I. Physical bases of the photovoltaic effect

The photovoltaic effect regards the transformation of the electromagnetic radiation (of light,

named phos in Greek) in electricity/potential difference (which is measured in Volt, in honor

of the Italian physicist Alessandro Volta, one of the promoters of electricity studies). The

figure below illustrates this effect. The photovoltaic effect was discovered by E. Becquerel in

1839, and the first device was available in 1877. However, the photovoltaic effect in Si,

which paved the way for the ever-increasing use of this effect nowadays, was evidenced only

in 1954 at Bell Labs.

The devices that transform the solar energy in electricity are called solar cells, or

photovoltaic cells. The benefits of solar energy, which is ecologic and practically unlimited,

include energetic independence, minimum maintenance, a simple extension to larger areas of

the photovoltaic systems (the solar cells are supplied in modules), and the (non-negligible)

fact that it can be produced almost anywhere (the wind energy, for example, can be harvested

only in places with constantly high wind intensity, and hydroelectric plants can have an

undesirable impact on the environment).

2

An example of 6 modules connected in parallel, each consisting of 40 photovoltaic cells

connected in series, is presented in the figure above. This solar panel can be used for battery

recharging and/or for powering domestic appliances with 12 V dc.

To understand the operation of solar cells, one needs to know that

1) The solar radiation is composed of photons with a wide energy distribution. The photon

energy ωh=E depends on the frequency ω or on the wavelength == k/2πλ ωπ /2 c of the

electromagnetic radiation as follows: the energy decreases as λ increases. The energetic

distribution of the flux of photons emitted by the Sun is represented in the figure below.

Please note that the solar photon flux extends on a much wider energetic interval than the

visible range, and has a maximum in the infrared spectral region.

2) The photons interact with the atoms of a semiconducting material (for example), i.e. are

absorbed, and, if the photon energy is high enough, it can break the bond between the valence

electron and atom. As a result, the valence electron is excited on higher energy levels, so that

a negatively charged conduction electron and a positively charged hole in the valence band

appear simultaneously. These charges move freely in the crystal, until they recombine. The

electron and the hole are the constituents of electricity. For example, the energy necessary for

an electron in the valence band to reach the conduction band of Si corresponds to a

wavelength of 1.12 μm (in infrared!), 70% of photons in the solar radiation being able to

excite an electron in the conduction band of silicon (see the figure below). If the photon

energy is much higher than that corresponding to λ = 1.12 μm, the excess energy is

transformed into heat, which degrades the performance of solar cells.

3

3) To have efficient solar cells, it is necessary to collect the electron-hole pair before

recombination. This can be done by the separation of electrons and holes by a built-in electric

field, the current produced by the electrons and holes generated by photon absorption being

called photocurrent.

For a quantitative treatment of the photovoltaic effect we study in the first part of this

course the formation of energy bands in solids, which determines the classification of

materials in metals, semiconductors and isolators, the doping process of semiconductors, the

junctions between differently-doped semiconductors and, respectively, the heterojunctions

between different materials, the absorption of light and, finally, the photovoltaic effect.

Electron dynamics in crystalline solids Crystal structure The properties of crystalline solids, especially the electron dynamics, are determined by the

symmetry of the crystalline lattice. The crystal lattice is a periodic arrangement of atoms over

large distances, of at least thousands of periods. The materials used in the fabrication of solar

cells include also non-crystalline materials, which have no long-range order, but their optical

properties (in particular, the absorption) are similar to that of crystalline materials because the

wavelength of the incident photons (of the order of 1 μm) is much larger than the

inhomogeneities of the material, so that the photons “see” an effective homogeneous medium.

The macroscopic, perfect crystal is formed by adding identical building blocks (unit

cells) consisting of atoms or groups of atoms. A unit cell is the smallest component that

reproduces the whole crystal by translational repetition along three directions in space. The

periodicity of the resulting crystalline structure is confirmed by X-ray diffraction experiments.

4

A crystal is a combination of a basis and a lattice. The basis is the group of atoms or

molecules that forms, by infinite repetition, the macroscopic crystal, while the lattice (also

called Bravais lattice) is a set of mathematical/abstract points in which the basis is positioned.

(a) (b) Examples of crystals in which the basis consists of (a) one atom and (b) two atoms

In three dimensions, all Bravais lattice points can be obtained as superposition of

integral multiples of two non-planar primitive translation vectors , and : 1a 2a 3a

321 aaaR pnmmnp ++= (1)

(m, n, and p are arbitrary integers). If the basis consists of n atoms, their positions can be

described by the set of vectors 321 aaar jjjj pnm ++= , j = 1,2,…,n, defined with respect to

one point of the Bravais lattice. In general, 1,,0 ≤≤ jjj pnm . The volume of the primitive

unit cell, constructed starting with the primitive translation vectors, is 321 )( aaa ⋅×=Ω . The

primitive unit cell is the unit cell with the smallest volume.

Every point of a Bravais lattice is equivalent to every other point, the Bravais lattice

being invariant under the operation of discrete translation 321 aaaT srqqrs ++= along integer

multiples q, r and s of the primitive translation vectors because =+= mnpqrsmnpqrs RTRT )(

is also a Bravais lattice point. Besides discrete translations, the Bravais lattice is

invariant to the point group operations, which are applied around a point of the lattice that

remains unchanged. These operations are: a) the rotations by an angle

psnrmq +++ ,,R

nC n/2π , n = 1, 2, 3,

4 and 6, about a specific axis, b) the inversion I, which is defined by the operation rr −→ if

applied about the origin, c) the reflection jσ , which can be applied around the horizontal (j =

h), vertical (j = v), or diagonal planes (j = d), and d) the improper rotation , which consists

of the rotation followed by reflection in the plane normal to the rotation axis.

nS

nC

5

(a) (b)

(c)

(d)

(e) Examples of symmetry operations: (a) translations, (b) rotation, (c) inversion, and reflection with respect to (d) a vertical plane, and (e) a horizontal plane.

C2 C3 C4 C6

Examples of two-dimensional figures with different rotation symmetries with respect to an axis normal to their plane.

Depending on their symmetry, all crystals can be classified in 7 crystal systems, which

contain 14 Bravais lattices represented in the figure below, where the primitive translation

vectors are denoted by a, b, c (their lengths a, b, and c being denoted lattice constants), and α,

β, γ are the angles between b and c, c and a, and a and b, respectively. The lattice constant

values range between 3 Å and 6 Å, depending on the crystal.

6

Examples of simple crystal structures One of the most simple crystal structures is that of salt, or NaCl (sodium chloride). As can be

seen from the figure below, its corresponding lattice is face-centered cubic, with a basis

consisting of one Cl− ion (blue, in the origin of the coordinate system) and one Na+ ion (green,

at the middle of the cube edge and in its center). Other crystals with the same structure are

LiF, LiBr, KBr, AgBr, MgO, MnO.

(a) (b) (c)

(d) (e) The crystal structure of (a) NaCl, (b) CsCl, (c) hexagonal close packed materials, (d) diamond (or Si), and (e) ZnS. Another common structure is that of CsCl (other crystals with the same structure are

AlNi, CaZn, AgMg, CsBr, CsI), which is characterized by a simple cubic lattice, with a basis

consisting of two ions: one Cs+ ion (red in the figure above, in the corners of the cube), and

one Cl− ion (green, at the center).

The hexagonal close-packed (hcp) crystal structure is a hexagonal Bravais lattice with

a basis consisting of two atoms (one in the bottom line and the other in the middle line in the

figure above), in which the atoms in one plane/layer, which touch each other, also touch the

atoms in adjacent planes. The atoms in the second layer are placed in the depressions left in

the center of every other triangle formed by the centers of the atoms in the first layer, the third

layer of atoms is placed exactly above the first, the fourth above the second, and so on. This

crystal structure is also encountered in He, Zn, Cd, Mg, Zr, Ti. If the distance between

7

adjacent layers of atoms is large enough so that the atoms in one layer do not touch those in

the adjacent layer, the hexagonal structure is no longer compact. Graphite, for example, has a

hexagonal non-compact crystal structure.

The crystal structure of diamond (and also of Si and Ge semiconductors), represented

in the figure above, is a face-centered cubic (fcc) lattice with a basis consisting of two

identical atoms, one in the corner of the cube and the other displaced with a quarter of a

diagonal. This crystal structure is usually encountered in materials where the covalent

bonding prevails, the atoms placed in the corners of a tetrahedron forming highly directional

covalent bonds with adjacent atoms by sharing one electron from a pair. If the crystal

structure of Si contains a basis of different atoms (for example, Zn and S in ZnS), we obtain

the zinc blende structure. It is encountered in SiC, ZnS, ZnSe, CdS, CdTe, AlAs, GaAs, GaP,

InAs, InSb, CuCl.

Reciprocal lattice Since the crystal is invariant under any translation with a Bravais lattice vector

321 aaaR pnmmnp ++= (2)

for any integers m, n or p, any function ϕ with the same periodicity as the crystalline lattice

must satisfy the relation

)()( mnpRrr += ϕϕ , (3)

where is an arbitrary position vector with coordinates , , and with

respect to the coordinate system determined by , , and . This means that

),,( 321 xxx=r 1x 2x 3x

1a 2a 3a

),,(),,( 332211321 paxnaxmaxxxx +++= ϕϕ (4)

or, for a function that can be expanded in a Fourier series

∑ ++=321 ,,

332211321 )](exp[),,(kkk

xkxkxkixxx kϕϕ (5)

8

it follows that, for any m, n, and p,

1)exp( 11 =aimk , , 1)exp( 22 =aink 1)exp( 33 =aipk . (6)

Thus, , with i = 1, 2, 3, can only take discrete values ik

iii ask /2π= , (7)

and the original function can be expressed as

∑ ⋅=k

k rkr )exp()( iϕϕ (8)

where

∫ ⋅−Ω= −cell dVi )exp()(1 rkrk ϕϕ , (9)

the integral being performed over the volume Ω of a primitive unit cell, and

332211 bbbk sss ++= (10)

is a vector in a coordinate system defined by the vectors , i = 1,2,3, such that ib

ijji πδ2=⋅ ab . (11)

Similar to the Bravais lattices that are constructed starting with the primitive vectors , one

can define a reciprocal lattice in terms of the primitive vectors , such that k in (10) are

points in the reciprocal lattice.

ia

ib

9

An example of a direct lattice and the corresponding reciprocal lattice in two

dimensions is given in the figure above at left and right, respectively. The primitive

translation vectors and the lattice constants are indicated in both cases.

In three-dimensional lattices, if 321 )( aaa ⋅×=Ω > 0, the vectors can be chosen as ib

),)(/2( 321 aab ×Ω= π ),)(/2( 132 aab ×Ω= π ))(/2( 213 aab ×Ω= π , (12)

and the volume of the primitive cell of the reciprocal lattice is given by

. The connections between the direct and reciprocal lattices for

the Bravais lattices in the cubic system are summarized in the table below

Ω=×⋅=Ω /)2()( 3321 πbbbrec

Real space (direct lattice) Reciprocal space

Lattice Lattice constant Lattice Lattice constant simple cubic a simple cubic a/2π

body-centered cubic a face-centered cubic a/4π

face-centered cubic a body-centered cubic a/4π

The first Brillouin zone The primitive unit cells of a Bravais lattice are not unique; different primitive cells can be

obtained choosing different sets of primitive translation vectors. The first Brillouin zone is a

primitive cell in the reciprocal space obtained by first drawing lines to connect a given lattice

point in the reciprocal lattice to all nearby lattice points, and then drawing planes at the mid

point and normal to the first set of lines.

The first Brillouin zone is then the volume in reciprocal space defined by the

intersection of these planes. Higher-order Brillouin zones, say the nth Brillouin zone, can be

defined as the volume in reciprocal space that can be reached from the origin by crossing

exactly planes. The first Brillouin zone is important in the study of electron dynamics in

the crystal. All Brillouin zones have the same volume, and higher-order zones can be reduced

to the first Brillouin zone by applying the symmetry operations specific to the crystal.

1−n

The first Brillouin zone for a) a body-centered cubic lattice in the direct space (the

reciprocal lattice is a face-centered cubic structure!) and b) for a face-centered cubic lattice in

the direct space (the reciprocal lattice is a body-centered cubic structure!) are represented in

the figure below.

10

Usually, as can be seen from the figures below, the directions with high symmetry in the

reciprocal space have specific notations.

Electron dynamics in the crystalline lattice In an isolated atom (see the figure below), the electrons orbit around the nucleus along

discrete circular trajectories, the attraction force of the nucleus being weaker for the electrons

on higher energy levels.

11

The electrons on these higher energy levels can then be easily delocalized (“knocked” from

the immediate neighborhood of the nucleus and set in motion in the whole crystal, so that a

positive ion is left behind) by an external electric field. These electrons become conduction

electrons. Unlike them, the electrons on lower energy levels contribute to the stability of atom

bonding in crystal and are called valence electrons. The dynamics of the conduction electron

in a crystal lattice is determined by a periodic potential and is determined by the Schrödinger

equation

)()()(2

22

rrr ψψ EUm

=⎟⎟⎠

⎞⎜⎜⎝

⎛+∇−

h , (13)

where the potential energy

)()( Rrr += UU (14)

describes the electrostatic interaction with all other electrons as well as the positive ions, and

R is a vector of the Bravais lattice. Due to periodicity (symmetry at translations), the electron

wavefunction can be expressed a Bloch functions, i.e. as

)exp()()( rkrr kk ⋅= iuψ (15)

where k is a wavevector of the reciprocal lattice and )()( Rrr kk += uu is a function with the

same periodicity as the crystal lattice. In addition, the electron energy is an even function of k,

)()( kk −= EE , (16)

and has the same periodicity as the reciprocal lattice. More precisely,

)()( Gkk += EE , (17)

where G is a vector of the reciprocal lattice. Therefore, the energy takes distinct values only

for k vectors in the first Brillouin zone. In general, the energy can take several values, ,

for the same k.

)(knE

12

Observation: The electron in crystal differs from a free electron. For a free electron in

volume V, which satisfies the Schrödinger equation with 0)( =rU , and has the wavefunction

)exp()( 2/1 rkrk ⋅= − iVψ and the momentum kp h= , all points in space are equivalent since

the probability to find the electron at position r, , is independent of r. On the

other hand, for Bloch functions const., and only points that differ

through a translation vector R are equivalent. In this case is the quasi-momentum of the

electron in the crystal and k is the quasi-wavevector; for simplicity, in the following we refer

to k as wavevector.

V/1|)(| 2 =rkψ

≠= 22 |)(||)(| rr kk uψ

kh

Similar to the free electron, for which we can define the velocity as ,

or since the dispersion relation of the free electron is so that

, the same formula can be utilized for the velocity of the electron in the crystal:

mm // pkv == h

Ekv ∇= −1h mE 2/22kh=

mE /2kk h=∇

Ekv ∇= −1h . (18)

Moreover, the effect of external forces F on the electron in the crystal can be separated from

the effect of the Coulomb interaction with all other electrons and ions in the crystal. The

influence of these electrostatic interactions is described by the concept of effective electron

mass. So, by defining the acceleration of the electron in crystal when the force F is applied as

)(1112 Fkva kkkkk ⋅∇∇=⎟

⎠⎞

⎜⎝⎛∇∇=⎟

⎠⎞

⎜⎝⎛∇== E

dtdE

dtdE

dtd

hhh (19)

or, on components μ,ν = x,y,z,

∑∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛=

∂∂∂

==ν

ν

μνν

ννμ

μμ F

mF

kkE

dtdv

aef

11 2

2h, (20)

the effective mass can be introduced as

νμμνkk

Emef ∂∂

∂=⎟⎟

⎠

⎞⎜⎜⎝

⎛ 2

2

11h

. (21)

13

The effective mass is a symmetric tensor, νμμν )/1()/1( efef mm = , its tensorial character

indicating that the acceleration of an electron in the crystal has, in general, a different

direction than that of the applied force. The effective mass is a parameter that describes the

effect of the crystalline lattice upon the electron motion, and not a characteristic of the

electron, as is the free electron mass. The effective mass can take positive, negative, or even

infinite values at inflexion points, where the curvature of the dispersion relation vanishes.

In the neighborhood of an extremum (maximum or minimum) of the energy

dispersion relation in the k space, the energy takes the form

0k

⎟⎟⎠

⎞⎜⎜⎝

⎛+++=

3,

23

2,

22

1,

21

2

0 2)(

efefef mk

mk

mkEE hk , (22)

where . If the iso-energetic surfaces are ellipsoids, the effective mass along the

axis of the ellipsoid is called longitudinal effective mass, while the common value of the

effective mass along the other two directions is called transverse effective mass. In isotropic

crystals the iso-energetic surfaces are spheres.

00 )( EE =k

Energy bands The energy spectrum of the electron in the crystal can be calculated either in the

approximation of quasi-free electrons (or weak binding approximation, in which the state of

the electron in the crystal is modeled as a perturbed state of a free electron, the perturbation

being determined by the periodic crystal potential) or in the approximation of quasi-bound

electrons (or tight-binding approximation, in which the state of the electron in the crystal is

modeled as a perturbed state of the electron in an isolated atom, the perturbation being

determined by the periodic crystal potential). In both cases the periodicity of the crystalline

lattice leads to the formation of allowed and forbidden energy bands.

14

The formation of energy bands, separated by forbidden bands, from the s and p atomic

orbitals of Si, as the distance between atoms decreases is represented in the figures above; a is

the lattice constant.

In the tight-binding approximation, the potential felt by the electron is a sum of the

periodic potential of the crystalline lattice and the potential of the isolated atom

(see the figure below), and the energy can be written as

)(rV )(raU

∑=

⋅−−=vecinil

la iACEE )exp( Rkk (23)

Ua(r)

V(r)

where is the energy of the isolated atom, and A and C are constants that depend on the

atomic orbital from which the energy band forms; the sum is over the nearest neighbors of a

given atom.

aE

15

For example, in a simple cubic lattice with lattice constant a,

)]cos()cos()[cos(2 321 akakakACEE a ++−−=k , (24)

and all possible values are obtained for wavevector components in the first Brillouin zone, i.e.

for aka i // ππ ≤≤− . The discrete energy levels in the isolated atom transform into a band

that extends between and , where, if A > 0, minE maxE

,6min ACEE a −−= for , (25a) 0=ik

,6max ACEE a +−= for aki /π±= . (25b)

mef

The figure above, left, represents several energy bands that form from different atomic

orbitals for which, alternatively, the parameter A is positive and negative. The interval in the k

space for this representation is the first Brillouin zone. If the different energy bands are

indexed by i, the forbidden intervals between them are denoted by . For example, the

width of the first forbidden energy band, i.e. the first energy gap, is

giE

)(6)()( 21121,2,1max,2min,1 AACCEEEEE aag +−−−−=−= . (26)

16

Note (see also the figure above, right) that the effective mass near the center of the

first Brillouin zone, for which 1<<aki and

, is positive (for A positive), and given by ,

while at the edges of the first Brillouin zone, for

=+++−−= )(6 23

22

21

2 kkkAaACEE ak

22min kAaE + 22122 2/)( AaEmef hh =∇= −

kk

ii kak −±= /' π , , and 1' <<aki ='kE

, the effective electron mass is negative,

.

−=++−+− max2

32

22

12 )'''(6 EkkkAaACEa

22 'kAa22122 2/)( AaEmef hh −=∇= −

kk

Electrons and holes

If an electron in the crystal, with electrical charge e− and positive effective mass

(near the minimum of the energy band), is placed in an electric field E, its acceleration

0>efm

efef memdtd /// EFva −=== (27)

is similar to that of the free electron. However, if the electron energy is close to the maximum

value in the band, 0<efm , and its equation of motion becomes

)/()/(/ efef memdtd −−=−== EFva . (28)

In this case, unlike for the free electron, the electric field slows down the electron instead of

accelerating it. To avoid this interpretation, the motion of the electron with a negative

effective mass in the electric field E is considered equivalent to the motion of a quasi-particle

with electrical charge e+ and positive effective mass. This quasi-particle, called hole, exist

only in the crystal (not in free space, as the electron) and reflects the absence of an electron

More precisely, if an electron from the upper part of an energy band (the valence band)

gathers enough energy from thermal vibrations or an external field to reach empty states from

the higher energy band, called conduction band, the remaining empty state in the valence

band is a hole. Thus, an energy band filled with electrons with the exception of its upper part

can be considered as partially filled with holes. Instead of following the movements of the

whole system of electrons, we can focus only on the movement of a much smaller number of

holes. In general, the holes differ from electrons not only through the opposite sign of the

electrical charge, but also through a different effective mass.

17

Classification of solid-state materials Although the structure of energy bands is similar in all materials, their occupation is different.

In an energy interval dE the number of states that can be occupied by an electron with a

dispersion relation is efmkEE 2/) =k( 220 h+

2/1032

2/1

)(4

)2()( EE

mED ef −=

hπ. (29)

These states are occupied according to the Pauli principle, which states that at low

temperatures (in particular, at T = 0 K) the states are occupied in the order of increasing

energy, so that only two electrons (with opposite spins) can exist on an energy state with a

given k. The energy value below which all states are occupied is called Fermi energy, or

Fermi level and is denoted by . The distribution function of electrons according to their

energy at a temperature T is given by the Fermi-Dirac formula

FE

]/)exp[(11)(

TkEEEf

BF−+= (30)

where E is the energy of the electron and is the Boltzmann constant. Bk

As follows from the figure above, at T = 0 K, the Fermi-Dirac distribution function is step-

like, with value 1 for energies less than the Fermi energy and 0 otherwise.

18

Two situations occur depending on the position of the Fermi level:

1) At low temperatures the Fermi level is inside an energy band (see the figure below, left),

i.e. the electrons fill partially the (last) energy band. The materials is then a metal and can

easily conduct electricity because the electrons near FE can occupy empty states with

higher energy if accelerated by a small applied electric field.

2) At low temperatures the occupy completely a number of energy bands, so that the Fermi

level is inside the forbidden band between the last occupied band, called valence band,

and the next empty band, called conduction band (see the figure below, right). In this case

a small electric field does not provide enough energy to the electrons to reach empty

states in the conduction band, so that no electric current passes through the material. The

material is a dielectric.

At higher temperatures, T ≠ 0 K, the dielectric materials are classified as isolators and

semiconductors depending on the width of the energy gap that forms between the valence and

conduction bands. If < 3 eV, thermal fluctuations can excite electrons from valence band

into the conduction band, where they can participate at electrical conduction, and the material

is called semiconductor. As a consequence, in an undoped semiconductor the number of

electrons in the conduction band is equal to the number of holes in the valence band. The

number of charge carriers is, however, small. In the presence of an electric field, both

electrons and holes contribute to the electrical current, these charge carriers being displaced in

opposite directions. In isolating materials > 3 eV and the electrical current cannot pass

even at higher temperatures.

gE

gE

19

Intrinsic and doped semiconductors

In an undoped crystalline semiconductor, or intrinsic semiconductor, with a perfect crystalline

lattice, the number of electrons in the conduction band, n, equals the number of holes in the

valence band, p, their concentrations depending on temperature according to the expressions

∫∞

=cE

n dEEDEfn )()(2 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

TkEEN

B

Fcc exp , (31a)

∫∞−

=vE

p dEEDEfp )()(2 ⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

TkEEN

B

vFv exp , (31b)

where the double spin degeneration, which states that two electrons/holes can occupy the

same energy level according to the Pauli exclusion principle, was taken into account, and

and are the density of states of electrons and holes, respectively, which are

obtained from (29) if the effective mass is replaced with that of electrons and holes, and

, respectively, and is replaced with the bottom of the conduction band and,

respectively, with the top of the valence band . In the expressions above the Fermi-Dirac

distribution function has been approximated with the Maxwell-Boltzmann distribution

function

)(EDn )(EDp

nm

pm 0E cE

vE

1)/exp()/exp()( <<−= TkETkEEf BBF , approximation valid for small

concentrations of charge carriers, and the parameters

32

2/3

4)2(

hππ TkmN Bn

c = , 32

2/3

4)2(

hππ Tkm

N Bpv = , (32)

are the effective densities of states in the conduction and valence bands, respectively. The

intrinsic concentration in undoped semiconductors is defined as . npni =2

From the neutrality condition pn = one can determine the position of the Fermi level,

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=

n

pB

vcF m

mTkEEE ln

43

2 (33)

which is located in the middle of the energy gap at T = 0 K and moves towards the energy

band with the smaller effective mass as the temperature increases.

20

The undoped semiconductors have weak electrical properties. For example, the Si

atom has 4 valence electrons, each of them forming a covalent bond with another valence

electron from an adjacent Si atom. This bond is very strong and there are no free electrons in

the perfect crystal. Therefore, the Si crystal is a very weak electricity conductor.

To improve the electrical conduction of semiconductor materials, these are doped.

More precisely, atoms of a different type than those of the crystalline lattice are introduced in

the material. These impurity atoms are donors, if electrons that can move freely in the crystal

are released through the substitution of a lattice atom by the impurity atom, or acceptors, if a

free hole appear after the substitution process. For example, if a Si atom is replaced by a P

atom, 4 of the 5 valence electrons of P form strong bonds with the closest electrons in Si, the

remaining electron being weakly attached to the positive nucleus of the P atom. This last

electron can become easily free, i.e. a conduction electron, if a weak electric field is applied. P

is thus a donor impurity for Si, the semiconductor with a large number of donor atoms being

of type n since the negative electrical charges assure the electrical conduction.

+ B

Ec

Ev

EF

Ec

Ev EF

If the electrons in the conduction band originate predominantly from donor impurities with

concentration (which become, as a result, ionized), i.e. if dN dNn ≅ and p << n, the position

of the Fermi level shifts toward the conduction band (see the figure above, left) and is given

by

21

)/ln( cdBcF NNTkEE += . (34)

Similarly, if a Si atom is replaced by B, all 3 valence electrons form strong bonds with

the electrons of adjacent Si atoms, but there is still one electron in Si that cannot form a pair

since it has no electron to bond with. This missing electron is in fact a hole, which is only

weakly bonded since the strong covalent bond in Si cannot be formed. This hole can easily

become free in the presence of a weak electric field, and the impurity atoms (in this case, B)

with a missing valence electron are called acceptors. The semiconductor with a large number

of acceptor atoms is of type p, the electrical conduction taking place with the help of positive

electrical charges. For Si, the typical B concentration is of 2×1016/cm3, and that of P is

1019/cm3. If the hole concentration equals that of the ionized acceptor atoms, and n

<< p, the Fermi level shifts toward the valence band (see the figure above, right) and has the

expression

aNp ≅

)/ln( avBvF NNTkEE += . (35)

P-n junctions and semiconductor heterojunctions

A p-n junction forms if an n-doped semiconductor is in close proximity with the same, p-

doped, semiconductor material. In the p region, the hole concentration is higher than that of

electrons due to the acceptor impurities, while in the n region the electrons dominate due to

donor impurities. As a result, there are gradients of hole and electron concentrations in the

neighborhood of the junction, so that the holes tend to diffuse from the p into the n region

while the electrons diffuse in the opposite direction. The electrons that reach the p region due

to diffusion recombine immediately with the holes. Similarly, the holes that reach the n

recombine with electrons, so that, in the neighborhood of the junction plane, a negatively

charged region forms in the p-type semiconductor due to the ionized acceptors that are no

longer compensated by the holes, while in the n-type semiconductor a positively charged

region forms because of the ionized donors that remain non-compensated due to electron

diffusion. These charged regions generate an electric field E oriented from the n to the p

region, which extends across a narrow region around the interface between the two materials

22

(across in the p region and across 0<≤− xl p nlx ≤<0 in the n-type semiconductor). This

electric field opposes further diffusion of charge carriers (see the figure below).

++ +

+ +

ln -lp

n p x

E +

+++

+ +

There is a potential difference/potential barrier associated to this internal

electrical field, named contact potential or diffusion potential. If a voltage with the same

polarity as the internal field of the junction (− at the p region and + at n) is applied from an

exterior source, the potential barrier increases and the junction is inverse (or backward)

polarized. The current that passes through the junction is very small since the holes in the p

region and the electrons in the n-type semiconductor (the majority carriers) cannot overcome

the increased potential barrier and the concentrations of minority charges in the two regions

are very small. On the contrary, if an opposite voltage is applied (+ at the p region and – at n)

the junction is direct (or forward) polarized and the height of the potential barrier decreases,

allowing the diffusion of charge carriers from the regions where they are in the majority to

those where they are in minority.

deV

The charged region in the immediate neighborhood of the junction, which is depleted

in free carriers, is called depletion layer or space charge region. The width of this layer,

, is determined by the concentration and spatial distribution of the impurities in the

p and n regions, as well as by the polarization of the junction. At direct polarization the

depletion layer narrows, while at inverse polarization it widens. The non-compensated electric

charges on either side of the junction are equivalent to a capacitance, called the capacitance of

the depletion layer. In a circuit, the current that passes through the directly polarized p-n

junction when a positive voltage V is applied is due to the injection of minority carriers over

the potential barrier. Because the number of electrons with energy higher than is

proportional to , the current through the junction depends exponentially

on the direct voltage. At the inverse polarization, the current is carried by majority carriers; its

value is small and practically independent of the voltage. Thus, the p-n junction is strongly

nonlinear, and has rectifying abilities.

np lll +=

)( VVe d −

]/)(exp[ TkVVe Bd −

23

To find the expression of the current, we assume that the p-n junction is steep, all

impurities are ionized, and that there are no free charge carriers in the depletion layer. Under

these circumstances, the impurity distribution is step-like, and the distribution of space

charges ρ is shown in the figure below, left. and denote the concentrations of

acceptor and donor impurities, respectively.

aN dN

x

ρ

eNa

eNd

ln -lp

x

E

Em

ln -lp

x

ψ Vd

ln -lp

We have

⎩⎨⎧

≤<<≤−−

=nd

pa

lxeNxleN

x0,

0,)(ρ (36)

And the electrostatic potential and the electric field are determined by solving the one-

dimensional Poisson equation in a material with dielectric permittivity ε

ερψ )(

2

2 xdxd

−= , dxdE ψ

−= , (37)

with suitable boundary conditions. The neutrality condition for the total charge in the

depletion layer imposes that . The spatial dependence of the electrical field and

electrostatic potential are represented in the figure above, center and right, respectively.

ndpa lNlN =

In the absence of an applied exterior voltage, the redistribution of electric charge in the

p-n junction is equivalent to the bending of the conduction and valence bands near the

interface, so that the Fermi level, situated at different energies in the p and n regions, becomes

the same in the whole device (see the figure below). The concentrations of charge carriers at

equilibrium (in the absence of an exterior voltage) are also illustrated in the figure below.

24

EF Ec

Ev

x

p,n

np

nn

ln -lp

pp

pn

Δp

Δn

n

p

These concentrations are given by )/exp( TkNn Bncn Δ−= , )/exp( TkNn Bpcp Δ−= ,

)/exp( TkeVnn Bdnp −= , )/exp( TkeVpp Bdpn −= (38)

where and the subscripts n and p refer to the respective regions; the

concentrations of holes can be determined in a similar way, or directly from ,

. From (38) it follows that the interface potential depends on the dopant

concentrations: .

Fc EE −=Δ

2inn npn =

2ipp npn =

)/ln()/( 2ipnBd npnETkV = )/ln()/( 2

idaB nNNETk≅

Analogous to the p-n junction, a heterojunction forms at the interface between two

different semiconductors, with different energy gaps and Fermi levels (see the figure below,

left). The bending of the energy bands near the interface is caused again by the redistribution

of the electric charge. If , the electrons diffuse from material 1 leaving behind

positively charged donors. This space charge at the interface generates an electrostatic

potential that bends the energy bands so that, at equilibrium, the Fermi level is constant

throughout the structure (see the figure below, right). Unlike in p-n junctions, in

heterojunctions there are discontinuities in the conduction and valence bands, and

21 FF EE >

cEΔ vEΔ ,

respectively.

Ec1

Ev1

EF1 Ec2

Ev2 EF2 Eg1 Eg2

ΔEc

ΔEv

ΔEv

+ + + +

ΔEc

- - - - EF

25

The I-V characteristics of a p-n junction

In the presence of an exterior voltage, the system of electrons is no longer in equilibrium, and

we must define Fermi quasi-levels for electrons in the p region, , as well as for those in

the n region, , since the concept of Fermi level makes sense only when the system of

electrons is at equilibrium. In the figure below we have represented the position of the Fermi

quasi-levels at inverse (left) and direct (right) polarizations.

FPE

FnE

EFn Ec

Ev

EFp eV

EFn Ec

Ev

EFp eV

n n

p

p

When an exterior voltage V is applied, minority carriers with concentrations and are

injected in the p and n regions, respectively, so that the carrier concentrations at the sides of

the depletion layer are

nΔ pΔ

)()/exp()( nnBnn lppTkeVplp Δ+== , )()/exp()( ppBpp lnnTkeVnln −Δ+==− . (39)

If and 0)( =−<<Δ plxn 0)( =>>Δ nlxp , the concentrations of injected carriers determined

by the diffusion equation are

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=Δ

p

n

Bn L

lxTk

eVpxp exp1exp)( , ⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=Δ

n

p

Bp L

lxTk

eVnxn exp1exp)( , (40)

where , are the diffusion lengths in the two regions. For a steep junction and a narrow

depletion layer compared to the diffusion lengths,

nL pL

pn LLl ,<< , so that the recombination in

the layer can be neglected, the minority carrier currents in the p and n regions are,

respectively,

dxpdeDEppej pnpp

Δ−Δ+= )(μ ,

dxndeDEnnej npnn

Δ+Δ+= )(μ . (41)

26

The first terms on the right-hand sides represent the drift currents (currents induced by the

electric field E with velocities Ev nn μ= , Ev pp μ= along the direction of the applied field,

where nμ are pμ are the mobilities of electrons and holes), and the last terms describe the

diffusion currents, characterized by the diffusion coefficients and . nD pD

At weak electric fields and small injection levels the diffusion currents dominate,

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

Δ−=

p

n

Bp

nppp L

lxTk

eVL

peDdx

pdeDxj exp1exp)( , (42a)

⎟⎟⎠

⎞⎜⎜⎝

⎛ +⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

Δ=

n

p

Bn

pnnn L

lxTk

eVL

neDdx

ndeDxj exp1exp)( , (42b)

and the total current, which is constant along the junction and for which )()( pnnn ljlj −= ,

, is )()( nppp ljlj =−

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=+= 1exp1exp)()(

TkeVj

TkeV

LpeD

LneD

xjxjjB

sBp

np

n

pnpn , (43)

where is the saturation current. The I-V characteristic is represented in the figure below:

for positive voltages the current increases exponentially with the voltage, while

for negative voltages, for , the current is independent of the voltage, .

sj

eTkV B />

eTkV B /< sjj −≅

I

V -Is

If other phenomena are taken into account, for example the generation and recombination of

carriers in the depletion layer and the high injection levels, an exponential dependence of the

form is obtained, the general dependence (with ]1)2/[exp( −= TkeVjj Bs 21 ≤≤ β ) being

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= 1exp

TkeVjj

Bs β

. (44)

27

Light absorption The absorption of light in a material is characterized by the absorption coefficient α, which

represents the fraction of electromagnetic radiation absorbed per unit length in the material.

This can be expressed as

IdxdI α=− (45)

where I is the intensity of flux of light that propagates along the x direction and is normally

incident on the layer with width dx in the material, the – sign indicating that the intensity I

decreases as x increases. By integrating the equation above we obtain optical absorption law

)exp()1()( 0 xIRxI α−−= (46)

where is the intensity of light flux incident on the surface, and 0I 1<R is the reflection

coefficient of light. α represents the penetration depth of electromagnetic radiation at which

the flux intensity decreases e = 2.718 times. The average intensity of radiation that penetrates

a distance d inside the material is obtained by integrating the formula above, with the result

)]exp(1][/)1[()exp(]/)1[()( 00

0 ddIRdxxdIRdId

m ααα −−−=−−= ∫ (47)

I

x

(1-R)I0

dx

dI

I0

d

Im

This law is valid only for monochromatic light, because the absorption coefficient depends on

the light wavelength λ. The dependence of α on λ is characteristic for each material and is

determined by the physical processes that take place inside it.

Irrespective of the nature of physical processes, if the incident electromagnetic

radiation is considered to consist of photons with frequency ω, energy ωh=E and

wavevector k, the photon-illuminated material interaction occurs with the conservation of

both energy and momentum, which impose that

28

ωh=− if EE , (48)

kkk =− if , (49)

where and are the energy and momentum in the illuminated material in the final

(subscript f) and initial (subscript i) states.

ifE , if ,k

The nature of the interaction between photons and the illuminated material depends on

the photon energy. In semiconductors, the dominant absorption mechanism is the band-to-

band or fundamental absorption, in which an electron is promoted from the valence to the

conduction band as a result of photon absorption. In this case |||| if kkk −<< and the

electron momentum is conserved, the transition from the valence to the conduction band

being “vertical” in the k space (see the figure below).

E Ec

Eg

k

Ev

foton

Ei

Ef

α

Eg hω

If both conduction and valence bands are isotropic, parabolic and have extrema at , the

band-to-band absorption is direct, and the electron energy around the extrema of the two

bands can be written as , . Energy conservation

implies in this case the equality , where is the

reduced mass, defined through

0=k

ngf mkEE 2/22h+= pi mkE 2/22h−=

ωω hhh −+==−− rgif mkEEE 2/0 22rm

pnr mmm /1/1/1 += .

GaAs, GaSb, InP, InAs, CdS are examples of semiconductors with direct energy

bands. From (45) it follows that the absorption coefficient can be expressed as

,/)/)(/(/)/( IdtdIcnIdxdI refr−=−=α (50)

29

where is the refractive index of the semiconductor, c is the light velocity in vacuum, refrn

ωhNI = , with N the photon number in the incident radiation, and )/(/ dtdPdtdI totωh= is

the absorbed energy per unit time, with hω the photon energy and the transition

probability per unit time. A quantum calculation of the transition probability per unit time

leads to the following expression of the direct band-to-band absorption coefficient:

dtdPtot /

ωω

ααh

h gE−= 0 , (51)

with 0α an energy-independent term. The dependence of the absorption coefficient on the

frequency of incident photons is represented in the figure above. Note that the absorption has

a threshold at gp E=ωh , typical α values being of the order of 104–105 cm–1.

For impurified semiconductors with shallow energy levels and low concentrations of

the impurities, the discrete donor and acceptor levels, and , respectively, and the

possible transitions between these levels and between the impurity levels and the valence and

conduction bands are illustrated in the figure below, left.

dE aE

In this case the transition of charge carriers from the impurity levels on states in the

conduction or valence bands takes place as a result of absorption of one photon with energy

less than , so that the absorption spectrum has a lower threshold than for the fundamental

absorption mechanism. If, for example, there are only acceptor levels in the semiconductor,

the absorption spectrum overimposed on the fundamental absorption is represented in the

figure below, right. The absorption on impurities has a smaller intensity (the density of states

and hence the number of carriers on the acceptor level is much smaller than the carrier

number in the valence or conduction bands) and is shifted toward lower energies in

comparison to the fundamental absorption.

gE

Ec

Ed

Ea

Ev

α

Eg - EaEg

hω

30

The absorption on impurities can be observed for moderate concentrations of

impurities in doped semiconductors and for low impurity concentrations in compensated

semiconductors. For shallow impurity levels this absorption mechanism is observed in far

infrared (λ < 25 μm) and at low temperatures, in order to avoid the ionization of impurities.

The absorption coefficient α is proportional to the impurity concentration.

If the valence and conduction bands have extrema along different directions in the k

space (see the figure below), the conservation condition of the total momentum requires that

transitions take place with the involvement of phonons or impurities. In this case, the

absorption mechanism is called band-to-band indirect; Si and Ge are examples of

semiconductors with indirect energy bands. If phonons participate to light absorption, they

can be emitted or absorbed in this process, and the electron transition is modeled as occurring

in two steps: 1) a photon is absorbed, and the process is represented as a vertical transition in

the k space, 2) a phonon with frequency fω is absorbed or emitted, the process being

represented by an oblique transition in the k space.

Quantum calculations show that, in this case

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−−−

+−

−+=+=

)/exp(1)(

1)/exp()( 22

0

TkE

TkE

Bf

gf

Bf

gfemabs ω

ωωω

ωωω

ααααh

hh

h

hh

h (52)

and the )( ωωγ hh f= dependence is approximately a straight line for fgE ωω hh +> , as

well as for fgfg EE ωωω hhh +<<− , where phonon absorption is much more probable

than phonon emission (see the figure below).

E

fonon emis

fonon absorbit

k

Eg

Ev

Ec

fonon

T2 T1 < T2

hω

(αem)1/2

(αhω)1/2

(αabs)1/2

Eg-hωf Eg+hωf

31

Because there are several types of phonons in the crystal and more than one phonon

can be involved in the process, the absorption characteristics can have several discontinuities

that appear when the threshold at which new phonons can participate at the transition is

reached. For h hω ω> −Eg f the term in (52) related to phonon absorption dominates, while

for h hω ω> +Eg f phonon emission prevails. The absorption coefficient is smaller than at

direct transitions: ≅α 10–20 cm–1, and its temperature dependence is much more significant.

In conclusion, for solar cells it is desirable to use materials with direct energy bands.

The spectral dependence of the absorption coefficient of some semiconductors is presented in

the figure below.

band-to-band indirect absorption band-to-band

direct absorption

Problem: Using the absorption coefficient characteristic above, estimate the thickness of a

GaAs solar cell, such that 90% of the incident radiation at 800 nm is absorbed, if R = 1. What

would be the necessary thickness if R = 0.6 ?

32

The photoelectric effect

At the absorption of a photon with energy higher than or equal to the bandgap of a

semiconductor material, a bond between a pair of electrons or between an electron and a

positively charged nucleus is broken, so that a pair of electric charges consisting of a negative

electron and a positive hole is generated. These charges attract each other and recombine

unless they are spatially separated. A method for electron-hole separation is the application of

an electric field, due to which the electrons and holes are drifted in different directions so that

an electricity flux/a photocurrent appears. This is the working principle of solar cells. The

electric field can be, for example, the built-in field in a p-n junction.

Let us consider a typical solar cell represented in the figure below, left, in which the

electromagnetic radiation is incident on the n-type layer.

eVcd

l

If the n layer is sufficiently thin, we can assume that the photons are absorbed in the vicinity

of the depletion layer, so that the resulting electrons and holes are spatially separated by the

built-in electric field in the junction. The electrons are displaced toward the n regions and the

holes toward the p-type layer (see the figure above, right), the electric charges in these regions

increasing until a stationary, equilibrium regime is reached between the generation and

recombination processes. These electric charges generate an electric field, opposed to that of

the junction, which decreases the potential barrier in the depletion layer, as in the case of a

direct polarization of the junction. The equivalent direct voltage in an open circuit is called

open-circuit voltage . Its polarity is + at the p region and − at n. If the device is connected ocV

33

in series with a load , a photocurrent flows in the exterior circuit in an opposite direction

than in the case when the non-illuminated (dark) junction would have been polarized to an

external source at the same (direct) voltage. In consequence, the photocurrent has the same

direction as the inverse (saturation) current of the non-illuminated junction, the solar cell

generating the current (see the figure below). The total current is then

sR

LI

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛+−= 1exp

TkeVIII

BsL β

. (53)

I

V

-IL

dark

illumination

-Is

V=-RsI

V Voc

P

I

Isc

Vm

Im

A closer look at the characteristic of the photoelement (see the figure above,

right) for V > 0, I < 0, allows us to define two important parameters of the solar cell: the open-

circuit voltage,

VI −

)/1ln()/( sLBoc IIeTkV += β , which is the voltage that corresponds to I = 0,

and the short-circuit current, Lsc II −= , which corresponds to V = 0. In applications, it is

common to consider the dependence with the opposite current (with I > 0). The open-

circuit voltage can not be smaller than the energy bandgap of the semiconductor (divided by

the electron charge), whereas the photocurrent is determined by the intensity of the solar

radiation (by the number of photons with energy higher than the bandgap).

VI −

Moreover, the performance of a solar cell is determined by the (output) power IVP =

on the load. The working point (i.e., the voltage and current) are found from the intersection

of the characteristics and the straight line defined by VI − IRV s−= , the power being given

by the are of the rectangle formed from the voltage and current values in the working point.

The power dependence on voltage (the blue line in the figure above, right) has a maximum

34

defined by (see the gray area) for which 0/ =dVdP mII = , mVV = . The maximum power

is generated on the load mmm VIP = mmsm IVR /−= . To optimize the power a compromise

between a high current and a high voltage is generally necessary.

A solar cell is characterized by the filling factor (or filling ratio)

ocscmm VIVIFF /= (54)

and the conversion efficiency

=η output power/input power (55) radm PP /=

where is the power of the incident solar radiation. Both parameters have values less than

unity. The typical conversion efficiency has values of 12−20% for standard solar cells, while

a typical filling factor is 75%.

radP

In addition, for a better comparison of the materials from which the solar cell is

fabricated, one can define the external quantum efficiency as the number of collected

electrons per incident photon,

=extη collected electrons/incident photons. (56)

This parameter is within the 60−90% range, depending on the reflectivity, wavelength and

illumination conditions. Another useful parameter is the internal quantum efficiency, defined

as the number of collected electrons per absorbed photon,

iη = collected electrons/absorbed photons (57)

which has typical values of 80−95%, depending on the wavelength.

Efficient solar cells have high conversion efficiency, hence high external quantum

efficiency, and can transform in electric energy photons in a wide spectral range. In the next

parts of the course we will analyze the performances of solar cells from different generations

and will discuss the solutions for improving their efficiency.