Amorphous Semiconductors

Properties

Amorphous materials have attracted much attention. The first reason is potential industrial applications as suitable materials for fabricating devices, and the second reason is a lack of understanding of many properties which are very different from crystalline materials. An ideal crystal is defined as an atomic arrangement that has infinite translational symmetry in all three dimensions, whereas such a definite definition is not possible for an ideal amorphous solid. An amorphous solid is defined as one that does not maintain long-range translational symmetry or has only short-range order, it does not have the same precision in its definition, because long- or short-range order is not precisely defined. In addition to surface atoms in amorphous materials, however, there are also present other structural disorders due to different bond lengths, bond angles, and coordination numbers at individual atomic sites.

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The theory of amorphous systems is relatively difficult, because some of the techniques of simplification for deriving analytical results in crystals cannot be applied to amorphous structures. One needs to depend heavily on numerical simulations using computers, which itself is a relatively new field.

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It is commonly established that there are three types of structural disorders in an amorphous solid which do not exist in crystalline solids. These are stated below.

  • Different bond lengths
  • Different bond angles
  • Under- and over-coordinated sites

Bond States

In amorphous silicon, all silicon atoms are bonded covalently but it is not necessary that all atomic sites are of the same coordination number four. Some are under coordinated, which means that one or more covalent electrons on a silicon atom cannot form covalent bonds with neighboring atoms. These uncoordinated bonds are called dangling bonds. The density of dangling bonds in amorphous silicon is very high, which reduces the photoconductivity of the material and also prevents it from doping. The hydrogenation of amorphous silicon saturates many dangling bonds and makes it more suitable for fabricating devices.

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The presence of strained and weak bonds gives rise in a-Si:H to band tail states, which are also found in other a-semiconductors and insulators. A crystalline semiconductor has quite well defined valence and conduction band edges, and hence a very well defined electron energy gap between the top of the valence band and bottom of the conduction band.

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In amorphous semiconductors, the neutral dangling bond states lie in the middle of the energy band gap, and bonding and anti-bonding orbital of weak bonds lie above the valence band and below the conduction band edges respectively.

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In addition to the neutral dangling bond states, there may also exist charged dangling bond states. If the charge carrier-phonon interaction is very strong in an amorphous solid, due to the negative <math>U</math> effect, then the positive charged dangling bond states would lie above the neutral bond state, but below the conduction band edge. The negative charged dangling bond states would lie below the neutral dangling bond state, but above the valence band edge. In the case of weak charge carrier-phonon interactions, these positions are reversed on the energy scale. These energy states found within the energy gap in amorphous solids are localized states and any charge carrier created in these states will be localized on some weak, strained or dangling bonds.

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An interesting point is that as the band tail states lie above the valence and below the conduction band edges, these states are usually the highest occupied and lowest empty energy states in any amorphous semiconductor. Band tail states play the dominant role in most optical and electronic properties of a-semiconductors, particularly in the low temperature region.

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The electronic energy states of amorphous semiconductors consist of delocalized states like valence and conduction bands, commonly known as the extended states and the localized states like band tails and dangling bond states. The extended states arise due to short-range order, and tail states due to disorder.

Band tail, mobility edge, and dangling bond states

In addition to the fully coordinated covalent bonds, there are also many weak, strained and even uncoordinated or dangling bonds in amorphous semiconductors. The bonding and anti-bonding states of weak and strained bonds lie close the valence and conduction band state edges, respectively, because the bond lengths are relatively larger than those giving rise to the extended states. This is clear from the fact that the energy gap between bonding and anti-bonding states reduces if the distance between the bonding atoms increases. For larger separation between atoms the energy band gap, which is similar to the separation between bonding and anti-bonding states, is smaller.

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The presence of weak and strained bond states gives rise to band tail states or tail states within the energy gap and near the extended state edges in amorphous semiconductor. These states are localized states.

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The edge separating the conduction extended states and tail states is called the mobility edge. As the tail states are localized energy states, no conduction is expected to occur when excited electrons occupy these states. At <math>0 K</math> only conduction can occur when excited electrons are in the extended states above the conduction tail states, and that defines the mobility edge, which is the energy above which the electronic conduction can occur at <math>0 K</math>. One can define a similar edge separating the valence extended states from valence tail states. Thus there are two mobility edges, electron mobility edge at the bottom of the conduction extended states and hole mobility edge at the top of valence extended states.

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Dangling bonds contribute to non-bonded states, and therefore they can be regarded as equivalent to the states of isolated atoms, which lied in the middle of the bonding and anti-bonding energy states. The dangling bond states lie in the middle of the energy gap between the edges of the valence and conduction extended states. The energy gap in amorphous semiconductors is not well defined as that in crystalline semiconductors due to the presence of tail states. The dangling bond states are also localized energy states.

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The influence of the presence of dangling bonds on the electronic and optical properties of amorphous semiconductors can be very significant, depending on their numbers, because these bonds do not facilitate electronic conduction except through hopping or tunneling.

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Structure

A fundamental understanding of the properties of condensed matter, whether electronic, optical, chemical, or mechanical, requires a detailed knowledge of microscopic structure (atomic arrangement). The structure of a crystalline solid is determined by studying its structure within the unit cell. The structure of the crystal as a whole is then determined by stacking unit cells. Such a procedure is impossible in determining the structure of amorphous solids. Due to the lack of long range periodicity in amorphous solids, unlike crystalline solids, determination of structure is very difficult. There is no technique to provide atomic resolution in amorphous solids compared with that in crystals.

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The diffraction measurements give the structure <math>S(\vec Q)</math> with scattering vector <math>\vec Q</math>. The Fourier transform of <math>S(\vec Q)</math> produces the radial distribution function (RDF). The RDF studies show that the structure of many amorphous solids is non-random and there is a considerable degree of local ordering despite the lack of long-range order. The RDF, <math>J(r)</math>, is defined as the number of atoms lying at distances between <math>r</math> and <math>r+dr</math> and is given by the expression below, where the density function , <math>\rho (r)</math>, is an atomic pair correlation function. The function exhibits oscillatory behavior, because peaks in the probability function represent average interatomic separations.

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<math>J(r) dr = 4 \pi r^2 \rho (r) dr</math>

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The RDF oscillates about the average density parabola given by the curve <math>4 \pi r^2 \rho_o</math>. The position of the first peak in the RDF produces the average nearest-neighbor bond length <math>r_1</math> and the position of the second peak gives the next-nearest-neighbor distance <math>r_2</math>. The knowledge of <math>r_1</math> and <math>r_2</math> yields the bond angle <math>\theta</math>.

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<math>\theta = 2 \sin^{-1} \frac

Unknown macro: {r_2}
Unknown macro: {2 r_1}

</math>

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The area under a peak gives the coordination number of the structure. The second peak is generally wider than the first for covalent amorphous solids, which can be attributed to a static variation in the bond angle <math>\theta</math>. If no bond-angle variation exists, then the width of the first two peaks should be equal.

Density of States

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<math>\mbox

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<math>\frac

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= g(\epsilon) = \frac

Unknown macro: {m}
Unknown macro: {hbar^2 pi^2}

\sqrt{ \frac

Unknown macro: { 2 m epsilon }

{ \hbar^2}}</math>

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<math>\mbox

Unknown macro: {Crystalline Material}

</math>

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<math>g_n(\epsilon) = \frac

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\int_{S_{n(\epsilon)}} \frac

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Unknown macro: { | vec nabla_k epsilon (k) | }

</math>

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Band Diagram

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Van Hove

Van Hove singularities occurr where there is a non smooth density of state. It seems there won't be Van Hove singularities in an amorphous semiconductor.

Anderson

Milestones in a-Si development

  • There is poor electronic properties of pure a-Si. A dramatic increase in photoconductivity is a result of hydrogenation (10 atomic %). This was discovered in the late '60s.
  • a-Si is typically made by glow discharge plasma decomposition of silane (<math>SiH_4</math>) gas.
  • Doping of a-Si is achieved by adding <math>PH_3</math> (n-type) or <math>B_2H_6</math> (p-type).
  • Typical carrier mobilities are ~<math>1 \frac
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    </math> versus <math>1000</math> in crystalline silicon, while minority carrier diffusion lengths are about <math>100 nm</math>.
  • By the mid 80's, <math>10%</math> conversion efficiency was achieved in a-Si cells. Depending on technology <math>5-13%</math> is possible.
  • The typical solar cell design is a <math>p-i-n</math> junction. The individual layers are about <math>50 nm</math>. They serve the purpose of establishing the <math>V_
    Unknown macro: {bi}
    </math> but do not provide photocarriers.
  • Amorphous silicon is a more efficient absorber of light compared to crystalline silicon and therefore can be significantly thinner (<math>x100</math>).
  • The first large scale a-Si commercial installation was in Davis, CA.

Cause and Effect Relations Between Atomic Structure and Electronic Properties

  • The existence of short range order results in a similar overall electronic structure of an amorphous material compared to a crystal with a similar stoichiometry.
  • Silicon is semiconducting while <math>SiO_2</math> is an insulator both in the crystalline and amorphous forms
  • The abrupt band edges are replaced by broadened tails of states extending into the forbidden gap. These originate from the deviations in bond length and angle. The band tails are very important in determining the electronic properties as most of the electronic transport in semiconductors occurs near the band edges and is thus due to these states.
  • Electronic states deep in the gap occur because of coordination defects. These defects dominate the electronic properties by acting as recombination centers.
  • The presence of localized states due to the disorder- Anderson Localization

Notes

  • There is no k-conservation in amorphous materials due to the lack of long range order and lack of discrete translational symmetry.
  • short range order and correlation exists
  • The basic description of the electronic states is in terms of the DOS function.
  • Since k is not a "good quantum number" the distinction between direct bandgap and indirect bandgap semiconductors is lost. Consequently optical translations are allowed in materials such as silicon without the participation of a phonon
  • Disorder reduces carrier mobility due to scattering
  • Conductivity is due to extended as well as localized states

Calculate Eigenfunctions

Use the variational approach and a trial wavefunction that consists of a linear combination of atomic orbitals.

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<math>\overline E = <\hat H></math>

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<math>\overline E = \frac{\int_{-\infty}^

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\chi^* \hat H \chi dV}{\int_{-\infty}^

\chi^* \chi dV} \ge E_

Unknown macro: {ground state}

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<math>\chi = \sum_

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^N c_n \phi_n</math>

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Search for coefficients that minimize <math>< \hat H ></math>.

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<math>\frac

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< \hat H> = 0</math>

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The nature of eigenfunctions of amorphous semiconductors

References

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