© m.n.a. halif & s.n. sabki chapter 2: energy bands & carrier concentration in thermal...

© M.N.A. Halif & S.N. Sabki

CHAPTER 2: ENERGY BANDS CHAPTER 2: ENERGY BANDS & CARRIER CONCENTRATION & CARRIER CONCENTRATION IN THERMAL EQUILIBRIUMIN THERMAL EQUILIBRIUM


OUTLINEOUTLINE

2.1 INTRODUCTION:

2.1.1 Semiconductor Materials

2.1.2 Basic Crystal Structure

2.1.3 Basic Crystal Growth technique

2.1.4 Valence Bonds

2.1.5 Energy Bands

2.2 Intrinsic Carrier Concentration

2.3 Donors & Acceptors


2.1.1 SEMICONDUCTOR MATERIALS2.1.1 SEMICONDUCTOR MATERIALS

To understand the characteristic of semiconductor materials – you need PHYSICS.

Basic Solid-State Physics – materials, may be grouped into 3 main classes:

(i) Insulators,

(ii) Semiconductors, and

(iii) Conductors- Refer to your basic electronic devices knowledge (EMT

111/4).- Electrical conductivity : =1/.


Figure 2.1.Figure 2.1. Typical range of conductivities Typical range of conductivities for insulators, semiconductors, and for insulators, semiconductors, and conductors.conductors.


SEMICONDUCTOR’S ELEMENTSSEMICONDUCTOR’S ELEMENTS

The study of semiconductor materials – 19th century.

Early 1950s – Ge was the major semiconductor material, and later in early 1960s, Si has become a practical substitute with several advantages:

(i) better properties at room temperature,

(ii) can be grown thermally – high quality silicon oxide,

(iii) Lower cost, and

(iv) Easy to get, silica & silicates comprises 25% of the Earth’s crust.


COMPOUND SEMICONDUCTORSCOMPOUND SEMICONDUCTORS

Please refer to the Table 2 (in Sze, pg. 20)

Types of compounds:

(i) binary compounds

- combination of two elements.

- i.e GaAs is a III – IV.

(ii) ternary and quaternary compounds

- for special applications purposes.

- ternary compounds, i.e alloy semiconductor AlxGa1-xAs (III – IV).

- quaternary compounds with the form of AxB1-xCyD1-y , so-called combination of many binary & ternary compounds.

- more complex processes.

GaAs – high speed electronic & photonic applications


2.1.2 BASIC CRYSTAL STRUCTURE2.1.2 BASIC CRYSTAL STRUCTURE

Lattice – the periodic arrangement of atoms in a crystal.

Unit Cell – represent the entire lattice (by repeating unit cell throughout the crystal)

The Unit Cell

3-D unit cell shown in Fig. 2.2.

Relationship between this cell & the lattice – three vectors: a, b, and c (not be perpendicular to each other, not be equal in length).

Equivalent lattice point in 3-D:

m, n, and p - integers.

Fig. 2.2.Fig. 2.2. A generalized A generalized primitive unit cell.primitive unit cell.

cpbnamR

(1)


Figure 2.3.Figure 2.3. Three cubic-crystal unit cells. ( Three cubic-crystal unit cells. (aa) Simple ) Simple cubic. (cubic. (bb) Body-centered cubic. () Body-centered cubic. (cc) Face-centered ) Face-centered

cubic.cubic.

UNIT CELLUNIT CELL

• simple cubic (sc) – atom at each corner of the cubic lattice, each atoms has 6 equidistant nearest-neighbor atoms.• body-centered cubic (bcc) – 8 corner atoms, an atom is located at center of the cube. - each atom has 8 nearest-neighbor atoms.• face-centered cubic (fcc) – 1 atom at each of the 6 cubic faces in addition to the 8 corner atoms. 12 nearest-neighbor atoms.

Large numberof elements


Fig. 2.4Fig. 2.4. (. (aa) Diamond lattice. () Diamond lattice. (bb) Zincblende ) Zincblende lattice.lattice.

THE DIAMOND STRUCTURETHE DIAMOND STRUCTURE

• Si and Ge have a diamond lattice structure shown in Fig. 2.4.

• Fig. 2.4(a) – a corner atom has 1 nearest neighbor in the body diagonal direction, no neighbor in the reverse direction.

• Most of III-IV compound semiconductor (e.g GaAs) have zincblende lattice structure.

a/2

a/2


Figure 2.6.Figure 2.6. Miller indices of some important planes in a cubic Miller indices of some important planes in a cubic crystal.crystal.

CRYSTAL PLANES & MILLER INDICESCRYSTAL PLANES & MILLER INDICES

• Crystal properties along different planes are different – electrical & other devices characteristics can be dependent on the crystal orientation – use Miller indices.• Miller indices are obtained using the following steps: (i) find the interfaces of the plane on the 3 Cartesian coordinates in term of the lattice constant. (ii) Take the reciprocal of these numbers and reduce them to smallest 3 integers having the same ratio. (iii) Enclose the result in parentheses (khl) as the Miller indices for a single plane.


BASIC CRYSTAL GROWTH TECHNIQUEBASIC CRYSTAL GROWTH TECHNIQUE

• Refer to EMT 261.• 95% electronic industry used Si.• Steps from SiO2 or quartzite.• Most common method called Czochralski technique (CZ).• Melting point of Si = 1412ºC.• Choose the suitable orientation <111> for seed crystal.

Si ingot

Figure 2.8.Figure 2.8. Simplified schematic drawing of the Simplified schematic drawing of the Czochralski puller. Clockwise (CW), Czochralski puller. Clockwise (CW),

counterclockwise (CCW).counterclockwise (CCW).

Prof. J. Czochralski, 1885-1953


Figure 2.11.Figure 2.11. ( (aa) A tetrahedron bond. () A tetrahedron bond. (bb) ) Schematic two-dimensional Schematic two-dimensional representation of a tetrahedron bond.representation of a tetrahedron bond.

VALENCE BONDSVALENCE BONDS

• Recall your basic electronic devices - EMT 111/4.• Fig. 2.11(a) – each atom has 4 ē in the outer orbit, and share these valence ē with 4 neighbors.• Sharing of ē called covalent bonding – occurred between atoms of same and different elements respectively.

Example:• GaAs – small ionic contribution that is an electrostatic interactive forces betweeneach Ga+ ions and its 4 neighboring As- ions-means that the paired bonding ē spend more time in the As atom than in the Ga atom. Figure 2.12.Figure 2.12. The basic bond The basic bond

representation of intrinsic silicon. (representation of intrinsic silicon. (aa) A ) A broken bond at Position A, resulting in a broken bond at Position A, resulting in a conduction electron and a hole. (conduction electron and a hole. (bb) A ) A broken bond at position B.broken bond at position B.


ENERGY BANDSENERGY BANDS

• Energy levels for an isolated hydrogen atom are given by the Bohr model*:

eVnnh

qmEH 2222

0

40 6.13

8

mo – free electron mass (0.91094 x 10-30kg)q – electronic charges: 1.6 x 10-19 C0 – free space permittivity (8.85418 x 10-12 F/m)h – Planck constant (6.62607 x 10-34 J.s)n – positive integer called principle quantum number

1 eV = 1.6 x 10-19 J

• For 1st energy state or ground state energy level, n = 1, EH = -13.6eV.

• For the 1st excited energy level, n = 2.

• For higher principle quantum number (n ≥ 2), energy levels are split.* Find Fundamentals Physics books @ KUKUM Library!!

(2)

Neils Bohr, 1885-1962 Nobel Prize in physics

1922


For two identical atoms:

When they far apart – have same energy.

When they are brought closer:

– split into two energy levels by interaction between the atoms.

- as N isolated atoms to form a solid.

- the orbit of each outer electrons of different atoms overlap & interact with each other.

- this interactions cause a shift in the energy levels (case of two interacting atoms).

- when N>>>, an essentially continuous band of energy. This band of N level may extend over a few eV depending on the inter-atomic spacing of the crystal.

ENERGY BANDS (cont.)ENERGY BANDS (cont.)


Figure 2.13.Figure 2.13. The splitting of a The splitting of a degenerate state into a band of degenerate state into a band of

allowed energies.allowed energies.

Equilibrium inter-atomic distance of the crystal.

Figure 2.14.Figure 2.14. Schematic presentation Schematic presentation of an isolated silicon atom.of an isolated silicon atom.



Figure 2.15.Figure 2.15. Formation of energy Formation of energy bands as a diamond lattice bands as a diamond lattice crystal is formed by bringing crystal is formed by bringing isolated silicon atoms together.isolated silicon atoms together.

• Fig. 2.15: schematic diagram of the formation of a Si crystal from N isolated Si atoms.

• Inter-atomic distance decreases, 3s and 3p sub shell of N Si atoms will interact and overlap.

• At equilibrium state, the bands will split again: 4 quantum states/atom (valence band) 4 quantum states/atom (cond. band)



At T = 0K, electrons occupy the lowest energy states: Thus, all states in the valence band (lower band) will be full, and all states in

the cond. band (upper band) will be empty.

The bottom of cond. band is called EC, and the top of valence band called EV.

Bandgap energy Eg = (EC – EV).

Physically, Eg defined as ‘the energy required to break a bond in the semiconductor to free an electron to the cond. band and leave the hole in the valence band’.

(Please remember this important definition). This is one of a hot-issue in scientific research in the world!!!

Bandgap is one of the factors that affect the devices performance.



Energy & Momentum Energy & Momentum

Please recall back your memory in the basic of fundamental PHYSICS!!

What is the definition of an Energy in PHYSICS???The capacity of a physical system to do work

What is the definition of Momentum in PHYSICS???The product of the mass times the velocity of an object


Figure 2.16.Figure 2.16. The parabolic The parabolic energy (energy (EE) vs. momentum ) vs. momentum ((pp) curve for a free ) curve for a free electron.electron.

0

2

2m

pE

The Energy-Momentum Diagram

• The energy of free-electron is given by

p – momentum, m0 – free-electron mass

• effective mass = mn

1

2

2

dp

Edmn

• for the narrower parabola (correspond to the larger 2nd derivative) – smaller mn

• for holes, mn = mp

(3)

(4)


Figure 2.17.Figure 2.17. A schematic energy-momentum A schematic energy-momentum

diagram for a special semiconductor diagram for a special semiconductor with with mmnn = 0.25 = 0.25mm00 and and mmpp = = mm00..

• Fig. at the RHS shows the simplified energy-momentum of a special semiconductor with: - electron effective mass mn = 0.25m0 in cond. band. - hole effective mass mp = m0 in the valence band.

• electron energy is measured upward • hole energy is measured downward

• the spacing between p = 0 and minimum of upper parabola is called bandgap Eg.

• For the actual case, i.e Si and GaAs – more complex.

The Energy-Momentum Diagram


Figure 2.18.Figure 2.18. Energy band structures of Si Energy band structures of Si and GaAs. Circles (and GaAs. Circles (ºº) indicate holes in the ) indicate holes in the valence bands and dots (•) indicate valence bands and dots (•) indicate electrons in the conduction bands.electrons in the conduction bands.

bandgap

Si GaAs

• Fig. 2.18 is similar to Fig. 2.17.

• For Si, max in the valence band occurs at p = 0, but min of cond.band occurs at p = pc (along [100]direction).

• In Si, when electron makestransition from max point (valenceband) to min point (cond. band) itrequired:Energy change (≥Eg) + momentumchange (≥pc).

• In GaAs: without a change in momentum.

• Si – indirect semiconductor.• GaAs – direct semiconductor.


GaAs - very narrow conduction-band parabola, using (8), we may expected that GaAs – smaller effective mass (mn = 0.063m0).

Si – wider conduction band parabola, using (8), effective mass mn = 0.19m0.

The different between direct and indirect band structures is very important for applications in LED and semiconductor laser. This devices require the types of band structure for efficient generation of photons (you may learn this applications in Chapter 9 – PHOTONIC DEVICES).


CONDUCTIONCONDUCTION

Metals/Conductors

Semiconductors

Insulators

(Recall your basic knowledge in EMT 111)


Metals/ConductorsVery low resistivity.Cond. band either is partially filled (i.e Cu) or overlaps in valence band (i.e Zn, Pb).No bandgap. Electron are free to move with only a small applied field. Current conduction can readily occur in conductors.

Insulators

Valence electrons form strong bonds between neighboring atoms (i.e SiO2).No free electrons to participate in current conduction near room temperature.Large bandgap. Electrical conductivity very small – very high resistivity.

SemiconductorsMuch smaller bandgap ~ 1eV.At T=0K, no electrons in cond. band. Poor conductors at low temperatures.

Eg = 1.12eV (Si) and Eg=1.42eV (GaAs) – at room temp. & normal atmosphere.

CONDUCTIONCONDUCTION


Figure 2.19.Figure 2.19. Schematic energy band representations of ( Schematic energy band representations of (aa) a ) a conductorconductor with two with two possibilities (either the partially filled conduction band shown at the upper possibilities (either the partially filled conduction band shown at the upper

portion or the overlapping bands shown at the lower portion), (portion or the overlapping bands shown at the lower portion), (bb) a ) a semiconductorsemiconductor, and (, and (cc) an ) an insulatorinsulator..

(a)

(b)

(c)

(b)


2.2 INTRINSIC CARRIER 2.2 INTRINSIC CARRIER CONCENTRATION CONCENTRATION

In thermal equilibrium:

- at steady-state condition (at given temp. without any external energy, i.e light, pressure or electric field).

Intrinsic semiconductor – contains relatively small amounts of impurities compared with thermally generated electrons and holes.

Electron density, n – number electrons per unit volume.

To obtain electron density in intrinsic s/c – evaluate the electron density in an incremental energy range dE.

Thus, n is given by integrating density of state, N(E), energy range, F(E), and incremental energy range, dE, from bottom of the cond. band, EC = E = 0 to the top of the cond. band Etop.


Figure 2.20.Figure 2.20. Fermi distribution function Fermi distribution function FF((EE) versus () versus (EE – – EEFF) for various ) for various

temperatures.temperatures.

top topE E

dEEFENdEEnn0 0

)()()(

• We may write n as

• Fermi-Dirac distribution function (probability that ē occupies an electronic state with energy E)

kTEEEF

F /)exp(1

1)(

n is in cm-3, and N(E) is in (cm3 .eV)-1

k Boltzmann constant ~ 1.38066 x 10-23 J/K, and T in Kelvin.

EF – energy of Fermi level (is the energy at which the probability of occupation by electron is exactly one-half)

(5)

(6)


Enrico Fermi (1901 – 1954)Nobel Prize in Physics 1938

“There are two possible outcomes: If the result confirms the hypothesis, then you've made a measurement. If the result is contrary to the

hypothesis, then you've made a discovery”

Paul Adrien Maurice Dirac (1902 – 1984)Nobel Prize in Physics 1933

“Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field”


For energy, 3kT above or below Fermi energy, then

kTEEeEF

kTEEeEF

FkTEE

FkTEE

F

F

3)(for 1)(

3)(for )(

/)(

/)(

• > 3kT the exponential term larger than 20, and < 3kT – smaller than 0.05.

• At (E – EF) < 3kT – the probability that a hole occupies a state located at energy E.

(7)

(8)


Figure 2.21.Figure 2.21. Intrinsic semiconductor. ( Intrinsic semiconductor. (aa) Schematic band diagram. () Schematic band diagram. (bb) Density of ) Density of states. (states. (cc) Fermi distribution function. () Fermi distribution function. (dd) Carrier concentration.) Carrier concentration.

• N(E) = (E)1/2 for a given electron effective mass.• EF located at the middle of bandgap.• Upper-shaded in (d) corresponds to the electron density.


From Appendix H in Sze, pg. 540, density of state, N(E) is defined as

Eh

mEN n

2/3

2

24)(

(9)

• substituting (9) and (8) into (5), thus

kTEENpkTEENn FVVCFC /)(exp and ,/)(exp

2/3

2

2

h

kTmXN n

C

GaAsfor 2 and Si,for 12 XX

2/3

2

22

h

kTmN pV

(10)

• NC and NV effective density of state in cond. band & valence band respectively.

h – Planck constant ~ 6.62607 x 10-34 J.s

• At room temperature, T = 300K;

Effetive density Si GaAs

NC 2.86 x 1019 cm-3 4.7 x 1017 cm-3

NV 2.66 x 1019 cm-3 7.0 x 1018 cm-3


Figure 2.22.Figure 2.22. Intrinsic carrier densities in Si and Intrinsic carrier densities in Si and GaAs as a function of the GaAs as a function of the reciprocal of temperature.reciprocal of temperature.

• Intrinsic carrier density is obtained by:

)2/exp(

)/exp(2

2

kTENNn

kTENNn

nnp

gVCi

gVCi

i

(11)

Where Eg = EC - EV

(mass action law)


2.3 DONORS & ACCEPTORS2.3 DONORS & ACCEPTORS

Recall your basic knowledge in EMT 111 – Electronic Devices.

When semiconductor is doped with impurities – become extrinsic and impurity energy levels are introduced.

Donor : n-type

Acceptor : p-type

Ionization energy:

Hn

SD E

m

mE

0

2

0

(12)

• 0 – permittivity in vacuum ~ 8.85418 x 10-14 F/cm

• S – semiconductor permittivity, and EH ~ Bohr’s energy model


Figure 2.23.Figure 2.23. Schematic bond pictures for ( Schematic bond pictures for (aa) ) nn-type Si with donor (arsenic) and -type Si with donor (arsenic) and ((bb) ) pp-type Si with acceptor (boron).-type Si with acceptor (boron).


Figure 2.24.Figure 2.24. Measured ionization energies (in eV) for various impurities in Si and Measured ionization energies (in eV) for various impurities in Si and GaAs. The levels below the gap center are measured from the top of the valence GaAs. The levels below the gap center are measured from the top of the valence

band and are acceptor levels unless indicated by band and are acceptor levels unless indicated by DD for donor level. The levels for donor level. The levels above the gap center are measured from the bottom of the conduction band and above the gap center are measured from the bottom of the conduction band and

are donor levels unless indicated by are donor levels unless indicated by AA for acceptor level. for acceptor level.


NON-DEGENERATE SEMICONDUCTORNON-DEGENERATE SEMICONDUCTOR

In previous section, we assumed that Fermi level EF is at least 3kT above EV and 3kT below EC. Such semiconductor called non-degenerate s/c.

Complete ionization – cond. at shallow donors in Si and GaAs where they have enough thermal energy to supply ED to ionize all donor impurities at room temperature, thus provide the same number of electrons in the cond. band.

Under complete ionization cond., electron density may be written as

n = ND (13)

And for shallow acceptors;

p = NA (14)

Where ND and NA – donor and acceptor concentration respectively.

From electron and hole density (10) and (13) & (14), thus

)/ln(

)/ln(

AVVF

DCFC

NNkTEE

NNkTEE

(15)

* Higher donor/acceptor concentration – smaller ∆E.


Figure 2.25.Figure 2.25. Schematic energy band representation of extrinsic Schematic energy band representation of extrinsic semiconductors with (semiconductors with (aa) ) donor ionsdonor ions and ( and (bb) ) acceptor ions.acceptor ions.

Non-degenerate Semiconductor


Figure 2.26Figure 2.26. . nn-Type semiconductor-Type semiconductor. (. (aa) Schematic band diagram. () Schematic band diagram. (bb) Density ) Density of states. (of states. (cc) Fermi distribution function () Fermi distribution function (dd) Carrier concentration. Note ) Carrier concentration. Note that that npnp = = nnii

22..

*** p-type semiconductor???

Much closer to cond. band



To express electron & hole densities in term of intrinsic part (concentration and Fermi level) – used as a reference level when discussing extrinsic s/c, thus;

kTEEnp

kTEEnn

Fii

iFi

/)(exp

/)(exp

(16)

• In extrinsic s/c, Fermi level moves towards either bottom of cond. band (n-type) or top of valence band (p-type). It depends on the domination of types carriers.

• Product of the two types of carriers will remains constant at a given temp.



• The impurity that is present is in greater concentration, thus it may determines the type of conductivity in the s/c.• Under complete ionization:

2

2

2

4)(with

/

2

1

/

2

1

iAD

pip

DAp

nin

AD

DA

nNNB

pnn

BNNp

nnp

BNNn

NpNn

(17)

Solve (17) with mass action law, thus

(18)

(19)

Subscript of n and p refer to n and p-type.



EXAMPLEEXAMPLE

A Si ingot is doped with 1016 arsenic atom/cm3. (a) Find carrier concentration and the Fermi level at room

temperature (T = 300K). (b) Eg = 1.12eV (Si). Sketch the level of energy.


Figure 2.28.Figure 2.28. Fermi level for Si and GaAs as a function of temperature and Fermi level for Si and GaAs as a function of temperature and impurity concentration. The dependence of the bandgap on temperature is impurity concentration. The dependence of the bandgap on temperature is

shown.shown.

Real fab.> 1015


Figure 2.29.Figure 2.29. Electron density as a function of temperature for a Si Electron density as a function of temperature for a Si sample with a donor concentration of 10sample with a donor concentration of 101515 cm cm-3-3..


For a very heavy doped n-type and p-type s/c, EF will be above EC or below

EV – this refereed to “degenerate semiconductor”.

Approximation of (7) and (8) are no longer use. Electron density (5) may solved numerically.

Important aspect of high doping ~ bandgap narrowing effect (reduced the bandgap), and it given by (at T=300K);

DEGENERATE SEMICONDUCTORDEGENERATE SEMICONDUCTOR

meVN

Eg 10

2218

(20)


CONCLUSION REMARKSCONCLUSION REMARKS

The properties of s/c are determined to a large extend by the crystal structure.

Miller indices to describe the crystal surfaces & crystal orientation.

The bonding of atoms & electron energy-momentum relationship – connection to the electrical properties of semiconductor.

Energy band diagram is very important to understand using physics approach why some materials are good and some are poor in term of conductor of electric current.

Some external/ internal changing of s/c (temperature and impurities) may drastically vary the conductivity of s/c.

The understanding Physics behind the semiconductor behaviours is very important for Microelectronic Engineer to handle the problems as well as to produced a high-speed devices performance.


“I was born not knowing and have had only a little time to change that here

and there”

Richard P. Feynman (1918-1988)Nobel Prize in Physics 1965

MotivationMotivation


Next Lecture: Next Lecture:

Carrier Transport Phenomena

© m.n.a. halif & s.n. sabki chapter 2: energy bands & carrier concentration in thermal...

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