Superlattice: Definition from Answers.com

Superlattice is a periodic structure of repeating quantum wells that sets up a new set of selection rules which affects the conditions for charges to flow through the structure. This nanostructure consists of two different semiconductor materials, which are deposited alternately on each other to form a periodic structure in the growth direction. Since the first proposal by Leo Esaki and Raphael Tsu of synthetic artificial superlattices in 1970,^[1] great advances in the physics of such ultra-fine semiconductors, presently called quantum structures, have been made within the past two decades. The concept of quantum confinement has led to the observation of quantum size effects in isolated quantum well heterostructures and is closely related to superlattices through the tunneling phenomena. Therefore, these two ideas are often discussed on the same physical basis, but each field has its own intrigue and different physics useful for applications in many electric and optical devices.

1 Superlattice types
2 Superlattice Materials and Growth Technology
3 Miniband structure
4 Bloch states
5 Wannier Functions
6 Wannier–Stark ladder
7 Superlattice Transport
8 References
9 See also

Superlattice types

The superlattice miniband structures are divided into two different types, called type I and type II. For the type I heterostructures the bottom of the conduction subband and the top of the valence subband are formed in the same semiconductor layer. In the type II the conduction and valence subbands are staggered in both real and reciprocal space, so that electrons and holes are confined in different layers.

There also exists a class of quasiperiodic superlattices named after Fibonacci. A Fibonacci superlattice can be viewed as a one-dimensional quasicrystal, where either electron hopping transfer or on-site energy takes two values arranged in a Fibonacci sequence.

Superlattice Materials and Growth Technology

GaAs/AlAs surperlattice and potential profile of conduction and valence bands along the growth direction (z).

Semiconductor materials, which are actually used to fabricate the superlattice structures, may be divided by the element groups, IV, III-V and II-VI. While the group III-V semiconductors have been extensively studied, group IV heterostructures such as the Si_xGe_1-x system are much more difficult to realize because of the large lattice mismatch. Nevertheless, the strain modification of the subband structures is interesting in these quantum structures and has attracted much attention.

So far mostly the III-V compound semiconductors represented by the GaAs/Al_xGa_1-xAs heterostructures have been investigated. In particular, one distinguished merit of the GaAs/Al system is that the difference in lattice constant between GaAs abd AlAs is very small despite the highly corrosive nature of AlAs. In addition, the difference of their thermal expansion coefficient is also small. Thus, the strain remaining at room temperature can be minimized after cooling down from the higher epitaxial growth temperatures. The first compositional superlattice was realized using the GaAs/Al_xGa_1-xAs material system.

Superlattices can be produced using various techniques, but the most common are molecular-beam epitaxy (MBE) and sputtering. With these methods, layers can be produced with thicknesses of only a few atomic spacings. An example of specifying a superlattice is [Fe₂₀V₃₀]₂₀. It describes a bi-layer of 20Å of Iron (Fe) and 30Å of Vanadium (V) repeated 20 times, thus yielding a total thickness of 1000Å or 100nm. The MBE technology as a means of fabricating semiconductor superlattice is of primary importance. In addition to the MBE technology, metal-organic chemical vapor deposition (MO-CVD) has also contributed to the development of superconductor superlattices, which are composed of quarternary III-V compound semiconductors like InGaAsP alloys. More recently, a combination of gas source handling and ultrahigh vacuum (UHV) technologies such as metal-organic molecules as source materials, are becoming popular as well as gas-source MBE using hybrid gases like arsine (AsH₃) and phosphine (PH₃).

Generally speaking MBE is a method of using three temperatures in binary systems, e.g., the substrate temperature and the source material temperature of the group III and the group V elements in the case of III-V compounds.

The structural quality of the produced superlattices can be verified by means of X-ray diffraction or neutron diffraction spectra which contain characteristic satellite peaks. Other effects associated with the alternating layering are: giant magnetoresistance, tunable reflectivity for X-ray and Neutron mirrors, neutron spin polarization, and changes in the elastic and acoustic properties. Depending on the nature of its components, a superlattice may be called magnetic, optical or semiconducting.

Miniband structure

The schematic structure of a periodic superlattice is shown below, where A and B are two different semiconductor materials of respective layer thickness a and b (period: $d = a + b$ ). When, a and b are not too small compared with the interatomic spacing, an adequate approximation is obtained by replacing these fast varying potentials by an effective potential derived from the band structure of the original bulk semiconductors. It is straightforward to solve 1D Schördinger equation in each of the individual layers, whose solutions $ψ$ are linear combinations of real or imaginary exponentials.

For a large barrier thickness, tunneling is weak perturbation with regard to the uncoupled dispersionless states, which are fully confined as well. In this case the dispersion relation $E z (k z)$ , periodic over $2π / d$ with over $d = a + b$ by virtue of the Bloch theorem, is fully sinusoidal

$E_z(k_z)=\frac{\Delta}{2}(1-\cos(k_z d))$

and the effective mass

${m^* = \frac{\hbar^2}{\part^2 E / \part k^2}}|_{k=0}$

Changes sigh for $2π / d$ . In the case of minibands this sinusoidal character is no longer preserved. Only high up in the miniband for wavevectors well beyond $2π / d$ , is the top actually 'sensed' and does the effective mass change sign. The very shape of the miniband dispersion obviously influences miniband transport profoundly, and accurate dispersion relation calculations are indispensable in the case of wide minibands. It is worth mentioning here that the condition for observing single miniband transport is the absence of any interminiband transfer by any process. This means in particular that the thermal quantum k_BT should be much smaller than the energy difference $E 2 - E 1$ between the first and second miniband, even in the presence of the applied electric field.

Bloch states

For an ideal superlattice a complete set of eigenstates states can be constructed by products of plane waves $e^{ i \mathbf{k} \cdot \mathbf{r} }/ 2\pi$ and a z-dependent function $f k (z)$ which satisfies the eigenvalue equation

$( E_c(z) - \frac{\part }{\part z} \frac{\hbar^2}{2 m_c (z)} \frac{\part }{\part z} + \frac {\hbar^2 \mathbf{k} ^2}{2m_c (z)} ) f_k (z) = E f_k (z)$ .

As $E c (z)$ and $m c (z)$ are periodic functions with the superlattice period d, the eigenstates are Bloch state $f_k (z)= \phi _{q, \mathbf{k}}(z)$ with energy $E^\nu (q, \mathbf{k})$ . Within first-order perturbation theory in k², one obtains the energy

$E^ \nu (q, \mathbf{k}) \approx E^ \nu(q, \mathbf{0}) + < \phi _{q, \mathbf{k}}| \frac{\hbar^2 \mathbf{k}^2}{2m_c (z)} | \phi _{q, \mathbf{k}} >$ .

Now, $\phi _{q, \mathbf{0}} (z)$ will exhibit a larger probability in the well, so that it seems reasonable to replace the second term by

$E_k = \frac{\hbar^2 \mathbf{k}^2}{2m_w}$

where $m w$ is the effective mass of the quantum well.

Wannier Functions

By definition the Bloch functions are delocalized over the whole superlattice structure. This may provide difficulties if electric fields are applied or effects due to the finite length of the superlattice are considered. Therefore, it is often helpful to use different sets of basis states which are better localized. A tempting choice would be the use of eigenstates of single quantum wells. Nevertheless such a choice has a severe shortcoming: the corresponding states are solutions of two different Hamiltonians, each neglecting the presence of the other well. Thus these states are not orthogonal which provides complications. Typically, the coupling is estimated by the transfer Hamiltonian within this approach. For these reasons, it is more convenient to use the set of Wannier functions.

Wannier–Stark ladder

If an electric field F is applied to the superlattice structure the Hamiltonian exhibits an additional scalar potential eφ(z) = −eFz which destroys the translational invariance. In this case, we can easily see that if there exists an eigenstate with wavefunction $φ 0 (z)$ and energy $E 0$ , then the set of states corresponding to wavefunctions $Φ j (z) = Φ 0 (z - j d)$ are eigenstates of the Hamiltonian with energies E_j = E₀ − jeFd as well. These states are equally spaced both in energy and real space and form the so-called Wannier–Stark ladder. This feature has to be considered with some care, as the potential $Φ 0 (z)$ is not bounded for the infnite crystal, which implies a continuous energy spectrum. Nevertheless, the characteristic energy spectrum of these Wannier–Stark ladders could be resolved experimentally in semiconductor superlattices.

Superlattice Transport

Overview of the different standard approaches for superlattice transport.

The motion of charge carriers in a superlattice is different from that in the individual layers: mobility of charge carriers can be enhanced, which is beneficial for high-frequency devices, and specific optical properties are used in semiconductor lasers.

If an external bias is applied to a conductor, such as a metal or a semiconductor, typically an electrical current is generated. The magnitude of this current is determined by the band structure of the material, scattering processes, the applied field strength, as well as the equilibrium carrier distribution of the conductor.

References

^ L. Esaki and R. Tsu, "Superlattice and negative differential conductivity in semiconductors", IBM Journal of Research and Development, vol. 14, no. 1 (January 1970), pp. 61-65.

H.T. Grahn, "Semiconductor Superlattices", World Scientific (1995).
Ivan K. Schuller, "A New Class of Layered Materials", Phys. Rev. Lett. 44, 1597 (1980).[1]
Morten Jagd Christensen, "Epitaxy, Thin Films and Superlattices", Risø National Laboratory, (1997).[2]
C. Hamaguchi, "Basic Semiconductor Physics", Springer (2001).
A. Wacker, Phys. Reports 357 (2002).
H.J. Haugan, et al. InAs/GaSb type-II superlattices for high performance mid-infrared detectors (Journal of Crystal Growth. Volume 278, Issues 1-4, 1 May 2005, Pages 198-202)[3]

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