POT1 Basics

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Semiconductor physics


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The Kronig-Penney model

The Kronig-Penney model demonstratesthat a simple one-dimensional periodicpotential yields energy bands as well asenergy band gaps.

The potential assumed in the modelis shown in the Figure

The periodic potential assumed in the Kronig-Penney model.The potential barriers (region I) with width, b, are spaced bya distance (region II), a-b , and repeated with a period, a .

References:[1]http://ecee.colorado.edu/~bart/book/book/chapter2/ch2_3.htm#2_3_

Solutions for k and E are obtained whenthe following equation is satisfied[1]

Where


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This transcendental equation can be further simplified for the case wherethe barrier is a delta function with area, V 0 b, for which it becomes

Solutions are only obtained if the function, F , is between -1 and 1 since ithas to equal cos( ka ).

Graphical solution to the Kronig-Penney model for a = 1nm and V 0 b = 0.2 nm-eV. Shown is theenergy, E , versus ka /p and F , which has to equal cos( ka ), from which one can identify theallowed energies.



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Energy versus ka /pi as presented in Figure (black curves) compared to that of a free electron (gray curves). Shown are: a)the E (k ) diagram, b) the E (k ) diagram combined with the reduced-zone diagram and c) the reduced-zone diagram only.

The E (k ) relation resembles a parabola except that only specific ranges of

energies are valid solutions to Schrdinger's equation and therefore areallowed, while others are not.

The range of energies for which there is no solution is referred to as an energyband gap.

The transitions between allowed and forbidden energies occur at non-zerointeger multiples of ka /p.



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Temperature dependence of the energy bandgap

Temperature dependence of the energy bandgap ofgermanium (Ge), silicon (Si) and gallium arsenide(GaAs)

Example

Calculate the energy bandgap ofgermanium, silicon and galliumarsenide at 300, 400, 500 and 600

K.

SolutionSimilarly:


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Doping dependence of the energy bandgap

High doping densities cause the bandgapto shrink. This effect is explained by thefact that the wavefunctions of the electronsbound to the impurity atoms start tooverlap as the density of the impuritiesincrease.

Doping dependence of the energy bandgap of germanium (Ge), gallium arsenide (GaAs), andsilicon (Si).


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Density of States

The density of states in a semiconductor equals the density per unit volumeand energy of the number of solutions to Schrdinger's equation.We will assume that the semiconductor can be modeled as an infinite quantum well in which

electrons with effective mass, m*, are free to move.The energy in the well is set to zero.The semiconductor is assumed a cube with side L. (This assumption does

not affect the result since the density of states per unit volume should notdepend on the actual size or shape of the semiconductor.)

The solutions to the wave equa tionwhere V ( x ) = 0 are sine and cosinefunctions


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The density per unit energyis then obtained using the

chain rule:

Density of States Cont...

The kinetic energy E of a particle with mass m* is related to thewavenumber, k , by:

And the density of states per unit volume and per unit energy, g (E ),becomes

The same analysis also applies toelectrons in a semiconductor. Theeffective mass takes into account theeffect of the periodic potential on theelectron. The minimum energy of theelectron is the energy at the bottom ofthe conduction band, E

c, so that the

density of states for electrons in the


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Calculation of the density of states in 1, 2 and 3 dimensions

We will here postulate that the density of electrons in k space is constantand equals the physical length of the sample divided by 2p and that for

each dimension. The number of states between k and k + dk in 3, 2 and 1dimension then equals:

Why?

We now assume that the electrons in a semiconductor are close to a bandminimum, E min and can be described as free particles with a constanteffective mass

Elimination of k using the E(k) relation above then yields the desired density ofstates functions, namely:


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Calculation of the density of states in 1, 2 and 3 dimensions

Density of states per unit volume and energy for a 3-D semiconductor

(blue curve), a 10 nm quantum well with infinite barriers (red curve)and a 10 nm by 10 nm quantum wire with infinite barriers (greencurve). m * /m 0 = 0.8.


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Fermi-Dirac distribution function

The Fermi-Dirac distribution function, alsocalled Fermi function, provides theprobability of occupancy of energy levelsby Fermions. Fermions are half-integerspin particles, which obey the Pauliexclusion principle. The Pauli exclusionprinciple postulates that only one Fermioncan occupy a single quantum state.Therefore, as Fermions are added to anenergy band, they will fill the availablestates in an energy band just like water fillsa bucket. The states with the lowest energyare filled first, followed by the next higherones.

The Fermi function at three differenttemperatures

Electrons are Fermions.Therefore, the Fermi functionprovides the probability that anenergy level at energy, E , inthermal equilibrium with a largesystem, is occupied by anelectron. The system is

characterized by itstemperature, T , and its Fermi


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Impurity distribution functions

The distribution function of impurities differs from the Fermi-Diracdistribution function even though the particles involved are still Fermions.

The difference is due to the fact that an ionized donor energy level stillcontains one electron with either spin. The donor energy level cannot beempty since this would leave a doubly positively charged atom, which wouldhave an energy different from that of the singly ionized donor level. Thedistribution function for donors therefore differs from the Fermi function andis given by:

The distribution function for acceptors differs also because of the differentpossible ways to occupy the acceptor level. The neutral acceptor contains noelectrons. The ionized acceptor contains one electron, which can have eitherspin, while the doubly negatively charged state is not allowed since this wouldrequire a different energy. This restriction would yield a factor of 2 in front ofthe exponential term. In addition, one finds that most commonly usedsemiconductors have a two-fold degenerate valence band, which causes thisfactor to increase to four, yielding:


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Carrier densities Cont...

The carrier density integral. Shown are the density ofstates, g c(E ), the density per unit energy, n(E ), and theprobability of occupancy, f (E ). The carrier density, no, equalsthe crosshatched area.

This general expression isillustrated with Figure for a

parabolic density of statesfunction with E c = 0. The figureshows the density of statesfunction, g c(E ), the Fermifunction, f (E ), as well as theproduct of both, which is thedensity of electrons per unitvolume and per unitenergy, n(E ). The integralcorresponds to thecrosshatched area.The actual location of the top of the

conduction band does not need to beknown as the Fermi function goes tozero at higher energies. The upperlimit can therefore be replaced byinfinity. We also relabeled the carrierdensity as no to indicate that the

carrier density is the carrier densityin thermal equilibrium.


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The density of states and carrier densities in the conduction and valence band. Shown are theelectron and hole density per unit energy, n(E ) and p (E ), the density of states in the conduction andvalence band, g c(E ) and g v(E ) and the probability of occupancy, f (E ). The crosshatched area indicatesthe electron and hole densities.


Similarly for holes one obtains:


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While These integral can not be solved analytically at non-zerotemperatures, we can obtain either a numeric solution or an approximateanalytical solution.


Non-degenerate semiconductors

where N c is the effective density ofstates in the conduction band . The Fermienergy, E F , is obtained from:

Similarly


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The effective density ofstates in the conductionband of germaniumequals:

Note that the effective density of states istemperature dependent and can be obtainfrom:


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Intrinsic semiconductors

Intrinsic semiconductors are usuallynon-degenerate, so that theexpressions for the electron and hole

densities in non-degeneratesemiconductors apply. Labeling theFermi energy of intrinsic materialas E i, we can then write two relationsbetween the intrinsic carrier densityand the intrinsic Fermi energy,namely:

It is possible to eliminate the intrinsic Fermi energy from both equations,simply by multiplying both equations and taking the square root. Thisprovides an expression for the intrinsic carrier density as a function of theeffective density of states in the conduction and valence band, and the

bandgap energy E g = E c - E v.


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The temperature dependenceof the intrinsic carrier densityis dominated by theexponential dependence onthe energy bandgap. Inaddition, one has to considerthe temperature dependenceof the effective densities ofstates and that of the energybandgap.

Intrinsic carrier density versus temperature in gallium arsenide(GaAs), silicon and germanium. Compared is the calculateddensity with (solid lines) and without (dotted lines) the

temperature dependence of the energy bandgap.

Intrinsic semiconductors


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A l i f d l


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Analysis of non-degeneratelydoped semiconductors

The total charge density is therefore given by

Using Mass Action law, exact solution is:

Fermi energy of n-type and p -type silicon, E F,n and E F,p , as afunction of doping density at 100, 200, 300, 400 and 500 K.Shown are the conduction and valence band edges, E c and E v.The midgap energy is set to zero.

A l i f d l


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Analysis of non-degeneratelydoped semiconductors

N ilib i i d i i


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Non-equilibrium carrier densities

To describe a system that is not inthermal equilibrium we assumethat each of the carrier

distributions is still in equilibriumwith itself. Such assumption is

justified on the basis that electronsreadily interact with each other andinteract with holes only on a muchlonger time scale. As a result theelectron density can still becalculated using the Fermi-Diracdistribution function, but with adifferent value for the Fermienergy. The total carrier density fora non-degenerate semiconductoris then described by:

Where d p is the excess hole density and F p isthe quasi-Fermi energy for the holes.

Where d n is the excess electrondensity and F n is the quasi-Fermi energy forthe electrons


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