11.1 Band Theory of Solids 11.2 Semiconductor Theory 11.3 Semiconductor Devices 11.4 Nanotechnology

CHAPTER 11Semiconductor Theory and Devices

It is evident that many years of research by a great many people, both before and after the discovery of the transistor effect, has been required to bring our knowledge of semiconductors to its present development. We were fortunate to be involved at a particularly opportune time and to add another small step in the control of Nature for the benefit of mankind.

- John Bardeen, 1956 Nobel lecture

11.1: Band Theory of Solids

Three categories of solids, based on electrical conductivity conductors semiconductors insulators

Energy

Insulator Metal Semimetal Semiconductor Semiconductor

Can be classified by Energy band Theory

Electrical Resistivity and Conductivity at 293 K

The electrical conductivity at room temperature is quite different Metals and alloys have the highest conductivities followed by semiconductors and then by insulators

Semiconductor Conduction The free-electron model from Chapter 9 does not apply to

semiconductors and insulators, since these materials simply lack enough free electrons to conduct in a free-electron mode.

There is a different conduction mechanism for semiconductors than for normal conductors.

Typical conductor Semiconductor

The resistivity increases dramatically as T → 0.

Resistivity vs. Temperature

Band Theory of Solids In order to account for decreasing resistivity

with increasing temperature as well as other properties of semiconductors,a new theory known as the band theory is introduced.

How Energy Band formed?

In solids with a large number of atoms,

the energy levels are split into nearly continuous energy bands.

Each band consists of a number of closely spaced energy levels.

Consider initially the known wave functions of two hydrogen atoms far enough apart so that they do not interact.

Interaction of the wave functions occurs as the atoms get closer

Band Theory of Solids

An atom in the symmetric state has a nonzero probability of being halfway between the two atoms,

while an electron in the antisymmetric state has a zero probability of being at that location.

Symmetric state Antisymmetric state

Band Theory of Solids

In the symmetric case the binding energy is slightly stronger resulting in a lower energy state, than antisymmetric.

In a real solid with a large number of atoms, nearly continuous energy bands, with each band consisting of a number of closely spaced energy levels.

When more atoms are added,further splitting of energy levels.

Symmetric

Antisymmetric

Those Energy splitting occurs at all possible energy levels (1s, 2s, and so on)

0.367 nm

Solid Sodium

Solid Sodium (1S2 2S2 2P6 3S1) Conductor

원자가 N개있으면,

3S 궤도에는원자당 2개의전자총 2N 개수용가능Na 원자는 1 개의전자만있으므로 N개의전자만존재

페르미에너지는띠중간에생긴다.

전기장을걸어주면 운동에너지를가지고전자들이 쉽게움직인다.

Conductor

EF

Solid: Carbon Diamond (2S2 3P2) and Silicon (3S2 3P2)

Carbon Silicon Internuclear distance

Conduction Band

Valence Band

Forbidden

4N atoms in (2 x 4N) possible levels Insulator

4N

4N

Valance Band, Conduction Band, Forbidden Gap

Silicon (3S2 3P2): Insulator, but Semiconductor

Silicon

The bandgap energy (1 eV) is small, Many Electrons can easily be excited to the conduction band in Room T. Fermi Energy is placed at the gap center. Its conductivity is in between conductor and insulator, “Semiconductor”

1 eV

EF

Kronig-Penney Model for Energy gap An effective way to understand the energy gap in semiconductors is to

model the interaction between the electrons and the lattice of atoms.

In 1931 Kronig and Penney assumed that an electron experiences an infinite one-dimensional array of finite potential wells.

Each potential well models attraction to an atom in the lattice, the well size must correspond roughly to the lattice spacing.

Kronig-Penney square-well potentialLattice spacing (2a)

Kronig-Penney Model

Since the electrons are not free, their energies are less than the height V0 of each of the potentials, but the electron is essentially free in the gap 0 < x < a, where it has a wave function of the form

E In the region between a < x < a + b the electron can tunnel through the wave function loses its oscillatory solution and it becomes exponential:

where

where

Matching solutions at the boundaries; x = 0 and x = a,

K is another wave number.

There exist restricted forbidden zones

-1 < < +1

The gaps occur regularly at ka = n

forbidden

forbidden

Kronig-Penney Model

The allowed ranges k are referred to as Brillouin zones.

Condition for Bragg Diffraction:

Energy Band Theory in Momentum (k) Space- Brillouin Zones defined by Bragg Diffraction -

In the case = 0:

It was Forbidden Zone Boundary in K-P Model.

일차원운동을하는전자에서

|k| =/a 일때파동은앞뒤로반사를하고따라서격자의주기와같은정상파에해당하는파만Schrodinger 방정식의해가된다.

In the case = 0:

Origin of Forbidden band

ForbiddenEnergy Band(Energy Bandgap)

|k| =/a 일때다른해가존재하지않으므로, E1, E2사이의에너지를가질수없다.

Brillouin Zones

1st and 2nd Brillouin zonesin 2-D square lattice.

Brillouin Zones

1st and 2nd Brillouin zonesin 2-D square lattice.

Brillouin Zone and Electron Energy

에너지가작을때전자의에너지는k2에비례하므로 2차원공간에서에너지등고선은단순한원이된다.

그러나 k가증가함에따라등고선은점차가까워지고더욱찌그러진다.

전자는영역경계에가까워질수록반사기더쉬워지는데, 이는 입자적관점에서보면양이온의주기적배열때문에생긴다.

Equi-Energy Contour mapin eV unit.

Origin of Forbidden band

AllowedEnergy Band

ForbiddenEnergy Band(Energy Bandgap)

Forbidden Energy Band(Energy Bandgap)

|k| =/a 에서두에너지값을가진다. 낮은한개는첫번째 BZ,높은값은두번째 BZ.

이두값들사이에불연속이존재하고금지된띠 (Forbidden Band)에해당.

Origin of Forbidden band

ForbiddenEnergy Band(Energy Bandgap)

에너지가낮을때는자유전자이론과일치

에너지가높아짐에따라곡선이찌그러지므로에너지상태의수는더많아진다.

에너지가더커지게되면가능한에너지상태들은더제한을받게되고영역의구석으로가게되어

약 6.5 eV 이상이되면어떤에너지상태도존재할수없다.

Brillouin 영역에서의전자에너지의분포. (점선은자유전자이론에의한분포)

Forbidden band

주어진한방향을따르는인접한 BZ들사이에는반드시 Energy gap이존재하지만어떤결정에서다른방향으로의여러간격들이중첩되면금지된띠가없어질수있다.

두종류의결정에서세방향의 k에대한 E의곡선. (a) 금지된띠가있는경우 (b) 허용된에너지띠들이중첩되어금지된띠가존재하지않는경우.

Conduction

Valence

ForbiddenInsulator

Conductor(s1 , Na)

Conductor(s2 , Mg)

Brillouin Zone and Solids

Effective Mass• 전자들이결정과상호작용을하므로외부에서가한힘에대한반응이자유전자의반응과다르다.즉,전자의질량이이반응때문에변화하는것으로볼수있고이것이유효질량이다.

• 유효질량에관한정의는파군의속도에 (Group velocity)관한식에서구할수있다.

F = m*a =

: Group velocity

k

k

11.2: Semiconductor Theory• Electrons and Holes

• Intrinsic Semiconductors

• Impurity Semiconductors by Doping

• p-type (doped by Acceptor) and n-type (doped by Donor)

Semiconductor Theory At T = 0 we expect all of the atoms in a solid to be in the ground

state. The distribution of electrons (fermions) at the various energy levels is governed by the Fermi-Dirac distribution:

β = (kT)−1 and EF is the Fermi energy.

When the temperature is increased from T = 0, more and more atoms are found in excited states. The increased number of electrons in excited states explains the temperature

dependence of the resistivity of semiconductors. Only those electrons that have jumped from the valence band to

the conduction band are available to participate in the conductionprocess in a semiconductor.

More and more electrons are found in the conduction band as the temperature is increased, the resistivity of the semiconductor therefore decreases.

Semiconductor TheoryFind the relative number of electrons with energies 0.10 eV, 1.0 eV, and 10 eV above the valence band at room temperature (293 K).

Conduction

Valence

Forbidden

Note that 1.0 eV = 1.60 x 10-19 J.The Fermi energy is at the top of the valence band, so the energy above the valence band is E - EF. EF

This implies that the number of electrons available for conduction drops off sharply as the band gap increases.

Holes and Intrinsic Semiconductors

When electrons move into the conduction band, they leave behind vacancies in the valence band. These vacancies are called holes.

Because holes represent the absence of negative charges, it is useful to think of holes as positive charges.

Whereas the electrons move opposite to the applied E field, the holes move in the direction of the electric field.

A semiconductor in which there is a balance between the number of electrons in the conduction band and the number of holes in the valence band is called an intrinsic semiconductor.

Examples of intrinsic semiconductors include pure carbon and germanium.

Impurity Semiconductor It is possible to fine-tune a semiconductor’s properties by adding a

small amount of another material, called a dopant, to the semiconductor creating what is a called an impurity semiconductor.

As an example, silicon has four electrons in its outermost shell (this corresponds to the valence band) and arsenic has five.

Thus while four of arsenic’s outer-shell electrons participate in covalent bonding with its nearest neighbors (just as another silicon atom would), the fifth electron is very weakly bound.

It takes only about 0.05 eV to move this extra electron into the conduction band.

Adding (or, Doping) only a small amount of arsenic to silicon greatly increases the electrical conductivity.

• Doping : 반도체에불순물을첨가하는것. (수-수백 ppm)• Dopant : donor(As, P, Bi, Sb), acceptor (Ga, In, Te)• 화합물반도체: GaAs, GaP, InSb, InP (III-V 화합물)

In addition to intrinsic and impurity semiconductors, there are many compound semiconductors, which consist of equal numbers of two kinds of atoms

n-type and Donor Impurity levels Addition of arsenic (V) to silicon (IV) creates what is known as

an n-type semiconductor (n for negative), because it is the electrons close to the conduction band that will eventually carry electrical current.

The new arsenic energy levels just below the conduction band are called donor levels because an electron there is easily donated to the conduction band.

When indium (III) is added to silicon (IV). Indium has one less electron one extra hole per indium atom. These holes creates extra energy levels just above the valence

band, because it takes relatively little energy to move another electron into a hole

Those new indium levels are called acceptor levels because they can easily accept an electron from the valence band.

Easier to think in terms of the flow of positive charges (holes) in the direction of the applied field, so we call this a p-type semiconductor (p for positive).

p-type and Acceptor Impurity levels

N-Type (Electrons) and p-type (Holes)

n-Type

p-Type

Hall effectThe magnitude and sign of the Hall voltage allow one to calculate the density and sign of the charge carriers in a conductor or semiconductor and verify that charges in carriers in p-type and n-type materials have different signs.

x

z

A thin strip of metal immersed in a magnetic field is used to test the Hall effect. (a) Here negative charge carriers are forced to the right. (b) In this configuration, the buildup of negative charge on the right side (with a corresponding positive charge on the left) sets up the electric field as shown. This creates an electric force on the charge carriers equal and opposite to the magnetic force. The voltmeter (reading VH) can detect the magnitudeand sign of the potential difference across the strip.

In equilibrium,

The sign of the Hall voltage define the sign of the charge carriers.

Electrons, Holes, and Dispersion curve (or, Dispersion relation)

A dispersion relation is the relationship between wavevector (k-vector) and energy (E) in a band structure.

E = E (k), or = (k)

The dispersion relation determines how electrons respond to forces (via the concept of effective mass).

An electron floating in space has the dispersion relation E= 2k2/(2m), where m is the (real) electron mass.

In the conduction band of a semiconductor, the dispersion relation is instead E= 2k2/(2m*) (m* is the effective mass), so a conduction-band electron responds to forces as if it had the mass m*.

Electrons near the top of the valence band behave as if they have negative mass:(When a force pulls the electrons to the right, these electrons actually move left.)

Electrons near the bottom of the band have positive effective mass.

The concept of holes is useful as a shortcut for calculating the total current of an almost-full band. Hole can be regarded as a positive-charge, positive-mass quasiparticle. A hole with positive charge and positive mass responds to electric and magnetic fields

in the same way as an electron with negative charge and negative mass. An analogy is a bubble underwater: The bubble moves the same direction as the water, not opposite.