgap

• Gradient in the band densities of states

The first three of these ideas are exploited in PV devices, through heterojunctions or homojunc-tions (pn-junctions, which use method #1). The pn-junction induces a gradient in the workfunction because the work function is defined as the energy required to remove the least tightlybound electron, or

Φw = Evac − EF .Since doping changes EF , it also changes the workfunction Φw. A nonzero electric field ~E isestablished at the interface between two materials with different workfunctions Φw, since EF isconstant in equilibrium, such that the potential energy difference across the junction is equal tothe difference in the work functions.

18


5.2 Metal-semiconductor junction

Let us explain how the ~E-field is established in a Schottky contact and Ohmic contact and how itcould cause photovoltaic action in a Schottky contact. A Schottky contact is a contact whichinhibits the flow of majority carriers, and an Ohmic contact is one which inhibits the flow ofminority carriers (and hence the current can easily pass in either direction). PV action dependson inhibiting the flow of majority carriers, so it is only possible in Schottky contacts. Let us fleshthis out in greater detail.

5.2.1 Schottky contact

Suppose we have an n-type semiconductor of workfunction Φn and a metal of workfunction Φm,such that Φm > Φn. The band will bend up when we get close to the metal. This will lower thedensity of charge carriers at the interface and hence the current which can cross the interface willbe very small. That is because the Fermi level EF will be farther away from the conduction band.

Figure 1: This ~E-field drives electrons to the right, holes to the left.

Similarly, we can construct a Schottky contact for p-type semiconductors with metal of lowerworkfunction Φm < Φp.

Let us consider how this Schottky contact could give us PV device physics. Electrons tend toflow to the right, so if we apply positive bias V < Φm − Φn to the left (which reduces the bandbending, because electrons have the opposite charge), then the electrons will still get swept to theright, which means the current flows to the left. Because the current flows in the direction of thebias (reverse of a resistor), we generate power.

However, Schottky contacts are not used for PV operation for a few reasons.

1. If the band bending is too large, then near the heterojunction, the valence band may getcloser than the conduction band to the Fermi level. This indicates a inversion layer whichsucks because then we cannot sustain a photovoltage.

2. Highly doped semiconductors may have an extremely thin depletion region, which meanstunneling effects could spoil the “sweeping” effect

3. The metal-semiconductor heterojunction tends to induce defect states.

19


5.2.2 Ohmic contact

An Ohmic contact can very easily pass current because the majority carrier density is highernear the junction; the band bends the other way. This occurs in an n-type semiconductor whenΦm < Φn, as shown below. It occurs in a p-type semiconductor when Φm > Φn.

Figure 2: This ~E-field drives electrons to the left, holes to the right.

Unfortunately, it is not possible to turn this into PV operation. That is because the voltagedifference from the edge of the semiconductor compared to the Fermi level of the metal is essentiallyzero, and hence there is no photovoltage.

5.3 Surface and interface states

Interfaces at heterojunctions and metal-semiconductor contacts always contain defects or impuri-ties. There are intrinsic defects due to the interruption of the crystal structure and extrinsicdefects due to adsorbed impurity atoms. Defects introduce new electronic states, and the ones ofinterest have energies lying within the bandgap. These surface states or interface states arelocalized near the interface and can trap charge. Because the important states lie in the bandgap,sometimes they are also called gap states.

5.3.1 Neutrality level, φ0

Gap states near the valence band tend to trap electrons and gap states near the conduction bandtend to trap holes. This is because of the distribution functions for electrons and holes.

We define the neutrality level, φ0, of a surface as follows: A surface may have many gap states,but when the surface is isolated from all external contact, the states are filled up to φ0, which canbe thought of as a Fermi level for the surface in isolation. If φ0 < EF , the surface is acceptor-likeand traps electrons; if φ0 > EF , the surface is donor-like and traps holes. See the figure below.(Question: how can we calculate φ0?)

20


Upon contact, the semiconductor undergoes band-bending just as it would as if it was in contactwith a metal.

There is negative surface charge density on the surface and positive volume charge density in thedepletion region of the semiconductor. These are supposed to add to zero, so there is no ~E-fieldin the bulk of the semiconductor. (Actually, the diagrams I drew here are only valid when thedensity of interface states is very high and the interface states can actually accomodate such largetransfer of charge. If the density of interface states is not so high, then we might have φ0 6= EFeven in direct contact. There will still be some nonzero band-bending; the bending will simplynot be enough to reach the φ0 = EF condition.)

5.3.2 Fermi-level pinning and other effects

The above considered a surface between a semiconductor and vacuum. Of course, the more impor-tant surfaces are those involved in heterojunctions. Surface states cannot alter the net potentialdifference across a heterojunction, which must always equal the difference in work functions.However, they can influence the way in which that potential difference is divided between the twomaterials.

Let us make two subtle arguments to understand the behavior of surface states at a (hetero)junction.

Claim: If the density of interface states is very large, then the Fermi level at the interface will bepinned at φ0 regardless of the value of the workfunctions.Reasoning: This is obvious from my picture above. Actually I think the point is that my pictureabove, where EF = φ0, only holds when the density of interface states is really really high, so wecan effectively treat the surface like a metal contact.

Applying this to metal-semiconductor contacts, we see that Fermi-level pinning implies that theinterface charge will screen the metal from the semiconductor and the built-in bias voltage will be

21


independent of the workfunction of the metal, called the Bardeen limit for metal-semiconductorcontacts. In the opposite limit (no surface states), any change in the metal workfunction causes anequal change in the built-in bias voltage, as we suggested above when we studied ideal Schottkycontacts.

Claim: If the density of interface states is very large and φ0 is low enough (i.e. quite near thevalence band), then the entire potential difference in a pn-junction can be dropped on the n side.Reasoning: What even does this mean? I think it means the following...

Figure 3: We can see that all the bending happens on the n side.

6 Analysis of the pn-junction

I know this stuff already from Razeghi, so I will skip much of it. Nelson gives more detail aboutnonequilibrium pn-junctions under bias and under illumination, so I will study that in depth.

As always, the equations of device physics consist of (1) Poisson equation for the potential (2)generalized drift-diffusion equations which account for various generation and recombination pro-cesses. The classical solution for the pn-junction can be found analytically with two approxima-tions: (1) depletion approximation (2) recombination in the doped material assumed to be linear.There is only one way in which the pn junction solution under illumination differs from the solutionin the dark: the generation of carriers via photogeneration.

6.1 A very complicated solution

The net current is composed of three parts: (1) electron current at the p-edge of the depletionregion (2) hole current at the n-edge of the depletion region (3) space charge region (SCR) currentinside the depletion region. Basically, the electron and hole currents are calculated in the n- andp-type regions respectively, and the SCR current is calculated in the depletion region. We splitthese over energy:

Jn(x) =

∫jn(E, x)dE in the n-type region,

Jp(x) =

∫jp(E, x)dE in the p-type region,

Jscr = q

∫ wn

−wp(U −G)dx =

∫dEdxjscr(E, x) = in the depletion region.

22


The total current density J is defined with an extra minus sign, since it is a photocurrent and isdefined to be positive when flowing from p to n through the external circuit. It is opposite to theregular engineering convention. The three parts arise because we shift one of the currents to the“other end” of the pn-junction, as described below:

J = −Jn(−wp)− Jp(−wp) = −Jn(−wp)− Jp(wn)− Jscr.

In other words, Jscr is the difference in hole current (or electron current) from one end of thepn-junction to the other. The ugly explicit solutions are given in the text, but I will give only theapproximate solutions. I think they will be good enough.

6.1.1 Why write it this way?

This form of the current J(V ) makes the dependence of the current on the light photogenerationmore obvious. Here is why:

• The electrons flow from the n-side to the p-side due to the built-in bias of the pn-junction.Therefore, we would like to find the electron flow at the p side, which is where the electronsexit. Those are the electrons that make it into the external circuit, so to speak. The holesflow from the p-side to the n-side, and they exit at the n side. We could also consider wherethe electrons “start,” but then we would have to account for recombination as the electronmakes its way through a treacherous region of mischevious holes. Actually, this is exactlythe contribution of the Jscr term, as you can see from its definition above.

• The density of conduction electrons is very small at x = −wp, since that is the edge of thep-doped region, where there are mostly holes. Similarly, the density of holes is very smallat x = wn, since that is the edge of the n-doped region, where there are mostly electrons.Therefore, Jn(−wp) and Jp(wn) are very small, because there are barely any carriers tocarry the current. The largest contribution to J , when we write it this way, is Jscr, whichmanifestly contains the photogeneration term inside the depletion region of the pn-junction.It will help us see, for example, why the current should be nearly voltage-independent in theoperating region of the PV device.

6.2 Approximate form in the dark

It turns out that

Jn(−wp) =qn2

iDn

NaLn(eβqV − 1) and Jp(wn) =

qn2iDp

NdLp(eβqV − 1),

where β = (kT )−1. Because both of these contributions are due to diffusion (i.e. from the ~E-fieldin the depletion region), we can group them together schematically as

Jdiff(V ) = J0diff(eβqV − 1).

We knew this already from Razeghi. The recombination current in the space charge region is

Jscr(V ) = J0scr(e

12βqV − 1), where J0

scr =qni(wn + wp)√

τnτp,

23


where wn,p are the widths of the depletion region in the respective halves of the pn-junction andτn,p are the recombination lifetimes and are related to the diffusivity.

The diffusive contribution is typically the most important one and dominates when very lit-tle recombination occurs in the depletion region. This is the Shockley diode limit. The scr-contribution hypothetically could dominate if there is a lot of recombination.

6.3 Approximate form in the light

When the junction is illuminated, light creates electron-hole pairs in all three regions: n, p, anddepletion region. In this scenario, n and p concentrations are enhanced in all regions and theelectron and hole quasi-Fermi levels get split (such that the product of their concentrations ishigher).

This follows because the drift-diffusion equation gets enhanced by a photogeneration (ignoring therecombination). For example,

∂xxn−n− n0

L2n

+g(E, x)

Dn

= 0 in the p-region, x < −wp.g(E, x) = (1−R(E))α(E)bs(E)e−α(E)x.

We match this to the boundary condition

n− n0 =n2i

Na

(eβqV − 1) at the p-depletion region boundary, x = −wp.

This follows because EFn − EFp = qV throughout the entire space charge region, and becausen2i /Na is the equilibrium electron charge density in the p region.

6.3.1 Form of PV device J(V )

If we write the short-circuit spectral photocurrent as

jsc(E) = −jn(E,−wp)− jp(E,wn)− jgen(E),

then the net photocurrent is

Jsc =

∫ ∞0

dEjsc(E).

That was for a short circuit. How about for a circuit connected to nontrivial load? It turns outthat in the depletion approximation, the dark current and short-circuit photocurrent add linearly.Therefore,

J(V ) = Jsc − J0diff(eβqV − 1).

The corresponding open-circuit voltage is

Voc =kT

qln(

Jsc

J0diff

+ 1).

For very negative voltage, this approaches the constant

J(V )→ Jsc + Jdiff, which justifies treating the PV device as a current source.

24


6.3.2 Effects of external parameters

In this subsection, we study how the J(V ) characteristic responds to the varying of certain externalparameters.

• Intensity of irradiation: If the photogeneration and recombination is assumed linear, thephotocurrent Jsc increases linearly with the intensity of light. The open-circuit voltagetherefore increases logarithmically and the cell efficiency also increases. (This is because thecell efficiency is roughly η ∝ VmJm

b, where b is the intensity.)

• Temperature: An increase in temperature decreases the photoefficiency. This is becauseincreasing the temperature increases the equilibrium population of carriers; this increasesJsc but decreases Voc. The decreases of Voc is stronger because the diffusion current (whichcontrols Voc goes like n2

i but the photocurrent (which controls Jsc) goes like ni.

• Parasitic resistance(s): We can introduce the effect of a parasitic internal resistance (i.e. aseries resistance) by introducing a distinction between voltage across the terminals, Vterm,and the ideal voltage V . Then

Vterm = V − JAr,

where r is the parasitic resistance.

6.3.3 Ideas for design

Now that we know how to analyze a pn-junction under illumination, how should we construct it?What regions should be heavily or lightly doped, how thick should they be, and how should weorient the PV device relative to the sun?

Claim: The optimal cell design is that the sunlight should hit the pn-junction at normal incidenceto the plane of the junction. The front surface should be very thin (thinner than the diffusionlength) and very highly doped. The back surface should be thick and lightly doped. Whether thep or n side should be at the front or back depends on the carrier mobilities, which essentially isdue to the different effective masses.Reasoning: For simplicity, let the front surface be the p-doped semiconductor. In the neutrallayers, only carrier electrons generated within one diffusion length Lp can survive and make itto the metal contact. Therefore if we make the width of the front region longer than a diffusionlength, we won’t get any current! We should make the width as short as possible, xp � Ln.

Also, the front region should be as short as possible so the pn-junction can be closer to the impactof the sunlight.

You would think we should also make the width of the back surface, xn, as short as possible.Wrong! This would let lots of the light escape. We can only trap the light if the total width of thepn-junction is long enough (recall the e−αx-dependence of the intensity, where x is the penetrationdepth into the solar cell). To overcome this problem, we make xn relatively long but also makethe doping Nd, or majority carrier electrons, very low. Because there are fewer mines to avoid,this means the diffusion length Lp of the holes in the n-type region can be quite long, and we canreduce the problem of having a large xn but still requiring it to be less than the diffusion length.

25


There is one more question to answer. Why should the doping in the front region be high? Ithink if nothing else, a higher doping in the front region can give us a stronger built-in voltageand hence more power.

6.4 Simple derivation of the J(V ) characteristic

???

7 Monocrystalline solar cells

The most common solar cell design is based on a monocrystalline pn-type homojunction. We likehomojunctions because there is no material interface at the junction, and thus avoid the problemof interface states, which could lead to very high levels of recombination in the most importantpart of the pn-junction.

We will study the two most important examples: (weakly absorbing) monocrystalline silicon and(strongly absorbing) gallium arsenide.

7.1 Characteristics of an ideal solar cell

• The reflectivity of the surface, R(E), should be small. Often this is remedied by introducinganti-reflection coatings and thin films.

• The optical depth of the device, described by α(E)(xp + xn), should be high for energiesE > Eg above the band gap.

• The built-in bias Vbi should be as large as possible. That means both the p and n regionsshould be heavily doped (but as we saw in the previous chapter, this can hurt us throughother effects such as a short diffusion length).

• Minority carrier lifetimes and diffusion lengths should be long. (Recall that majority carriersdon’t really have “diffusion legnths” because there are barely any minority carriers for themto bump into.) Surface recombination velocities should be small.

• The bandgap should be close to optimum.

7.2 Silicon: material properties

Though Si bandgap (1.1 eV) is fairly close to optimal, its bandgap is indirect, so a phonon mustassist in the photon absorption process (the direct transition would require a very high energy - 3eV). This makes Si absorb photons less efficiently. That is why its reflectivity, around R(E) ≈ 40%over visible wavelengths, is so high.

The fundamental impurities in silicon are of acceptor type and thus are more important in n-doped silicon than in p-doped silicon. Thus, the electron mobility in p-type is higher than the

26


hole mobility in n-type. The minority electron diffusion length is longer than the minority holediffusion length, so we prefer to create np-cells (first letter exposed directly to sunlight) ratherthan pn-cells. That is because recombination is really a problem in the thick “second” region.

The dominant form of recombination, in all but the very purest silicon, are trap-assisted nonra-diative processes. This includes SRH recombination.

7.3 Silicon: solar cell design

For the reasons described above, a silicon solar cell is an np-junction made in a wafer of p-typesilicon; the n-region is prepared by diffusing the n-type dopant (usually P) onto the surface of thep-type wafer. This means the junction will not be so abrupt, and our mental picture of pn-junctionwill get a bit smeared out.

In fact, carrier collection is negligible from the n-type emitter. Why? It turns out that when youdope the n-type silicon very heavily, you introduce so many impurities that the recombination issuper high. I think the attitude is to just give up on trying to produce current from the n-typebulk region and dope as much as possible to decrease the resistance.

Dimensions: The p-type wafer, or base, is perhaps 400µm thick and is doped to around Na ∼1016 cm−3. One cell is around 100 cm2 in area. The n-type emitter is doped to around Nd ∼1019 cm−3.

Treatment with extra clothes: The front surface is anti-reflection coated, and both front andback surfaces are contacted (i.e. you put the contacts on) and then put a thin glass covering ontop of Si-contact device. We also texture the front surface, as described below.

7.3.1 Improving the Si solar cell

The main challenges in crystalline solar cell design are to (1) maximize absorption (2) minimizesurface recombination (3) minimize series resistance of the device. We discuss the approaches totackling these problems below.

1. To maximize absorption, texture the front surface. This reduces the net reflection of light(by making light come in at an angle; i.e. with different Fresnel coefficients) and increasesthe optical depth of the cell (by making the light enter the PV device at an angle, whichmeans it can travel longer before hitting the back surface). We can do this by treating the〈111〉-Si surface with a chemical that etches away some of the crystal plane, leaving us witha superlattice of pyramids. (Even better is a superlattice of inverted pyramids.)

2. To maximize absorption, optimize the contacts. Obviously, the more surface area the contactstake up, the less area there is for PV action. A good solution is to make grooves in the n-region and put the contacts inside the grooves. In this way, one contact can have a muchlarger interfacial surface area with the semiconductor than if it was just sitting on top. Thegrooves are doped very heavily to try to improve conductivity over the junction, like anOhmic contact. However, “screen printing” is much cheaper.

3. To reduce rear surface recombination, heavily dope the back surface of the p-type base. Thisintroduces a p+− p type junction and presents a potential barrier to the minority electrons,

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so they travel more slowly once they approach the back contact. This lowers recombinationof the minority electrons with surface states due to the junction. (Also, the extra junctionmay enhance the built-in bias voltage!)

4. To reduce front surface recombination, coat the front with SiO2. A silicon-silicon dioxideinterface tends to have less defects than a hanging silicon surface or a silicon-metal interface.The silicon oxide, which has a large bandgap, prevents carriers from reaching the surfaceand recombining.

5. To reduce rear surface recombination, use point contacts. Basically, cover the entire backsurface with a thin film of silicon oxide, then poke holes in it and insert the metal contacts,which may be differentially doped (i.e. have a highly-doped p-type silicon on the end) toimprove conductivity. This allows most of the backside to be protected by the oxide coating,decreasing the number of trap states.

6. To reduce series resistance, apply differential doping around the contacts. Essentially, if themetal will touch an n-type semiconductor, expose the bare n-type semiconductor to highlyconcentrated, gas-phase n-type dopant before you insert the contact. This will make theimmediate vicinity of the metal very highly doped, improving conductivity, or equivalently,decreasing intrinsic parasitic resistance.

Incorporating all these improvements, record efficiencies for monocrystalline Si cells are around20-25% as of 2000. Because it is now 2019, hopefully people are doing even better (and loweringthe cost of state-of-the-art technologies). This is not far away from the theoretical maximum of29%.

I will not describe these types of cells, but let me just namedrop them: black cells, passivatedemitter cells, rear point contact cell, PERL cell (highest efficiency, perhaps).

7.3.2 Problems with Si solar cells

• Si has quite high reflectivity (due to the indirect bandgap). This means we must use quitethick layers (which is costly and bulky/heavy).

• Bandgap is a little off from the optimum.

7.4 GaAs: material properties

GaAs has a direct bandgap of 1.42eV at room temperature. Its theoretical conversion efficiency is31%, and over visible wavelengths the absorption coefficient α is around 10 times that of silicon.Therefore, only a few µm of GaAs are needed for a solar cell, rather than a few hundred µm. Thisis good when weight is important. However, currently a GaAs wafer that is 8 inches in diametercan run $5,000; in contrast, that same wafer, made out of silicon, can be as cheap as $5. Thismore than offsets the difference in thickness, perhaps by a factor of ten.

Recombination is a little different in GaAs compared to Si. In GaAs, SRH recombination in thedepletion region dominates (since absorption is high, so there are lots of “minority” carriers in the

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depletion region), so the dark current scales like eqV/2kT . In Si, SRH recombination in the neutralregions dominates so the dark current scales like eqV/kT .

Similar to Si, in GaAs the minority electrons have higher mobility than minority holes. However,the absorption in GaAs is so strong that we don’t need to worry about the width of the depletionregion exceeding the diffusion length, and hence both np- and pn-type devices are acceptable.However, usualy pn-type devices seem to perform better.

7.5 GaAs: solar cell design

Dimensions: Typically, the emitter is doped to 1018 cm−3 and has thickness around 0.2−0.5µm.The base is doped to around 1017 cm−3, which is still quite high, and has thickness around 4µm.

7.5.1 Improving the GaAs solar cell

The principal challenges with the GaAs solar cell are: (1) minimize front surface recombination(In this case, the front surface recombination is important because the absorption is so high. In Si,the back surface recombination dominated because the absorption was low, so light can more easilywriggle its way to the back.) (2) minimize junction recombination (3) minimize series resistance(4) minimize substrate cost.

Let us see what nifty tricks people have already thought of.

• Minimize front surface recombination by using a heavily doped layer in the front. This mirrorsthe back-doping process for Si.

• Minimize front surface recombination by using a window layer. This is a higher band-gapmaterial coating the front surface, which is supposed to reflect minority carriers away. Thisis slightly similar to coating with SiO2, except not so extreme. The most common materialsfor “window layers” are AlGaAs and InGaP, both of which are lattice-matched, and thus donot introduce so many interface states.

• Minimize substrate cost by choosing Ge as the substrate for growing GaAs, rather than GaAs.We must use a substrate to grow the very thin GaAs layers because they are so thin. Isuppose we don’t need a substrate for the thick Si layer and can instead whittle it downfrom mass-produced Si, for instance from Czochralski method of fabrication.

Actually, Ge is still very expensive. Apparently the cost of GaAs on Ge solar cells is morelikely to be limited by the cost of Ge rather than the cost of GaAs, even though GaAs ismore expensive from a per-volume standpoint! People are trying to find ways around this,such as by cleaving GaAs from a GaAs substrate and hence reusing the substrate. Thelayers, however, tend to be brittle.

8 Thin film solar cells

The previous chapter considered monocrystalline solar cells. However, single crystals are expensiveto produce, and we are interested in finding PV materials of less demanding material quality.

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The best candidates identified so far are amorphous Silicon (a-Si), polycrystalline cadmium tel-luride, polycrystalline copper indium diselenide, and microcrystalline thin-film silicon (p-Si). Be-cause they are thin-film materials, they can be produced by PVD or CVD which is much quickerthan growing something by Czochralski, for example.

Polycrystalline and amorphous semiconductors contain a much higher density of defects. Thisincreases the density of traps and recombination centers. Therefore (1) diffusion lengths areshorter, so such materials need to be very good absorbers (2) losses in front surface are verygreat, so it is advantageous to replace the emitter with a wide bandgap window material (3) thepresence of so many defects can make the materials difficult to dope, so the built-in bias may belimited through Fermi level pinning (4) grain boundaries and other defects increase the intrinsicresistance (5) the presence of defects implies that minority carrier diffusion lengths are carrier-density dependent. Hence, the transport equations become more complicated and are no longerlinear in the minority carrier density.

This chapter is maybe not so important - it just shows how the usual semiconductor theory breaksdown for non-monocrystalline materials. I will just summarize the main points.

8.1 Amorphous silicon (a-Si)

The absence of long-range crystalline order means that a-Si is a direct bandgap material, andhence a much stronger absorber than monocrystalline silicon. However, there are so many danglingbonds from the defects that it is impossible to dope unpassivated a-Si. The solution is to sendin interstitial Hydrogen, which forms a bond with the unpaired electron in a neutral defect.Passivation with H reduces the density of dangling bonds, such that dopants are less likely to gettrapped. After passivation, pn-junctions can be fabricated.

The basic a-Si solar cell is a p− i− n junction, where i stands for “intrinsic,” or equivalently anundoped region. The built-in bias is dropped over the width of the i-region. We insert an i regionbecause the diffusion lengths are so short, so most of the photoaction has to be done when thereis no doping (i.e. in the depletion region), so we would like the depletion region to be thicker andhence absorb more light. However, it can’t be too thick; otherwise the thickness will exceed thespace charge with.

Conclusion: all the current collection in an a-Si solar cell is due to Jscr rather than the currentsin the bulk n or p regions, since their diffusion lengths are so short.

8.1.1 Grain boundaries in polycrystalline materials

Actually I am sick of studying a-Si solar cells, but I think this information on the effects of grainboundaries is interesting. Let us see what grain boundaries do. They are relevant not in a-Si butin p-Si, polycrystalline silicon.

First of all, a grain boundary is basically a surface in the lattice which separates two “grains”of different orientation. Ideally, all the hanging bonds and stuff lie along grain boundaries. Thematerial is crystalline over the width of a grain, which is typically around 1 µm in p-Si. Thegrain is large relative to the wavelength λ of traveling electrons and holes, so the band structureof polycrystalline silicon is exactly the same as in monocrystalline silicon. However, the transport

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and recombination properties are strongly affected by the grain boundaries.

That is because the grain boundaries introduce extra electronic states which are spatially localizedand may have energies in the band gap. Unsurprisingly, a ton of trap states are located at thegrain boundaries. “Deep traps” are those near the center of the bandgap, which can captureelectrons or holes.

These intra-band gap states can trap charge, and therefore they influence the potential distributionnear the grain boundary. Generally, defect states at a surface in doped material tend to trapmajority carriers and establish an electrostatic field ~Eback opposing majority carrier flow. Thesame is true with grain boundaries, except it is kind of like the interface between two surfacesback-to-back. Suppose the majority carriers are holes. Similar to a 2DEG, this implies a planeof fixed positive charge at the interface and a layer of negative space charge on either side of theinterface, where the p-type material has been depleted (equivalently, where the holes have been“pushed away”). Conversely, minority carriers are pulled towards the trap states because theywould like to get married to the trapped majority carriers. Weird.

This may sound crazy, but a grain boundary of sufficient trap density can create a depletion regionaround it, in a doped material. If the material is doped n-type, the depletion region sandwichapproximately looks like

n− depletion region− n.

That is because after the trap states have captured n-carriers, all the other n-carriers tend torun away. Because the n-carriers are the majority carriers, the space charge region immediatelysurrounding the grain boundary interface is depleted. We can interpret this as a consequence ofFermi-level pinning, shown below.

Figure 4: Initial band diagram of isolated n-type semiconductors, not in contact with the interfacestates with neutrality level φ0.

Figure 5: The depletion region can be interpreted as the bending of doped semiconductor toaccomodate Fermi-level pinning, EF → φ0.

Grain boundaries also affect transport. Their effect actually depends on whether the current isflowing normal to the grain boundary or parallel to it.

• If the current flows across a grain boundary, the potential barrier slows down the transportof majority carriers and decreases conductivity.

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• If the current flows along a grain boundary, the minority carriers are affected, but majoritycarriers are not. That is because the majority carriers get pushed away so they don’t getslowed down. However, minority carriers get sucked in to the grain boundary trap states torecombine and find a significant other.

9 Managing light

We would like to find all kinds of tricks to increase the overall absorption of light. This givesus more power generated for a solar cell of fixed thickness. I am not interested in mathematicaldetails here; just the ideas.

Let us introduce (or review) our notation. Let the width of the active layer of semiconductor bew and let the incoming intensity spectrum of photons be bs(E). The number of photons reflectedaway from the top surface is R(E)bs(E). If the intensity at a distance x into the semiconductor isb(E, x), we found earlier that

b(E, x) = (1−R(E))bs(E)e−∫ x0 α(E,x̃)dx̃.

Here, α is the absorption coefficient, so if every photon participates in photogeneration, this meanselectron-hole pairs are generated at a rate g(E, x) = α(E, x)b(E, x). For a cell of thickness w, afraction

fabs(E) = 1−R(E)− b(E,w)

bs(E)= (1−R(E))(1− e−α(E)w)

of incident photons of energy E are absorbed, where the second equality specializes to the uniformcase. As we mentioned in an earlier chapter, increasing b(E, x) by a factor X approximatelyincreases the current by X and the open-circuit voltage by ln(X). Because power is current timesvoltage, the efficiency increases by ln(X).

We can increase the photon flux by three general processes: minimizing reflection, concentration,trapping, and photon recycling. We will describe these in detail in the next sections.

9.1 Minimizing reflection

Reflection can be minimized by covering the surface of the semiconductor with a layer of anti-reflective thin film. The optimal thickness of the thin film, for the specific indices of refraction ofthe semiconductor bulk and thin film, can be determined via the Fresnel equations.

Let us summarize these famous results. Let the indices of refraction of the air, thin film, andsemiconductor be n0, n1, ns, respectively. Let the thin film have thickness d and let the wavelengthof the photon in question be λ in vacuum. We can find, via elementary arguments (i.e. Hallidayand Resnick) that maximal destructive interference between light reflected from the surface of thethin film and the surface of the semiconductor occurs when the thickness of the thin film, d, isprecisely when the film is a quarter-wavelength long. The Fresnel equations turn out to give shouldthere be a square on the not-squared term in the denominator? I couldn’t find good referencesonline.

R =(n0 − ns)2 + (n0ns/n1 − n1)2 tan2 δ

(n0 + ns)2 + (n0ns/n1 + n1) tan2 δ, where δ =

2πn1

λd cos θ.

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R has a minimum when δ = π2. In this case,

R|δ=π2

=(n0ns/n1 − n1)2

(n0ns/n1 + n1)

which vanishes when n1 =√n0ns. Thus, the coating on the semiconductor should be n1 =

√ns,

with d chosen such that the wavelength which is totally reflected is in the middle of the rangeof wavelengths which can be usefully absorbed for that semiconductor. An anti-reflective (AR)coating on monocrystalline Si is designed to let in as much red light as possible. It happens toreflect blue or violet light, which is why solar cells look dark blue.

Better anti-reflective coatings can be achieved by using multiple coatings. The range of wavelengthswhich are almost perfectly transmitted can be much wider.

9.2 Increasing photon concentration

Concentration of light means using geometrical ray optics to increase the intensity of light hittingthe solar cell. This can be achieved using regular lenses or parabolic reflectors, for example. Itincreases the “effective capture area” of the solar cell. If the incident flux density at the cell surfaceincreases by factor X, the power increases by approximately the same factor, X.

The kind of concentration we use affects the solar cell design. For example, lenses can only focusparallel incoming light (around 85% of incoming light is not scattered by the atmosphere) and theuse of a lens necessitates a tracking system to follow the sun throughout the day and throughoutthe seasons. Certain kinds of parabolic reflectors can handle all kinds of incoming light, but onlywithin a certain angular half-“cone.” Kind of like the lightcone from special relativity, but inspace rather than spacetime. These reflectors do not require a tracking device.

In fact, concentration can increase the intensity delivered to the PV device by perhaps hundredsor thousands of suns. This means that photogeneration may be so strong that we might needto account for a nontrivially high concentration of minority carriers when thinking about the de-vice physics. The transport equations may look a little different. For example, the rate of SRHrecombination is still linear, implying a lifetime which is not directly dependent on carrier concen-tration (though it depends on carrier lifetime, which in turn depends on carrier concentration).However, the rates of radiative and Auger recombination go like n2 and n3, respectively, whichimplies lifetimes which go like n−1 and n−2, respectively.

In practice, concentration does not significantly improve the efficiency of typical Si or GaAs cells(measured in electrical energy per total photon energy), though it does of course improve thepower output per cell, since it increases the total photon energy. An issue with concentration isthe danger of raising the temperature of the solar cell. We recall from previous chapters that thiscould be a problem because it increases the intrinsic carrier density, which turns out to decreasesthe photoefficiency.

9.3 Photon trapping (photon confinement)

The simplest devices to use for photon trapping are mirrors and textured surfaces. In Si, wherecells are hundreds of microns thick, the characteristic size of the cell is much longer than the

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coherence length of the light, and hence ray-tracing (ignoring possible destructive interferenceeffects) is valid. The idea of photon confinement is to make the light ray stay in the PV devicefor longer, so it has more time to give more of its energy to the solar cell.

We can introduce a mirror at the back end of the solar cell. This effectively doubles the pathlength. A more sophisticated method is to attempt to exploit total internal reflection at the frontsurface; unfortunately, this is impossible for cells with parallel planar surfaces. This could beachieved by scattering the light within the cell. It can also be achieved by making the back end ofthe solar cell rough or textured, which makes the light scattered after it comes off the back end.This is typically achieved by using aluminum or gold.

Another, more creative way to do things is the “photovoltaic eye,” diagrammed below:

9.4 Photon recycling

Photon recycling is the reabsorption of photons emitted by radiative recombination inside thecell. In materials where nonradiative recombination dominates, we may ignore the spontaneousgeneration of light, but in highly nonequilibrium scenarios, this may not work.

Actually, photon recycling is an intrinsic process, unlike, say, light concentration. Photon recyclingclearly increases with average ray path length, and therefore increases with the degree of lighttrapping. However, the produced photons from spontaneous emission tend not to have a preferreddirection. Therefore, reflectors which simply reflect light straight back into the material will bemore effective than reflectors which have an operating principle predicated on directional influence.

10 Into the future: strategies for high efficiency

In this chapter, we investigate possible avenues of exploration for increasing the theoretical max-imum efficiency of solar cells. A hopeful by-product of such investigations in universities andcompany or government laboratories is the decrease in cost of existing solar cells. I am not inter-ested in mathematical details here; just the ideas.

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The theoretical limit on efficiency in a standard air mass 1.5 spectrum, or AM 1.5 spectrum, is33%. The majority of the sun’s energy is lost by:

• Failure to capture the energy of photons with energy smaller than the bandgap, E < Eg.This wastes around 23% of the total energy.

• Thermal dissipation of the carriers excited by photons with energy larger than the bandgap,E > Eg. This wastes around 33% of the total energy.

These problems are extraordinarily difficult to surmount. The main approaches are as follows.

• Tandem or “multi-band” solar cells. Use multiple bandgaps to try and capture differentphoton energies more efficiently.

• “Hot carrier” solar cells. Reduce the dissipation of thermal energy by photogenerated car-riers.

• Impact ionization solar cells. Increase the number of electron-hole pairs per photon.

Only tandem solar cells have been realized in practice (as of 2000). There is, however, no a priorireason that the other techniques cannot also be realized in the future. We can summarize theseefforts as an attempt to take the limiting theoretical efficiency higher than that of a simple singlepn-junction efficiency, and perhaps reach for the maximum Carnot efficiency.

Just for kicks, let’s see how high the Carnot efficiency of a perfect solar energy converter is, andcompare it to the maximum efficiency of a pn-junction based PV device.

Let the temperature of the sun be Ts, the temperature of the environment be Ta, and the temper-ature of the device be Tc. By Carnot’s argument, the work available is

W = (σsT4s − σsT 4

c )(1− TaTc

) =⇒ η =W

σsT 4s

= (1− (TcTa

)4)(1− TaTc

).

For Ts = 5760 K and Ta = 300 K, this has a maximum of 85% for the absurdly high devicetemperature Tc = 2470 K. For a single bandgap, the maximum theoretical efficiency is around30%.

10.1 Multiple band gaps

Actually, I wondered about multiple band gaps before reading about it in the book, and essentiallyI reinvented the idea of tandem cells myself. Just a few decades late.

The idea of multiple band gaps is to split the sunlight into different channels, and feed each channelinto the solar cell with the optimal bandgap to minimize the loss of energy due to thermalization.Each electron could be extracted with a chemical potential closer to the energy provided by theoriginal photon and a higher overall efficiency would result.

However, it is rather difficult to achieve efficient spectral splitting. A more practical strategyis to stack different bandgap junctions in series and let the wider bandgap materials filter out

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the high energy photons, while less energetic photons pass through (they are not as likely to beabsorbed because their energies are less than the bandgap; however, if their energy is really low,then they may be absorbed in intraband transitions, which are useless for PV power conversion).A schematic is provided below for the “two-terminal, two-band tandem solar cell:”

In practice, the middle contact between the two solar cells is often a highly-doped pn-junction(between two different semiconductors, of course), which creates an Ohmic contact between the pterminal of one cell and the n terminal of the other. So in real life, the above solar cell looks morelike two boxes stacked on top of the other.

10.2 Intermediate band and multiple band cells

The chief difficulty with the tandem cell idea is that the different cells must be made of differ-ent materials, since they have different bandgaps. This leads to issues with mismatch of latticeconstants, unavoidability of interface trap states, etc.

A theoretical idea to cure this disease is to achieve the idea of multiple band gaps, but with asingle junction in a single material. A hypothetical solution is to use a material which containsmore than two bands. For example, suppose there are three bands, and that at T = 0 only thebottom band is filled. Under illumination, the electrons can jump from the bottom to middle,bottom to top, or middle to top bands. If these are associated with different bandgaps, it may bepossible to pick up more light.

Theoretically, people think they could achieve such a bandstructure by introducing impuritiesinto a material with large bandgap or by exploiting low-dimensional physics. This is the idea of“quantum well” solar cells. Whatever feature produces such a bandstructure should, obviously,be periodic in space.

10.3 Hot carriers

The idea of hot carriers is that perhaps we could harness some of the excess kinetic energy of thephotogenerated carriers before they relax. This is because the current Jsc depends on the speed

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of the carriers. Higher speed, higher current. People have thought up two possible methods ofachieving this:

• Slow down electron-phonon interactions such that the photogenerated carriers can be col-lected while still “hot.” This would give us an increased voltage.

• Use the excess kinetic energy to knock electrons out of atoms and hence generate morecarrier pairs. This is known as impact ionization. It would give us an increased current.

Actually the theory of nonequilibrium transport of hot carriers is pretty interesting, so I shouldread some papers on it.

Let’s see how people plan to achieve hot carrier PV devices. We would like a “photoactive” materialwhere intraband cooling of photogenerated carriers is slower than transport to the contacts (eitherby slowing down the cooling, or cutting the transport time to the contacts) and also contactmaterials which permit selective electron or hole extraction through a narrow energy band(s).Why the latter? We need the contacts to allow adiabatic cooling of hot carriers during extraction.I think the point is that if the bands are narrow, then the electrons will not experience intra-bandscattering, and if the bands are well-separated, then the electrons will not experience inter-bandscattering by phonons. Why this is important in the contact is a little bit of a mystery to me.Perhaps it’s because if you have a regular metal or highly-doped semiconductor with a wide band,then there must be many dangling bonds in the bandgap near the interface. That would stop thecurrent at the junction. Thus, people like to use superlattice structures for the contacts, wherethe density of states is quantized into bands.

In fact, for the same reason, people would like to use superlattice structures of quantum dots orquantum wells / wires to create the bulk material. This is called the phonon bottleneck effectand means that if we tune the parameters of the low-dimensional device correctly, the energyspectrum of the phonons will lie outside the necessary energy to produce intraband or interbandtransitions.

An alternative strategy is to make use of structures where the charge separation is very fast. Ifthis is true, then the structure must be very thin, and hence have very high absorption. Fastcharge separation is observed in dye-sensitized solar cells, but not high optical absorption.

The design ideas for hot electrons also apply to impact ionization solar cells. In a big-pictureway, this is because if photogenerated conduction electrons are to knock out valence electronsinto the conduction band, they must be really hot. Impact ionization, kind of humorously, iscalled Auger generation and is the reverse process of Auger recombination. (Of course! Augerrecombination destroys an electron-hole pair, so the reverse process must create one.) Becausethe valence electron must gain energy Eg, the hot electron must have energy E > Ec + Eg. Theidea can be restated thusly: can we make the quantum efficiency QE(E) for light with E > 2Eggreater than unity?

Auger generation has been observed in Si photodiodes for Ephoton > 3.3 eV, which is certainlygreater than 2× Eg ≈ 2.2 eV.

There is another requirement: the electronic band structure should favor impact ionization. How?The material should have an indirect bandgap (such as Si), because there is a large transfer ofphonon energy ωq and presumably this comes with a large and nontrivial q. The material would

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preferably have valence and conduction bands parallel over a relatively wide range of wavevectork. This is because if we think about Auger generation,

the process can occur over a larger range of wavevectors if the middle region of the dispersionrelation is flatter. Dispersion relations are parabolic around their extrema, so this is equivalent tofinding a rather flat parabola. Engineering such a bandstructure is a real challenge and perhapssuccess will be achieved soon.

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Notes on Nelson’s The Physics of Solar Cells · 2019-07-27 · Notes on Nelson’s The Physics of Solar Cells Jimmy Qin Summer 2019 And God said, Let there be PV devices: and there