NOTES A. La Rosa Physics Department

P-N JUNTIONS

I. HARNESSING ELECTRICAL CONDUCTIVITY IN SEMICONDUCTOR MATERIALS

I.1 Intrinsic conductivity (pure silicon)

I.2 Extrinsic conductivity (silicon doped with selected different atoms)

II. ENERGY LEVELS DIAGRAM

II.1 Intrinsic materials

II.2 Extrinsic materials p-material, n-material

III. QUANTUM STATISTICS of ELECTRONS

III.1 The Fermi-Dirac Distribution and the Chemical Potential

III.2 Application of the Fermi-Dirac distribution to semiconductors

III.2.1 Intrinsic case. (Location of the chemical potential)

III.2.2 Extrinsic case. (Location of the chemical potential)

IV. p-n JUNCTIONS

IV.1 Diode: p and n materials placed together

IV.2 Space charge at a junction IV.3 Chemical potential at pn junctions

V. FORWARD BIAS, REVERSE BIAS

VI. COMPARISON of CHARGE-CARRIER POPULATION at EACH SIDE of the JUNCTION

_____________________________________________________________

I. HARNESSING ELECTRICAL CONDUCTIVITY IN SEMICONDUCTOR MATERIALS

Let’s take Silicon as a typical example of semiconductor material. A Silicon atom has 4 valence electrons. Joined by covalent bonds, Si atoms form a lattice structure constituted by two interpenetrating periodic fcc lattices, having 5x1022 Si-atoms/cm3 (8 atoms in a cube of side 5.43 Angstrons).

At zero temperature Silicon is an insulator because all its valence electrons participate in

Fig.1. Si atoms arranged in a crystal

network.

the bonding. It lacks electrons that can wander free around the crystal structure. At T=0,

conductivity = 0.

I.1 Intrinsic conductivity

Here we consider a Si material without foreign atoms (just pure silicon atoms).

Thermal generation of conduction electrons At room temperature (kT~ 0.025 eV), thermal agitation of the crystalline host provides energy to electrons and there is a non-vanishing probability (~ e-

Eg/2kT) that some of them will be able to break away from the (Eg=1.12 eV) bonding and, consequently, set almost “free” and wander around the crystal: they become conduction electrons. These electrons are able to participate in the conductivity of silicon under the presence of an external electric field. A typical value of the concentration of conduction electrons generated in this manner, at room temperature, is no ~1.4x1010 e-/cm3.

Generation of holes Notice that the creation of a new conduction electron leaves their Si atoms with one bonding vacancy. This bonding vacancy is called a “hole”.

Holes contribute to the conductivity. Indeed, under the presence of an external electric field, a bonded electron from a Si atom nearby will have a tendency to fill-in this bonding vacancy (the fact that the Si atom hosting the hole is a positive ion helps the process) and needs very little energy to do that. When that happens a new hole (with its associated positive ion) is left behind, which triggers the creation of another hole, and so on. The net result: a net positive charge is transported along the filling of holes.

Since a hole is created simultaneously with a conduction electron, the concentration of holes po equals the concentration of electrons: po = no. , both contributing to what is called intrinsic conductivity of silicon (that is, when free of foreign atoms.)

The intrinsic conductivity situation could be changed if selected foreigner atoms are introduced in the silicon crystalline structure, as described in the next section

I.2 Extrinsic conductivity

n-type: Si crystal doped with donor atoms (5 e- in their valence shell). Four e- participate in the bonding with Si atoms, while the fifth becomes almost free of any bonding. With a little bit of energy this fifth e- becomes free to wander around the entire silicon material. Example of doping level: 1016 arsenic atoms/cm3. Notice the ability to doped Si with foreigner atoms with concentrations in excess of the intrinsic regime.

Fig.2 Intrinsic conductivity. Electrons and holes contribute to electrical conductivity.

p-type: Si doped with acceptors atoms (3 valence e-). Boron atoms, for example. The three e- participate in the bonding with Si atoms, creating a bond vacancy to be filled-in. That is, acceptor atoms create holes, which also participate in the conductivity (as explained above).

Accordingly, the presence of donors and acceptors increases the electrical conductivity of the material.

Acceptors

Donors

Fig.3 Extrinsic conductivity. Harnessing electrical conductivity by inserting into the Si crystal matrix foreign impurities

II. ENERGY LEVELS DIAGRAM As isolated atoms are brought together to form a solid, various interactions occurs between neighboring atoms. In the process, important changes occur in the electron energy configurations. When the solid is formed (the forces of attraction and repulsion between atoms fins a balance at the proper interatomic spacing) the split energy levels display essentially continuum bands of energy.

2N states

2N states

6N states

Atomic separation

Energy

Eg

Conduction band

2p

2s

1s

Carbon atom

6e-/atom

4N states

4N states

Valence

band

N carbon atoms

brought together

Diamond

lattice spacing

brought together

2N states

Fig.4 Formation of energy bands by bringing together initially isolated carbon atoms.

At T = 00, semiconductor materials have basically the same structure as insulators:

a filled valence band separated from

an empty conduction band by a

a band-gap of non-allowed energy states

Empty

Filled

Empty

Filled

Eg Eg

Insulator Semiconductor

Fig.5 Band diagram at T = 0 K .

The difference between an insulator and a semiconductor lies in the size of the band-gap

Eg. The latter is much smaller in semiconductors. Compare, for example, 1.1 eV for Si

with 5eV for diamond.

Within the band structure model, even under the effect of an external electric field, filled

bands do not contribute to the electrical conductivity of the material.

At T≠ 0 K,

the relatively small bandgaps of semiconductors allow for excitation of electrons from the valence band to the upper conduction band via thermal excitation. This gives rise to the creation of electron-hole pairs, which are the carriers for electrical conductivity. The generation of e-h pairs can be qualitatively visualized as the breaking of covalent bonds in the crystal lattice (Fig.5). The energy required to break the bond is the band-gap energy Eg. [A shortcoming of the broken bond picture is that the free electron and the hole appear deceptively localized in the crystal lattice. Instead, their position spread over several lattice units undergoing probabilistic quantum mechanic jumps.]

The probability for such a jump in Si is proportional to e -1.1 eV/kT. The probability for such a jump in diamond is proportional to e -5 eV/kT. Thus, due to thermal excitation, in semiconductors the number of charge carriers available for electrical conduction is much greater than in insulators. (We will see later that there are other ways to increase the electrical conductivity.)

II.1 Intrinsic silicon A perfect semiconductor crystal with no impurities or lattice defects is called an intrinsic semiconductor.

In such materials there are no charge carriers at T=0 K. (The valence band is filled and the conduction band is empty.)

At T≠ 0 K, electron-hole pairs are generated (see also Fig. 6 above.)

Electron energy

Energy levels available to be

occupied by electrons

Silicon

Eg band-gap

Valence band

Forbidden levels

x

Conduction band

Fig.7 Energy level diagram for pure silicon. Number of electrons in the conduction band is equal to the number of holes in the valence band.

For intrinsic semiconductor material

Conduction band Valence band electron concentration = concentration of holes (1) n (electrons/cm3) p (holes/cm3)

II.2 Extrinsic silicon

Eg e-

4 4

4

4

4

4

4

4 4 4 4

Hole

4

Free

electron

At T = 300 K

Fig.6 Thermal generation of electron-hole pair carriers.

p p-material

n-material

n

Electron energies

e-

acceptors atoms

Electrically neutral material

Electrically neutral material

x

+ + + + + +

Acceptor’s energy levels

donor’s energy levels

e-

donor atoms + + + + +

Free electrons

Fig.8 In the p-material (n-material) the energy level of impurities are located near the valence (conduction) band.

III. QUANTUM STATISTICS of ELECTRONS

III.1 The Fermi-Dirac Distribution and the Chemical Potential

Figure 8 shows a system interacting with a (huge) thermal reservoir. The concept of thermal reservoir is an idealization conveying that no mater what amount of energy is transferred to or taken from the reservoir its temperature remains constant.

E

Reservoir at temperature

T

Energy exchange

Energy E

P(E)

Boltzmann

distribution

Small system A Fig.9 A system A interchanging energy with a thermal reservoir. The probability

of finding the system A in a state of energy E is proportional to TkE

Be/

.

We specialize to the case in which we have a system composed of N (non-

interacting) electrons. That is, N electrons are distributed among the different possible one-electron states.

(The figure below shows two possible particular configurations corresponding to the case N=3).

Since many configurations are possible (Fig.9 shows two of them), let’s call N

E

the energy associated to a given -configuration (or stationary state).

Energy

Energy E

1

Boltzmann

distribution 2

N Energy

1 2

N

1 2

3 4

5

6 7

8

1 2

3 4

5

6 7

8

Fig.10 Two possible stationary states of the N-electron system.

For simplicity we have chosen N=3.

The probability of finding a N-electron system in a particular state o of energy N

oE

is given by,

Tk

Tk

B

N

B

N

N

NE

E

EP

e

e o

o /

/

)(

)(

)( (2)

In the denominator the sum is all over the possible stationary states .

The denominator is known as the partition function and it is related to the Helmholtz

free energy F= U-TS (where U is the internal energy and S the entropy) by,

TkTk BNBN F

ee E //)

(

Definition of FN (3)

Thus,

TkB

N

NN

No

o

EFEP e

/)()(

We can omit the sub index o in the .

TkBN

NN

N

EFEP e

/)( )(

(4)

We have to incorporate the Pauli’s exclusion principle, which permits at most one

electron to occupy any single electron level. Accordingly, to construct an N-electron state one must fill N different one-electron levels.

Thus each N-electron stationary state can be specified by listing which of the N one-electron level are filled in that state. (For example [3,5,7] and [2,3,6] respectively in the examples given in Fig. 9 above, for N=3.)

A quantity very useful to calculate is N

if , the probability to find an electron in the

particular one-electron level i, when the N-electron system is in thermal equilibrium.

)( N

N

NEPfi (5)

Summation extends all over the N-electron states in

which there is an electron in the one-electron level i.

Notice,

)(1 NN

NEPfi (6)

Summation extends all over the N-electron states in

(10)which there is no electron in the one-electron

level i.

Notice also

)(1 1 i

NN

NEPfi (7)

Summation extends all over the (N+1)-electron states in

which there is an electron in the one-electron level i.

Using (4)

)) (1

/

1Tk

iE

NF BN

N

eif

Some manipulations

))

1

11 (

1

/

Tki

NE

NF

NF

NF BN

eif

))(1

//

1

1)) 1

( ][ TkTk BNBNNi

NEFFF

ee

Since the summation extends all over the (N+1)-electron

states in which there is an electron in the one-electron

level i, one obtains,

1)) 1

( /

][1

NBNNi

N

ii ffTkFF

e

(8)

The chemical potential at temperature T is defined by

= FN+1 – FN (9)

Expression (8) takes the form,

1)/(

)1

NBi

N

ii ffTk

e

(10)

Since N is of the order of 1022

, adding one electron will hardly alter the probability N

if .

Thus,

NBi

N

ii ffTk

e)/(

)

1

Hence,

))(

1

/1

T

Bk

ie

fN

i

(11)

Probability to find an electron in the particular one-

electron level i, of energy i, when the N-electron

system is in thermal equilibrium at temperature T.

Notice N

if is also the mean number of electrons in the one-electron level i. Hence,

iTk

i Bi

N

i

e

fN))(

1

/1

(12)

In subsequent expressions, the explicit reference to N is dropped (it will be implicitly

assumed.)

TkBef

/)(

1

1 )(

Fermi-Dirac distribution (13)

Implicitly, the chemical potential carries information of N.

T=0 T1 T2

f ()T2 > T1 > 0

1/2

1

Fig.11 The Fermi-Dirac distribution

III.2 Application of the Fermi-Dirac distribution to semiconductors

For any given system, it is (initially) hard to evaluate explicitly the value of the

chemical potential. For the case of semiconductors, we will follow an intuitive approach.

III.2.1 Intrinsic case. (Location of the chemical potential)

Let’s consider first the intrinsic case, where the concentration of electron n( e-/cm3) in

the conduction band is equal to the concentration of holes p( holes/cm3) in the valence

band. n (electrons/cm

3) = p (holes/cm

3)

Recall that we consider a hole as the absence of an electron.

Thus,

If f() is the probability to find an electron in a level of energy ,

then 1- f() is the probability that the level of energy is empty.

In other words,

1- f() is the probability to find a hole at the energy level . (14)

Notice also the symmetry of the Fermi-Dirac distribution f() around :

the probability f() that a state above is filled is the same as

the probability 1 - f() that a state below is empty (15)

Expression (14) and (15) imply that, for the intrinsic case, the chemical potential must be located near the middle of the band-gap.

Eg

T≠0

f () Probability to find a hole

1/2

1

Probability to find an electron

Carrier concentration

p n

Fig.12 Location of the chemical potential for the intrinsic case.

III.2.2 Extrinsic case. (Location of the chemical potential)

Eg

T≠0

f () Probability to find a hole

1/2

1

Probability to find an electron

Carrier concentration

p n

Fig.13 Location of the chemical potential for the n-type extrinsic case.

Eg

T≠0

f () Probability to

find a hole

1/2

1

Probability to

find an electron

Carrier

concentration

p n

Fig.14 Location of the chemical potential for the p-type extrinsic case.

IV. JUNCTIONS

IV.1 Diode: p and n materials placed together When the p and n samples are placed together, electrons and holes diffuse and

recombine, leaving behind unbalanced negative and positive regions that end up

constituting what is called the depletion region.

p p-material

n-material

n

Electron energies

Electrically neutral

material

Electrically neutral

material

x

+ + + + + +

NA acceptors /cm3

ND donors/cm

3

+ + + + +

Charged region

i = i (x)

p n

- - - - - -

- - - - - -

+

+

+

+

+

+

+

+

+

+

+

+

Case: NA > ND

W Depletion region

Built-in electric field

Practically no free carriers in this region

Cross section area A

Fig.9a Depletion region of width W formed around the junction.

IV.2 Space charge at a junction

“Within the transition region, electrons and holes are in transit from one side of the junction to the other.

Some electrons diffuse from n to p, and some are swept by the electric filed from p to n (and conversely for holes.)

There are, however, very few carriers within the transition region at any given time, since the electric filed serves to sweep out carriers which have wandered into W.

To a good approximation, we can consider the space charge within the transition region W as due only to the uncompensated donor and acceptor ions.

+

-

qND

qNA

xn0

-xp0

Q=Axn0ND

Q=-Axp0NA

Fig.9b .

where A is the cross section area of the junction.

+

W

o

xn0 -xp0

)(1

DqNdx

d

ε

)(1

AqNdx

d

ε

Electric field

Fig.9c

poAo

A

xqN

dxqNd

)(1

)(1

ε

ε

noDo

D

xqN

dxqNd

)(1

)(1

ε

ε

noDpoAo xNq

xNq

εε

The electric field )(x creates a potential difference V0 across the junction.

dx

dVx )( or

no

po

no

po

x

x

x

x

dxxdV )(

no

po

x

x

o dxxV )( = Wo2

1

Since

noDpoA xNxN and Wxx nopo

2

2

1W

NN

NNqV

DA

DAo

ε

This expression underscore the relationship between the voltage across the junction and

the width of the depletion layer.

IV.3 Chemical potential at pn junctions

At temperature T, semiconductor materials are characterized by their chemical potential, an energy level located somewhere in the band-gap. Its exact location depends on the doping (acceptors and donors) concentrations.

p n

Electron

energies

Chemical

potential

Chemical

potential

Energy levels before of the individual p-type and n-type silicon before joining them. Each material has its own chemical potential level.

When the materials are joined, electrons diffuse from n to p (leaving behind unbalanced positive donor ions) and recombine with hole in the p region (thus creating negative

acceptor ions. holes diffuse from p to n. While diffusion takes place, an electric field i

(that opposes the diffusion process) builds-up until a steady state is reached. In this situation a single Fermi level characterizes the whole (p-n junction) system.

p

n

Built-in internal electric field i = i(x)

Fermi level

-

- -

+

+

+ i

Energy (e- )

x

p n

i

e Vo Vo

p

n

Electrostatic potential

Energy of e-

x p

n

i

x

Fermi level

W

V. FORWARD BIAS, REVERSE BIAS

Forward bias

i

e Vo

Energy of e-

x

Energy of e-

Vf Vf

x

p n

i f

Vf

e (Vo –Vf )

Vf

+ -

p

n n p

Reverse bias

i

e Vo

Energy of e-

X

Energy of e-

VR VR

X

p n i

R

e (Vo +VR ) VR

VR

+ -

Origin of the drift current Io

Energy of e-

X

p n i

f

Vf

+ -

n p

i

i

Drift of thermally

generated electrons

Drift of thermally

generated holes

Thermally

generated e-h pairs Vf

Io

Thermally

generated e-h pairs

Io

Electrons thermally generated near the junction are swept by the internal electric field that exists across the junction. Similar situation happens to the thermally generated holes. The drift mechanism illustrated in the figure above constitute the origin of the unavoidable drift current I0. Notice, the electrons and holes will be swept by whatever electric field is

present across the junction, regardless of the actual value of . That is, the rate of charges crossing the junction is somehow independent of the forward or reverse bias voltages. Once they are generated nearby the junction, any electric field will sweep them across. Thus, I0 depends only on how fast are the e-h pairs generated, which is influenced by the band gap Eg and temperature.

VI. Comparison of charge-carrier population at each side of

the junction

nn: concentration of e- in the n-side

np: concentration of e- in the p-side

pn: concentration of holes in the n-side

pp: concentration of holes in the p-side

The sub-index 0 stands for “values when the system is in equilibrium”

npo = nno e-

qVo /kT

From thermodynamics considerations:

1

i

p n

i

e Vo

Energy

of e-

X

e- carrier

density population

n(p-side) = npo

n(n-side) = nno nno

npo

p-region n-region

Similarly:

Vo

p-region n-region

Electrostatic

potential

X hole+ carrier

density population

From thermodynamics considerations:

p(n-side) = pno

p(p-side) = ppo

pno

Ppo

p n

i

From 1 and 2 :

npo ppo = nno pno 3

In other words: nopo (p-side) = nopo (n-side)

Minority and majority carriers

pn: concentration of holes in the n-side

pp: concentration of holes in the p-side

The sub-index 0 stands for “values when the system is in equilibrium”

pno

i (x)

Vo

p-region n-region

Electrostatic

potential

x

From thermodynamics

considerations: 2

pno = ppo e-

qVo/kT

pno

ppo

ppo majority

carriers minority

carriers

http://www.chemicool.com /

References:

1. J. R. Cogdell, “Foundations of Electronics,” Prentice Hall, 1999.\Ben G. Streetman,

“Solid State Electronic Devices,”Prentice Hall, 1990.

2. Ashcroft and Mermin, “Solid State Physics,” Saunders College 1976.

3. Ben G. Streetman, “Solid State Electronic Devices,” Prentice Hall1990.