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Energy, E

Figure 9.10. Density ot states D(E) = dN(E)/dE plotted as a function of the energy E for conduction electrons delocalized in one (Q-wire), two (Q-well), and three (bulk) dimensions.

electrostatic forces becomes more pronounced, and die electrons become restricted by a potential barrier that must be overcome before they can move more freely. More explicitiy, the electrons become sequestered in what is called a potential well, an enclosed region of negative energies. A simple model that exhibits die principal characteristics of such a potential well is a square well in which die boundary is very sharp or abrupt Square wells can exist in one, two, three, and higher dimensions; for simplicity, we describe a one-dimensional case.

Standard quantum-mechanical texts show that for an infinitely deep square potential well of width a in one dimension, die coordinate x has the range of values — ¿a < x < inside the well, and the energies there are given by the expressions which are plotted in Fig. 9.11, where Eq = ¡'ZmcP is the ground-state energy and the quantum number n assumes the values n — 1,2, 3,____The electrons that are present fill up the energy levels starting from the bottom, until all available electrons are in place. An infinite square well has an infinite number of energy levels, with ever-widening spacings as the quantum number n increases. If the well is finite, then its quantized energies En all lie below the corresponding infinite well energies, and there are only a limited number of them. Figure 9.12 illustrates the case for a finite well of potential depth V0 = 7E0 which has only three allowed energies. No matter how shallow the well, there is always at least one bound state E,.

Figure 9.11. Sketch of wavefunctions for the four lowest energy levels (n = 1-4) of 8» one-dimensional infinite square wefl. For each level the form of the wavefunction is given on the left, and its parity (even or odd) is indicated on the right (From C. P. Poole, Jr., Handbook of Physics, Wiley, New York, 1998, p. 289.)

Figure 9.11. Sketch of wavefunctions for the four lowest energy levels (n = 1-4) of 8» one-dimensional infinite square wefl. For each level the form of the wavefunction is given on the left, and its parity (even or odd) is indicated on the right (From C. P. Poole, Jr., Handbook of Physics, Wiley, New York, 1998, p. 289.)

The electrons confined to the potential well move back and forth along the direction x, and the probability of finding an electron at a particular value of x is given by the square of the wavefunction |^s(x)|2 for the particular level n where the electron is located. There are even and odd wavefunctions \l>„(x) that alternate for the levels in die one-dimensional square well, and for the infinite square well we have the unnormalized expressions t¡/n = cos(nnx/a) n = 1,3,5,... even parity (9.7)

These wavefunctions are sketched in Fig. 9.11 for the infinite well. The property called parity is defined as even when >j/„(—x) = <l>„(x), and it is odd when

E2=4Eo

5.2Eo=E3

0.6Eo=E1

Figure 9.12. Sketch of a one-dimensional square well showing how the energy levels En of a finite well (right side, solid horizontal lines) lie below their infinite well counterparts (left side, dashed lines). (From C. P. Poole, Jr., Hancbook of Physics, Wiley, New York, 1998, p. 285.)

Another important variety of potential well, is one with a curved cross section. For a circular cross section of radius a in two dimensions, the potential is given by V — 0 in the range 0 < p < a, and has the value V0 at the top and outside, where p = (x2 and tan<£ — y/x in polar coordinates. The particular finite well sketched in Fig. 9.13 has only three allowed energy levels with the values Ex, E2, and E3. There is also a three-dimensional analog of the circular well in which the potential is zero for the radial coordinate r in the range 0 < r < a, and has the value V0 outside, where r = (x2 +y* + z2)1'2. Another type of commonly used potential well is the parabolic well, which is characterized by the potentials V(x) = jkx2, F(p) = \ kp2 and V(r) =\kr2 in one, two, and three dimensions, respectively, and Fig. 9.14 provides a sketch of the potential in the one-dimensional case.

Another characteristic of a particular energy state En is the number of electrons that can occupy it, and this depends on the number of different combinations of quantum numbers that correspond to this state. From Eq. (9.6) we see that the one-dimensional square well has only one allowed value of the quantum number n for each energy state. An electron also has a spin quantum number ms, which can take on two values, ms = +1 and ms = — for spin states up and down, respectively, and for the square well both spin states ms — ± \ have the same energy. According to the Pauli exclusion principle of quantum mechanics, no two electrons can have the same set of quantum numbers, so each square well energy state E„ can be occupied by two electrons, one with spin up, and one with spin down. The number of combinations of quantum numbers corresponding to each spin state is called its degeneracy, and so the degeneracy of all the one-dimensional square well energy levels is 2.

The energy of a two-dimensional infinite rectangular square well depends on two quantum numbers, nx — 0,1,2,3,... and ny = 0,1,2,3,..., where n1 — + ny. This means that the lowest energy state E, = E0 has two possibilities, namely, nx = 0,ny = 1, and nx = 1, ny = 0, so the total degeneracy (including spin direction) is 4. The energy state Es ~ 25E0 has more possibilities since it can have, for example, nx = 0, ny = 5, or nx = 3 and ny = 4, and so on, so its degeneracy is 8.

### 9.3.5. Partial Confinement

In the previous section we examined the confinement of electrons in various dimensions, and we found that it always leads to a qualitatively similar spectrum of discrete energies. This is true for a broad class of potential wells, irrespective of their dimensionality and shape. We also examined, in Section 9.3.3, the Fermi gas model for delocalized electrons in these same dimensions and found that the model leads to energies and densities of states that differ quite significantly from each other. This means that many electronic and other properties of metals and semiconductors change dramatically when the dimensionality changes. Some nanostructures of technological interest exhibit both potential well confinement and Fermi gas delocalization, confinement in one or two dimensions, and delocalization in two or one dimensions, so it will be instructive to show how these two strikingly different behaviors coexist

In a three-dimensional Fermi sphere the energy varies from E = 0 at the origin to E = E? at the Fermi surface, and similarly for the one- and two-dimensional analogs. When there is confinement in one or two directions, the conduction electrons will distribute themselves among the corresponding potential well levels that lie below the Fermi level along confinement coordinate directions, in accordance with their respective degeneracies dt, and for each case the electrons will delocalize in the remaining dimensions by populating Fermi gas levels in the delocalization direction of the reciprocal lattice. Table 9.5 lists the formulas for the energy dependence of the number of electrons N(E) for quantum dots that exhibit total confinement, quantum wires and quantum wells, which involve partial confinement, and bulk material, where there is no confinement The density of states formulas D(E) for these four cases are also listed in the table. The summations in these expressions are over the various confinement well levels i.

Figure 9.15 shows plots of the energy dependence N{E) and the density of states D(E) for the four types of nanostructures listed in Table 9.5. We see that the number of electrons N(E) increases with the energy E, so the four nanostructure types vary only qualitatively from each other. However, it is the density of states D(E) that determines the various electronic and other properties, and these differ dramatically for each of the three nanostructure types. This means that the nature of the

Table 9.5. Number of electrons N[E) and density of states D{E) = dH,E)/dE as a function of the energy £ for electrons <Mooaltz*c/oonfln«d in quantum dots, quantum wires, quantum wells, and bulk material*

Dimensions

Type Number of Electrons N(E) Density of States D<E) Delocalized Confined

Table 9.5. Number of electrons N[E) and density of states D{E) = dH,E)/dE as a function of the energy £ for electrons <Mooaltz*c/oonfln«d in quantum dots, quantum wires, quantum wells, and bulk material*

Dimensions

Type Number of Electrons N(E) Density of States D<E) Delocalized Confined

Dot |
N(E) = AoEW-^w) |
0(E) = •Ko H diâ(E - Em) |

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