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Most modern electronic devices are considered low-dimensional systems, i.e. systems which have one or more of their dimensions in the range of the wavelength of electrons. Note that the wavelength of the electrons in a given semiconductor depends on its effective

I ffj

mass through the simple relation X = X0. —;-, where is the wavelength of electrons in

V m

vacuum. We therefore have to extend our knowledge of eigenstates and density of states (DOS) for such systems before we begin our discussion of quantum transport. There are 3 types of low-dimensional systems and these are shown in figures 5.1a-c. These are a) the 2-dimensional systems in the form of a slab, see figure 5.1a, b) the 1-dimensional systems in the form of a wire, see figure 5.1b, and c) the O-dimensional systems in the form of a dot, see figure 5.1c. It should be clear that the numbers 2, 1, and 0 in front of the word dimensional refer to the number of extended (almost infinite) dimensions. Such systems do not stand in vacuum but are realized by conducting media surrounded by dielectric parts or semiconductors of higher band gap than the semiconductor where conduction takes place—see figure 5.2 for such an example.

The 2-, 1-, and O-dimensional nanosystems

FIGURE 5.1 The 2-, 1-, and O-dimensional nanosystems.

Prior to calculating the energy spectra of such systems we obtain another useful formula for the density of states (DOS). In chapter 2, the DOS was defined by the relationship 2.51a. We remind the reader that (in the limit A£ —» dE)

where dN(E) is the number of states between E and E+dE, V is the volume of the system, and g(E) is the DOS, that is the number of states per unit energy per unit volume. Note

Creation of a nanolength system by sandwiching a piece of a semiconductor between two semiconductors of a larger band-gap

FIGURE 5.2 Creation of a nanolength system by sandwiching a piece of a semiconductor between two semiconductors of a larger band-gap.

that the word density refers to how dense the eigenstates are in energy and not to volume V. For any system therefore we can write for the DOS

The spin is included by either considering that there is degeneracy in the £, (as we have done) or explicitly by inserting a factor 2 in the RHS of 5.1. We note that as g(E) is integrated over the energy E, everytime the latter passes over an eigenvalue E, the integration will yield a 1, so 5.1 is a correct expression for the DOS. We now return to the specific lowdimensional systems we were discussing. These systems will exhibit wavefunction localization effects over the dimension in which they are nanometric. Usually we are not interested in the wavefunction change from atom to atom but in the change this localization will effect on the energy spectrum. It is therefore advisable that we use the effective mass equation of section 2.8 which deals only with the envelop function.

5.2a: The 2-Dimensional Electron Gas

As already stated, such a system results most commonly when a slab of a semiconducting material of a band gap Egj is sandwiched between two slabs of semiconductor material of band gap Eg2 > Egl, see figure 5.2. As we also explained in section 2.8, the whole system may be thought as being produced by a confining-perturbing potential of depth equal to Vp„ = AE(: = EC1 - Е for electrons and AEv = EV2 — EVI for holes extending over the width L of the smaller band gap semiconductor, as shown in figure 5.2. Then assuming planar interfaces, we can write for the envelope function, see figure 5.1a

The effective mass equation 2.83 for electrons becomes

where in 5.3 we measure energies from the bottom of the unperturbed conduction band. Note that, in accordance with our notation for the Schroedinger equation, the symbol V here stands for potential energy, not potential. Using the method of separation of variables that we presented in section 1.3, the 3-dimensional equation 5.3 can be decomposed into the following two equations.



We have assumed an isotropic mass m', but if the material is not isotropic different effective masses mj[ and m*± should be used in 5.4 and 5.5 respectively.

The solutions of 5.4 are obviously plane waves and we can therefore write for the eigenvalues

The solutions of 5.5, on the other hand, are bound states in the confining well of Vpa just as those described in chapter 1, section 1.3. The exact mathematical form that these states take depends on the choice of origin of the coordinate system used. However, it suffices to know that they are discreet bound states and hence their eigenvalues can be labeled as where (i) is the order of the state. The total energy E can be labeled thus as

A graph of E in к space is shown in figure 5.3.

Since our system is macroscopic in only the x and у directions and nanoscopic in the z direction, a small modification is needed in rewriting the definition 2.51a (reproduced in the beginning of section 5.2). We must actually write for the DOS of 2-dimensional systems Energies as a function of the 2-dimensional h vector. The energies el_, £± are the discretized energies of the nanosystem

FIGURE 5.3 Energies as a function of the 2-dimensional h vector. The energies el_, £± are the discretized energies of the nanosystem.


where S is the area (in the x, у directions) of the 2-dimensional electron gas and the factor 2 was introduced in equation 5.10 to account explicitly for spin. Let the respective lengths in the x and у directions be Lx and Ly. Then S = Lxx Ly and the kx and ky are, after applying cyclic boundary conditions,

where nx,ny = 0,1,2,...

Turning the finite к summations in 5.10 to an integration and remembering to include the factor necessary to do so (which is S/(2n)~ here) we get


But dkxdky = kdkdQ = ddd where 0 is the polar angle. Integrating over this angle will

yield a factor of 2л, so that we only need to perform the integration over 7td(/c2). Equation 5.11 is then transformed into

where 0(x) is the step function (or Heaviside function) defined as follows: 0(x) = l for x > 0 and 0(x) = 0 for x < 0. If the 2-dimensional semiconductor is made of Si, the result above has to be multiplied by the generacy of the valleys in Si, /,„ if the energies e^are treated as non-degenarate.

Density of states of a 2-dimensional nanosystem

FIGURE 5.4 Density of states of a 2-dimensional nanosystem.

A graph of g2D (E) is shown in figure 5.4, which exhibits a staircase shape because as the energy E increases and moves over an eigenvalue el of the well, it gives a contribution (m*lnfi2) everytime it does so. Note that as the width of the well L in the г direction increases the energy distance between the various e'± decreases and the form of the curve in figure 5.4 approaches a parabola as we expect from a 3D system. 2-dimensional systems are technologically very important because many devices are of this form whereas the very latest generation of transistors are of the one-dimensional systems. We proceed to examine these systems now.

  • 5.2b: The 1-Dimensional Systems or Quantum Wires
  • 1-dimensional systems are equally important to 2-dimensional systems. Most state of the art transistors are 1-dimensional systems. The easiest way to produce such systems is to shrink the length of, say, the x direction which forms one dimension of a 2-dimensional system. This is actually the path that the microelectronics industry has followed for many years in pursuit of Moore’s law. A note of warning: conduction may not necessarily take place along an extended direction in many transistor configurations. In all cases discussed above a confining potential is “applied” which derives from the band-edge offset of a larger band gap semiconductor or a dielectric.

We consider the x direction to be the extended one and confinement to occur in the у and z directions, see figure 5.1b. Then we can write for the perturbing potential producing the 1-dimensional system (dropping the superscript el)

We can use the effective mass equation and write for the envelop wavefunction again

where v|/(x) are one-dimensional plane waves and X(y,z) denotes localized wavefunctions bounded by the potential of 5.13 above. The X(y,z) obey the equation

where relates to the total energy E by

Note that since the system is bound in 2 dimensions the energies £,,m and the wavefunctions X(i„,(y,z) of the localized states, must be labeled by 2 indices. Note also that again an isotropic effective mass m* has been used in 5.15. The density of states of a 1-dimensional system gw(E) can now be written in analogy with 5.9 in the form

where Lx is the length of the system in the extended x direction.

Adapting the general formula 5.1 to 1 dimension we get

Note that in the equations above, the extra factor 2 in going from 5.18b to 5.18c is due to


the change of the limits of integration. Changing the variable of integration у =-|- we

obtain after some trivial operations

The graph of glD (E) is shown in figure 5.5.

Density of states of a 1-dimensional system, £;^ are the discretized energies of the nanosystem

FIGURE 5.5 Density of states of a 1-dimensional system, £;^ are the discretized energies of the nanosystem.

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