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3-1-4. Direct & Indirect Semiconductors

• A single electron is assumed to travel through a perfectly periodic lattice.

• The wave function of the electron is assumed to be in the form of a plane wave moving.

xjkxk

xexkUx ),()( x : Direction of propagation k : Propagation constant / Wave vector : The space-dependent wave function for the electron

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U(kx,x): The function that modulates the wave function according to the periodically of the lattice.

Since the periodicity of most lattice is different in various directions, the (E,k) diagram must be plotted for the various crystal directions, and the full relationship between E and k is a complex surface which should be visualized in there dimensions.

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Eg=hνEg Et

k k

EE

Direct IndirectExample 3-1

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Example 3-1: Assuming that U is constant in

for an essentially free electron, show that the x-component of the electron momentum in the crystal is given by

xx khP Example 3-2

),()( xkUx xk xjkxe

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x

x

xjkxjk

x

khdxU

dxUkh

dxU

dxexj

heU

P

xx

2

2

2

2

)(Answer:

The result implies that (E,k) diagrams such as shown in previous figure can be considered plots of electron energy vs. momentum, with a scaling factor .

h

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3-2-2. Effective Mass

• The electrons in a crystal are not free, but instead interact with the periodic potential of the lattice.

• In applying the usual equations of electrodynamics to charge carriers in a solid, we must use altered values of particle mass. We named it Effective Mass.

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Example 3-2: Find the (E,k) relationship for a free electron and relate it to the electron mass.

E

k

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khmvp

222

2

22

1

2

1k

m

h

m

pmvE

Answer: From Example 3-1, the electron momentum is:

m

h

dk

Ed 2

2

2

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Answer (Continue): Most energy bands are close to parabolic at their

minima (for conduction bands) or maxima (for valence bands).

EC

EV

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• The effective mass of an electron in a band with a given (E,k) relationship is given by

2

2

2*

dkEd

hm

X

L

k

E

1.43eV

) ()( or** LXmm

Remember that in GaAs:

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• At k=0, the (E,k) relationship near the minimum is usually parabolic:

gEkm

hE 2

*

2

2In a parabolic band, is constant. So, effective mass is constant.

Effective mass is a tensor quantity.

2

2

dk

Ed

2

2

2*

dkEd

hm

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EV

EC

02

2

dk

Ed

02

2

dk

Ed

0* m

0* m2

2

2*

dkEd

hm

Ge Si GaAs

† m0 is the free electron rest mass.

Table 3-1. Effective mass values for Ge, Si and GaAs.

mn

*

mp

*

055.0 m 01.1 m 0067.0 m

037.0 m 056.0 m 048.0 m

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3-2-5. Electrons and Holes in Quantum Wells

• One of most useful applications of MBE or OMVPE growth of multilayer compou-nd semiconductors is the fact that a continuous single crystal can be grown in which adjacent layer have different band gaps.

• A consequence of confining electrons and holes in a very thin layer is that

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these particles behave according to the particle in a potential well problem.

GaAs Al0.3Ga0.7AsAl0.3Ga0.7As

50Å

E1

Eh

1.43eV1.85eV

0.28eV

0.14eV

1.43eV

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• Instead of having the continuum of states as

described by ,modified for

effective mass and finite barrier height.

• Similarly, the states in the valence band

available for holes are restricted to discrete

levels in the quantum well.

2

222

2mL

hnEn

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• An electron on one of the discrete condu-ction band states (E1) can make a transition to an empty discrete valance band state in the GaAs quantum well (such as Eh), giving off a photon of energy Eg+E1+Eh, greater than the GaAs band gap.

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