Order 1252715: Condensed Matter physics

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Lect-14-Semicond.pdf

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23/10/2017

Semiconductors

One shouldn’t work on semiconductors, that is a

filthy mess; who knows whether any

semiconductors exist.

(Über Halbleiter soll man nicht arbeiten, das ist

eine Schweinerei; wer weiss, ob es überhaupt

Halbleiter gibt.)

Wofgang Pauli, 1931

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Why semiconductors?

• SEMICONDUCTORS: They are here, there, and everywhere

• Computers Silicon (Si) MOSFETs, ICs, CMOS

laptops, anything “intelligent”

• Cell phones, pagers Si ICs, GaAs FETs, BJTs

• CD players AlGaAs and InGaP laser diodes, Si photodiodes

• TV remotes, mobile terminals Light emitting diodes (LEDs)

• Satellite dishes InGaAs MMICs (Monolithic Microwave ICs)

• Fiber networks InGaAsP laser diodes, pin photodiodes

• Traffic signals, car GaN LEDs (green, blue)

taillights InGaAsP LEDs (red, amber)

• Air bags Si MEMs, Si ICs

• and, they are important, especially to Elec.Eng.& Computer Sciences

Introduction Semiconductors are materials whose electrical

properties lie between Conductors and Insulators. Ex : Silicon and Germanium Difference in conductivity

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Semiconductor Materials • Elemental semiconductors – Si and Ge (column IV of periodic

table) –compose of single species of atoms

• Compound semiconductors – combinations of atoms of column III and column V and some atoms from column II and VI. (combination of two atoms results in binary compounds)

• There are also three-element (ternary) compounds (GaAsP) and four-elements (quaternary) compounds such as InGaAsP.

gap size

(eV)

InSb 0.18

InAs 0.36

Ge 0.67

Si 1.11

GaAs 1.43

SiC 2.3

diamond 5.5

MgF2 11

valence

band

conduction

band

Can a material with

μ in a band gap

conduct?

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Semiconductor materials

Semiconductor Materials • The wide variety of electronic and optical properties of these

semiconductors provides the device engineer with great flexibility in the design of electronic and opto-electronic functions.

• Ge was widely used in the early days of semiconductor development for transistors and diods.

• Si is now used for the majority of rectifiers, transistors and integrated circuits.

• Compounds are widely used in high-speed devices and devices requiring the emission or absorption of light.

• The electronic and optical properties of semiconductors are strongly affected by impurities, which may be added in precisely controlled amounts (e.g. an impurity concentration of one part per million can change a sample of Si from a poor conductor to a good conductor of electric current). This process called doping.

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Intrinsic semiconductors

• Pure, i.e. not doped, semiconductors are called intrinsic. • For the electronic properties of a semiconductor, “pure”

means pure within 1 ppm to 1 ppb.

Intrinsic Material A perfect semiconductor crystal with no impurities or lattice defects is called an

intrinsic semiconductor.

At T=0 K –

No charge carriers

Valence band is filled with electrons

Conduction band is empty

At T>0

Electron-hole pairs are generated

EHPs are the only charge carriers in

intrinsic material

Since electron and holes are created in

pairs – the electron concentration in

conduction band, n (electron/cm3) is

equal to the concentration of holes in the

valence band, p (holes/cm3).

Each of these intrinsic carrier

concentrations is denoted ni.

Thus for intrinsic materials n=p=ni

Electron-hole pairs in the covalent bonding

model in the Si crystal.

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Doped semiconductors

• A very small amount of impurities can have a big influence on the conductivity of a semiconductor.

• Controlled addition of impurities is called doping.

• There are two types of doping: n doping (impurities increasing #electrons) and p doping (impurities increasing #of holes).

• Typical doping levels are in the order of 1019 to 1023 impurity atoms per m3. Remember: Si has a concentration of 5*1028 atoms per m3 and an intrinsic carrier concentration of 1016 electrons/holes per m3 at room temperature.

- -

However, like all other elements it would prefer to have 8 electrons in its outer shell

The Silicon Atomic Structure

Silicon: our primary example and

focus

Atomic no. 14

14 electrons in three shells: 2 ) 8 ) 4

i.e., 4 electrons in the outer "bonding"

shell

Silicon forms strong covalent bonds with

4 neighbors

3s2 3p2 2s2 2p6 1s2

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Si and Ge are tetravalent elements – each atom of Si (Ge) has 4 valence

electrons in crystal matrix

T=0 all electrons are bound in

covalent bonds

no carriers available for

conduction.

For T> 0 thermal fluctuations can

break electrons free creating

electron-hole pairs

Both can move throughout the lattice

and therefore conduct current.

Electrons and Holes

Excite electron from valance

band to conduction band, e.g.,

absorbing a photon or thermal

excitation.

Absence of electron in

valence

band called a “hole” – treat

holes as elementary particles.

To conserve charge, if electron

is negative, hole is positive

charged.

Electron can fall back into

hole, releasing energy ,e.g.

emitting photon, and

annihilating

electron and hole.

Holes

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Effective Mass of Electrons

As before, describe curvature at

bottom of band in terms of effective

mass.

Near bottom of conduction band,

where k=k min

And the corresponding group

velocity is

The effective mass is defined as,

Recall, for free electron

Effective Mass of Holes

valence band

convension is:

“hole”

For the top of the valence band, can

write:

And define effective mass for holes,

Energy to move hole away from top of

band is:

And corresponding hole velocity is:

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Alternative definition is to define effective mass as being the

quantity that satisfies Newton’s second law

Effective Mass

A force is applied to an electron, then work done on electron

equal to its change in energy – consider work done per unit

time

Change in energy per unit time:

Equating:

used

(chain rule)

(since )

Then:

Effective mass

as a function of

momentum

The effective mass

m e

*/m e

m h

*/m e

InSb 0.014 0.4

InAs 0.022 0.4

Ge 0.6 0.28

Si 0.43 0.54

GaAs 0.065 0.5

Na 1.2

Cu 0.99

Sb 0.85

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Electrons and Holes

Electron-hole pairs in a semiconductor.

The bottom of the conduction band

denotes as Ec and the top of the valence

band denotes as Ev.

For T>0

some electrons in the valence band receive

enough thermal energy to be excited

across the band gap to the conduction

band.

The result is a material with some electrons

in an otherwise empty conduction band and

some unoccupied states in an otherwise

filled valence band.

An empty state in the valence band is

referred to as a hole.

If the conduction band electron and the

hole are created by the excitation of a

valence band electron to the conduction

band, they are called an electron-hole

pair (EHP).

Increasing conductivity by temperature

15 0 20 0 25 0 30 0 35 0 40 0 45 0 50 0 10 0

1 10 3

1 10 4

1 10 5

1 10 6

1 10 7

1 10 8

1 10 9

1 10 10

1 10 11

1 10 12

1 10 13

1 10 14

1 10 15

1 10 16

1 10 17

Carrier Concentration vs T emp (in Si)

T em p erature (K )

In tr

in si

c C

o n

ce n

tr at

io n

( cm

^ -3

)

ni T

Therefore the conductivity of a semiconductor is influenced by temperature

As temperature increases, the number of free electrons and holes created increases exponentially.

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Adding Electrons or Holes with Impurities: Doping

A phosphorous atom P replaces a silicon atom. The

P atom is like an Si atom plus an extra electron.

Extra electron goes in conduction band

P is an electron donor in silicon – also called an

n-type dopant. n is symbol for electron density

n- and p-doping

donor atom acceptor atom

Analogously, an Al replacing a silicon atom. The

Al atom has one fewer electrons than Si. Gives rise to a

hole. Al is an electron acceptor in silicon – also called an

p-type dopant. p is symbol for hole density.

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Adding Electrons of Holes with Impurities: Doping

All electrons

in covalent

bond with 2

electrons

Extra

electron

Extra hole

Donor and acceptors in covalent bonding model

In the covalent bonding model, donor and acceptor atoms can be visualized as shown in the Figure. An Sb atom (column V) in the Si lattice has the four necessary valence electrons to complete the covalent bonds with the neighboring Si atoms, plus one extra electron. This fifth electron does not fit into the bonding structure of the lattice and is therefore loosely bound to the Sb atom.

Donor and acceptor atoms

in the covalent bonding

model of a Si crystal.

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Donor and acceptors in covalent bonding model

A small amount of thermal energy enables this extra electron to overcome its coulombic binding to the impurity atom and be donated to the lattice as a whole. Thus it is free to participate in current conduction. This process is a qualitative model of the excitation of electrons out of a donor level and into the conduction band.

Similarly, the column III impurity Al has only three valence electrons to contribute to the covalent bonding, thereby leaving one bond incomplete. With a small amount of thermal energy, this incomplete bond can be transferred to other atoms as the bonding electrons exchange positions.

Donor and acceptor atoms

in the covalent bonding

model of a Si crystal.

Adding Electrons or Holes with Impurities: Doping Consider n-type dopant, e.g. P in Si

Extra electron in conduction band acts like

a free electron with mass, m*

but also have positive charge in nucleus of P

Forms a bound state like a H atom –

Attract with potential:

Energy eigenstates of H atom

Rydberg constant

Radius of wave function

mass of electron

Analogously, for a hydrogenic

Impurity state we have:

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n-doping Estimate binding energy with Bohr model:

using the modifications

phosphorus

penta-valent,

one electron too many

order of magnitude

The radius of this is quite big, 30 times the

Bohr radius

Adding Electrons or Holes with Impurities: Doping

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Extrinsic Material By doping, a crystal can be altered so that it has a predominance of either

electrons or holes. Thus there are two types of doped semiconductors, n-type (mostly electrons) and p-type (mostly holes). When a crystal is doped such that the equilibrium carrier concentrations n0 and po are different from the intrinsic carrier concentration ni, the material is said to be extrinsic.

Donor impurities (elements

of group V): P, Sb, As

Acceptor elements (group

III): B, Al, Ga, In

The valence and conduction bands of

silicon with additional impurity energy

levels within the energy gap.

When impurities or lattice defects are introduced, additional levels are created in the energy bands structure, usually within the band gap.

Total number of electrons

III – Al – 13

IV – Si – 14

V - P - 15

Extrinsic Material – donation of electrons An impurity from column V introduces an energy level very near the conduction band in Ge or Si. This level is filled with electrons at 0 K, and very little thermal energy is required to excite these electrons to the conduction band. Thus, at about 50-100 K nearly all of the electrons in the impurity level are "donated" to the conduction band. Such an impurity level is called a donor level, and the column V impurities in Ge or Si are called donor impurities.

Donation of electrons from

a donor level to the

conduction band

n-type material

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Extrinsic Material – donation of electrons

From figure we note that the material doped with donor impurities can have a considerable concentration of electrons in the conduction band, even when the temperature is too low for the intrinsic EHP concentration to be appreciable. Thus semiconductors doped with a significant number of donor atoms will have n0>>(ni,p0) at room temperature. This is n-type material.

Donation of electrons from

a donor level to the

conduction band

n-type material

Extrinsic Material – acceptance of electrons

Acceptance of valence band

electrons by an acceptor level,

and the resulting creation of

holes.

Atoms from column III (B, Al, Ga, and In) introduce impurity levels in Ge or Si near the valence band. These levels are empty of electrons at 0 K. At low temperatures, enough thermal energy is available to excite electrons from the valence band into the impurity level, leaving behind holes in the valence band.

P-type material

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Extrinsic Material – acceptance of electrons

Acceptance of valence band

electrons by an acceptor level,

and the resulting creation of

holes.

Since this type of impurity level "accepts" electrons from the valence band, it is called an acceptor level, and the column III impurities are acceptor impurities in Ge and Si. As figure indicates, doping with acceptor impurities can create a semiconductor with a hole concentration p0 much greater than the conduction band electron concentration n0 (this is p-type material).

P-type material

Statistical Mechanics of Semiconductors

Recall from Lecture 3 – density of states for free electrons

per unit volume

Electrons in conduction band like

free electrons but with mass m*, can write:

Similarly the density of states for holes

near the top of the valence band are: