Band Gap Engineering- Lecture 7

Sebastian Sebastian LourdudossLourdudoss

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Advanced Semiconductor Materials

Lecture 7, Modification of bandstructures

OutlineOutline

•• Properties of conduction and valence Properties of conduction and valence bandedgebandedge statesstates

•• BandstructureBandstructure modification by Alloyingmodification by Alloying

•• Heterostructure typesHeterostructure types

•• BandstructureBandstructure modification by modification by heterostructuresheterostructures

•• BandstructureBandstructure modification by strainmodification by strain

Part of the material and some figures are from:

1) J. Singh, Semiconductor Optoelectronics, McGraw Hill Int. Ed.,

Singapore, 1995,

2) J.H.Davies,The Physics of low dimensional semiconductors, Cambridge

University Press, NY, 1998.


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Properties of conduction and valence Properties of conduction and valence bandedgebandedge statesstates

Bloch theorem:Bloch theorem:

Wave function of electron Wave function of electron

= plane wave part + a periodic = plane wave part + a periodic

central cell partcentral cell part

Solution of the proper SchrSolution of the proper Schröödinger dinger

equation yields:equation yields:

Energy of the electron= Energy of the electron= f(kf(k) & ) & cell partcell part

* * Cell partCell part character determines character determines

selection rules for photonselection rules for photon--

electron interaction in electron interaction in

optoelectronic devicesoptoelectronic devices

* Valence bands and conduction * Valence bands and conduction

bands in semiconductors bands in semiconductors

composed of composed of ss and and pp orbitalsorbitals

* Therefore cell part of the wave * Therefore cell part of the wave

function made up of function made up of ss and and pp statesstates


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Properties of conduction and valence Properties of conduction and valence bandedgebandedge statesstates

Band structure of the Band structure of the conduction conduction

bandsbands

Direct:

E(k) = Ec + h2k2/2m*

Indirect:

E(k) = Ec + h2kl2/2ml*+ h2kt

2/2mt*

Band structure of the Band structure of the valence bandsvalence bands

E(k) = (- h2/2m0) {Ak2 ± [Bk4 + C(kx2

ky2 + ky

2 kz2 + kz

2 kx2 )]1/2}

A, B, C = dimensionless parameters

Upper sign => Heavy hole states

Lower sign => Light hole states

p-type states


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Band structure modification by alloyingBand structure modification by alloying

• Mixture of two or more binaries (e.g., A and B)

= alloys (e.g., AxB1-x) with different bandgaps and lattice

constants

• Lattice constant of the alloy = linear combination of the lattice constants

of binaries (called Vegard’s law);

aalloy = xaA + (1-x)aB

• Alloy Bandgap = linear combination of bandgap of the binaries (Virtual

Crystal Approximation)

Egalloy = xEg

A + (1-x)EgB

Not always true - A bowing factor is often involved

• Now there are rigorous equations for a and Eg for several alloy

semiconductors, e.g. for GaxIn1-xAsyP1-y:

a(x,y) = xyaGaAs + x(1-y)aGaP + (1-x)yaInAs + (1-x)(1-y)aInP

Eg(x,y) = 1.35 + 0.668x – 1.072y + 0.758x2 + 0.078y2 – 0.069xy – 0.322x2y + 0.03xy2


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Band structure modification by alloyingBand structure modification by alloying

Change of Change of aa’’ss, , EEgg’’ss and and bandstructuresbandstructures


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Band structure modification by Band structure modification by

heterostructuresheterostructures -- Concept of Concept of bandoffsetsbandoffsets or or

bandgap discontinuitybandgap discontinuity

Anderson model

(electron affinity model):

∆∆∆∆∆∆∆∆EECC = e(= e(χχχχχχχχAA -- χχχχχχχχBB))

∆∆∆∆∆∆∆∆EEVV = = ∆∆∆∆∆∆∆∆EEgg -- ∆∆∆∆∆∆∆∆EECC

χχ (chi) = electron affinity(chi) = electron affinity

There are several other There are several other

models and experimental models and experimental

methods.methods.


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Band structure modification by Band structure modification by

heterostructuresheterostructures (types I, II and III)(types I, II and III)

Type IIType I

Type III

Type I (straddling alignment), e.g.

InGaAs/ InP:

Sandwich traps both electrons and

holes

Type II (staggered alignment), e.g.

InP/InAlAs:

Sandwich favours electrons in InP

but holes in InAlAs

Type III (broken-gap alignment)

e.g., InAs/GaSb:

Conduction band of InAs overlaps

with the valence band of GaSb.

Consequence: Spontaneous transfer

of electrons and holes until the

‘built-in’ field resists (as in p-n

diode)


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Construction of band diagrams in Construction of band diagrams in

heterostructuresheterostructures


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Bandgap engineering by strain

Strain in mismatched materials improves certain properties:

Example: In0.53Ga0.47As and In0,52Al0.48As lattice matched

to InP. In electronic applications large ∆EC and small m* for electrons in

In0.53Ga0.47As is exploited. But if In in In0.53Ga0.47As is increased to above

0.53, both ∆EC and m* are improved at the cost of strain!

• Strain broadens the range of materials available for tuning the properties

of interest such as band offsets and effective masses

• Stain has strong effects on valence band => strong tool for bandgap

engineering

• Benefits of strain may allow to use desired layers on convenient but

usually mismatched substrates


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StrainStrain

•• Strain is allowed only Strain is allowed only uptoupto a particular thickness (critical thickness)a particular thickness (critical thickness)

Critical thickness Critical thickness ≈≈ aass/2/2||εε||

•• If the critical thickness is exceeded, defects are generatedIf the critical thickness is exceeded, defects are generated

•• Strained layer has the same lattice constant as the substrate oStrained layer has the same lattice constant as the substrate on the n the

plane of the substrate but different along the growth directionplane of the substrate but different along the growth direction

•• Compressive strain and tensile strainCompressive strain and tensile strain

Strain, ε =

(asubstrte- alayer)/alayer

εxx = εyy = ε;

εzz = -2C12ε/C11

C11 and C12 are the

force constants (in

Pa) of the material


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Strain in Strain in EEgg(x,y(x,y) ) –– a (a (x,yx,y) diagram for Ga) diagram for GaxxInIn11--xxAsAsyyPP11--yy

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x(Gallium)

y(Arsenic)

GaAs

3.67 % tshh eg: 1072 nmlh eg: 1349 nm

InAs

3.2 % cshh eg: 2861 nmlh eg: 1834 nm

InP

eg: 918 nm

GaP

7.1 % tshh eg: 597 nmlh eg: 740 nm

2%cs 1%cs lm 1%ts 2%ts

1500 nm1400 nm1300 nm

1200 nm1100 nm

1000 nm

Constant

bandgap

lines

Constant

lattice

constant

lines

lm = lattice

matched


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• Wave functions at the top of the

valence band has the symmetry of p

orbitals

• Wavefunctions overlap strongly in the

z- direction less in xy plane = free

movement of e-’s in the z-direction, less

free in the x- and y- directions

=> Energy of holes in z-direction

increases with k faster than in the x and

y-directions

• The other two p-orbitals (px and py)

behave in a similar manner

• To sum up the behaviour, we have one

”light” band and two degenerate

”heavy” bands


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• Strain affects pz orbital the most, i.e.,

along the z-direction or growth

direction

• Compression in xy plane, case (c) =

energy of pz falls

=> Top of the valence band arise

from px and py and hence these are

heavy along z but light along x and y

=> Implication: Mobility in the

plane of the junction is favoured due

to lighter hole mass

• Tension in xy plane, case (a) =

energy of pz increases

=> Top of the valence bands arise

from pz and hence these are light

along z but heavy along x and y

=> Implication: Mobility in the

plane of the junction determined by

heavier hole mass

Band structure modification due to strainBand structure modification due to strain

Strain, ε = (asubstrte- alayer)/alayer

εxx = εyy = ε; εzz = -2C12ε/C11

k = kx, ky => growth plane

kz = growth direction


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Band structure modification due to strainBand structure modification due to strain

•For QW:

• m*h (growth direction) determines the confinement energy

Confinement energy of electron (hole) = Energy between the first

quantisation level of electron (hole) to the conduction (valence) band

edge of the barrier layer and it depends upon the band offset (it also

depends on well composition, well thickness and barrier composition)

• m*h (in-plane) determines the density of states

E.g.: compressive InGaAs/GaAs :

m*h (in-plane) = 0.155 m0

For bulk: m*h (in-plane) = 0.5 m0


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Tangible effect on laser threshold

• Lasers have higher threshold if electron effective mass differs too much from the effective hole massRef.: Yablonovitch and Kane, J. Lightwave Tech., vol. 6, no. 8, pp. 1292-1299, 1988.

• Strained In0.25Ga0.75 As QW of thickness 5 nm on GaAs makes the hole effective mass = electron effective mass => decrease in threshold density by more than a factor of twoRef.: J.J.Coleman, Strained layer InGaAs Quantum-Well Heterostructure lasers, IEEE J. Seleted Topics in Quantum Electronics, vol. 6, no. 6, pp. 1008-1013, 2000.

Band Gap Engineering- Lecture 7

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