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Flashcards in Semiconductors Deck (41):
1

What is the origin of the term semiconductor?

-semiconductors are so called because their conduction properties are between that of metals and insulators
-they have a wide range of physical properties and are usually in crystalline form although amorphous semiconductors also have interesting properties
-they are versatile, can emit light, absorb light and are key components of transistors

2

Elemental Semiconductors

-the elemental semiconductors are drawn form the Group IV column of the periodic table: C, Si, Ge
-although the diamond form of C is more like n insulator with a gap of 5eV

3

Compound Semiconductors

-compounds formed from elements from two different groups of the periodic table
-there are two main classes, the III--V compounds and the II-VI compounds

4

Identifying a Semiconductor by its Band Structure

-in the energy band diagrams of semiconductors there will be a range of energies which no ban covers i.e. a band gap
-it is the size of this gap that distinguishes semiconductors from insulators, semiconductors have much smaller gaps ~1eV

5

Valence and Conduction Bands in Semiconductors vs Metals and Insulators

-semiconductors are different from metals and insulators because they contain 'almost empty' conduction bands and 'almost full' valence bands
-this implies that the transport of carriers must be seen from both bands
-to describe this we introduce holes in the 'almost full' valence band which are MISSING electron spaces

6

Valence and Conduction Bands and Effective Mass

-the valence band roughly has a negative quadratic shape whilst the conduction band has a positive shape
-this indicates that the effective mass has a different sign in each
-we discuss the convention of describing the conduction in the valence band by holes

7

Valence and Conduction Bands
Definitions

-the valence and conduction bands are the bands closest to the Fermi level and so determine the electrical conductivity
-in non-metals the valence band is the highest energy band below the Fermi energy and the conduction band is the lowest energy band above the Fermi energy

8

Valence and Conduction Bands
Conductivity in Semiconductors

-electrical conductivity in non-metals is determined by the susceptibility of electrons to excitation from the valence band to the conduction band
-due to thermal excitation, some electrons get enough energy to jump the band gap into the conduction band
-once an electron is in the conduction band, it can conduct electricity as can the hole it left behind in the valence band

9

Valence and Conduction Bands
Conductivity in Semimetals and Insulators

-in semimetals, there is some overlap between the energies covered by the valence and conduction bands so electrical conductivity is high since it is easy for an electron to move from the valence to the conduction band
-in insulators the gap between the valence band and conduction band is so large that under normal conditions flow of electrons from the valence to the conduction band is negligible

10

Valence and Conduction Bands
Conductivity in General

-in solids, the ability of electrons to act as charge carriers, to conduct, depends on the availability of vacant electronic states
-this allows electrons to increase their energy when an electric field is applied
-similarly holes in the valence band also allow for conductivity

11

Mobility
Definition

-semiconductors are often characterised by their mobility
-mobility is defined as the drift velocity per electric field:
μ = vd/E = eτ/m*

12

Express conductivity in terms of mobility

σ = n*e*μe + p*e*μh

-where μe is the mobility of the electrons and μh is the mobility of the holes
-n is the number density of electrons and p is the number density of holes
-e is the charge on an electron

13

Estimate the mobility of a metal compared to a semiconductor

-a metal has mobility ~20cm²/Vs whereas at 300K, Si has μe=1400cm²/Vs and μh=450cm²/Vs

14

Intrinsic Behaviour

-intrinsic is the name given to the properties of pure (undoped) semiconductors
-in a pure semiconductor, at low temperatures the conductivity approaches zero and the material behaves as an insulator because there isn't enough energy to excite carriers from the valence to the conduction band
-as the temperature increases the conductivity starts to increase, the number of carriers is a strong function of temperature and is determined by the Fermi-Dirac function

15

Number of Electrons in the Conduction Band
Starting Formula

N = ∫ g(E)f(E)dE
-where the integral is taken from Eg to infinity
-on this energy scale, the top of the valence band is zero and Eg is the beginning of the conduction band so Eg is equal to the width of the energy gap

16

Number of Holes in the Conduction Band
Starting Formula

H = ∫ g(E)[1-f(E)]dE
-where the integral is taken from Eg to infinity
-since holes are where electrons aren't, the probability of finding a hole 'occupied' is 1-f(E)

17

Do electrons in semiconductors behave classically?

-assuming that E-Ef>>kT, we can write:
f(E) = [e^((E-Ef)/kT)+1]^(-1)
~ e^(-(E-Ef)/kT)
-i.e. this approximation indicates that the probability of finding an electron in the conduction band is given by a Boltzmann factor, a classical function
-so unlike in metals where electrons behave quantum mechanically, electrons in semiconductors (within this approximation) behave classically

18

Consequence of Classical Behaviour of Electrons in Semiconductors

-velocity, unlike in a metal, depends on temperature due to the equipartition of energy:
1/2 m* v² = 3/2 kT

19

Free Electron Density of States in Semiconductors

g(E) =
V/2π² * [2m*/ℏ²]^(2/3) *
√(E-Eg)

20

How to derive formulae for n and p?

-sub into the starting formula
-change of variable:
x = E-Eg/kT , dx=dE/kT
-this gives a standard form integral:
∫ √(x)e^(-x) dx = √(π)/2
-this gives the expressions for n and p, they are valid for any semiconductor (doped or not)

21

Formula for n

n = no*e^[(Ef-Eg)/kT]
-where;
no = 2[me*kT/2πℏ²]^(3/2)

22

Formula for p

p = po*e^(-Ef/kT)
-where:
po = 2[mh*kT/2πℏ²]^(3/2)

23

np

-the product np at a given temperature is a constant:
np = 4(kT/2πℏ²)³ (me*mh*)^(3/2) e^(-Eg/kT)

24

Temperature Dependence of the Fermi Energy

Ef =
Eg/2 + 3/4 kTln(mh*/me*)

25

Conductivity in Terms of n and p

σ = σe + σh
= n*e*μe + p*e*μh
-recall:
μ = eτ/m

26

Conductivity in Terms of Temperature


σ = n*e*μe + p*e*μh
-and μ∝T^n with -2
σ = A*e^(-Eg/2kT)
-where A is constant

27

Extrinsic Behaviour

-the electronic properties of semiconductors are strongly affected by impurities, the behaviour resulting from impurities is known as extrinsic
-e.g. if impurity concentration was 1ppm and only 10% were ionised the concentration of electrons from impurities would be 10^20 /m³, ten times the intrinsic number of conduction electrons of Si at room temperature
-electronics properties are ENTIRELY dominated by impurtities
-impurities that add electrons to the conduction band are called donors
-impurities that add holes to the valence band are called acceptors

28

Doping
Definition

-the controlled addition of impurities to host semiconductor material

29

Bohr Model for Impurities
Purpose

-to understand the fundamental properties of doping, there is a simple model that we can apply to:
a) impurities that are added substitutionally, i.e. the impurity occupies the same lattice site that the host atom normally would
b) the impurity elements come from on column either to the left or the right of the host elements i.e. a valence difference between impurity and host of ±1

30

Bohr Model for Impurities
Description

-to model an impurity in a host semiconductor we use the Bohr model of the hydrogen atom with two minor changes:
I) since the impurity is in the electric field of the host and not in free space we modify the Coulomb potential by including the dielectric constant of the host (Ge-16, Si-12)
II) we need to use the effective mass of the electron since it is within the band structure of the host

31

Bohr Model for Impurities
Ionisation Energy and Impurity Object Bohr Radius

Ed = m*/mε² * 13.6eV
r = mε/m* * 0.5Å
-Ed is the ionisation energy of the donor (or acceptor) as a fraction of the ionisation energy of hydrogen
-Ed is measured DOWN from the conduction band and Ea is measured UP from the valence band
-r is the radius of orbit for the electron or hole about the impurity and is expressed as a fraction of the Bohr radius of the hydrogen atom

32

Conductivity in Extrinsic Semiconductors
Description

-follows the intrinsic case
-we will consider the case of a donor i.e. the impurity has one extra electron that will be donated to the system
-the donor density nd, is the number of impurity atoms per unit volume that have been affed to the host
-at any temperature the donors are either neutral (still have their extra electron) or have been ionised (their extra electron has been stripped away)
-the Fermi function indicates whether a donor has been ionised or not

33

Conductivity in Extrinsic Semiconductors
Defining Variables

ndo = neutral donor density
nd+ = ionised donor density
so in total:
nd = ndo + nd+

Ed = donor ionisation energy

34

Conductivity in Extrinsic Semiconductors
Derivation

-the Fermi function tells us which donors have been ionised:
nd0 = nd*f(Eg-Ed)
= nd / [exp[(Eg-Ed-Ef)/kT]+1]
-we will assume that n~nd+, where n is the number of conduction electrons in the conduction band per unit volume:
n = nd+ = nd - ndo
= nd / [exp[(Ef-Eg+Ed)/kT]+1]
-but n=no*e^[(Ef-Eg)/kT], which applies to all semiconductors:
no*e^[(Ef-Eg)/kT]
= nd*e^[-(Ef-Eg+Ed)/kT]
-this gives:
Ef = Eg - Ed/2 + 1/2 kTln(nd/no)
-and conductivity:
σ = μe*e√[no*nd] e^(-Ed/2kT)
= A*e^(-Ed/2kT)

35

Intrinsic Conductivity vs Extrinsic Conductivity

-intrinsic:
σ = A*e^(-Eg/2kT)
-extrinsic:
σ = A*e^(-Ed/2kT)
-for both, the constant A contains the mobility that is temperature dependent but only weakly so compared with to the exponential dependence of n
-the temperature dependence of n dominates the temperature dependence of σ

36

Temperature Dependence of the Mobility
High Temperatures

μ = eτ/m*
-the temperature dependence comes form τ which is the same as in metals
-electrons at the bottom of the conduction band can be treated classically:
1/2 m* v² = 3/2 kT
-mean free path depends on temperature because scattering rate does
-mainly electron-phonon scattering, hence mean free path, l∝1/T
-putting these together we have:
μ = el / [√(m*)[3kT]^(3/2)]
-so
μ ∝ T^(-3/2)

37

Temperature Dependence of the Mobility
Low Temperatures

μ = eτ/m*
-and for electrons in semiconductors we have:
1/2 m* v² = 3/2 kT
-scattering dominated by impurities as in metals BUT the impurities are ionised which makes them stronger scattering centres AND the scattering from these impurities depends on temperature
-we can estimate this as we did for scattering from impurities in a metal:
Σ = 1/nd*l = 1/nd*v*τ
= 1/nd*τ √(2m*/3kT)
~ 10^(-15) m²
-where Σ is the scattering cross section, this is equivalent to a diameter ~300Å
-it is not easy to calculate the temperature dependence of μ, but experimentally it is found that: μ ∝ T^(3/2)

38

Direct Bandgap Semiconductors
Definition

-maximum in the valence band directly opposite the minimum in the conduction band

39

Indirect Bandgap Semiconductors
Definition

-do not have the maximum in the valence band directly opposite the minimum in the conduction band

40

Direct and Indirect Transitions

-in direct bandgap semiconductors, transitions are direct and the conservation of momentum is obvious as the k vector of the electron is the same and p=ℏk
-in indirect transitions the k vector, and therefore the momentum, of the electron changes, there must be a source that can supply the missing k
-the momentum of a photon is 4 orders of magnitude too small to supply this missing momentum, only a phonon has sufficient momentum to participate

41

Detection of an Indirect Bandgap

-since the smallest energy required for a transition is Eg, the presence of an indirect bandgap can be detected by a tail in the absorption spectrum:
Eg - ℏ*ωphonon