M2 16 gg2t

where the symbols have the same meanings as above, and the exponential function provides the dominant dependence.

Trapped Particles in Two and Three Dimensions: Quantum Dot

The generalization of the Schrodinger equation to three dimensions is

where V2 = d2/dx2 + d2/dy2 + d2/dz2 and r is a vector with components x,y,z.

It is not difficult to see that the solutions for a particle in a three-dimensional infinite trap of volume I3 with impenetrable walls are given as ipn(x,y,z) = (2/I)3/2sinftt^jtx/L) sin(nyJty/I) sin(nzjtz/I), where nx = 1,2..., etc.,

These simple results can be easily adapted to two-dimensional boxes and also to boxes of unequal dimensions Lx Ly Lz.

Electrons Trapped in a Two-dimensional Box

A scanning tunneling microscope image, shown in Figure 4.3, reveals some aspects of the trapping of electrons in a 2D rectangular potential well. The well in this case is generated by the rectangular array of iron atoms (silver-colored dots in this image), which reflect electrons, much as in equation (4.55). The electrons are free electrons on the surface of (111) oriented single crystal copper. These electrons, by a quirk of the solid state physics of (111) copper, are essentially confined to the sur-

Figure 4.3 Electrons trapped in a small two-dimensional box on the (111) surface of copper. A rectangular array of iron atoms serves as a barrier, reflecting electron waves. [6]

y Figure 4.4 Geometry of a (111) plane, shown shaded. Copper is a face-centered cubic crystal, but only if the surface is cut to consist of the indicated (111) plane will the 2D electron effects be present face, and are further confined by the iron atoms to stay inside (or outside) the box. The traces of "ripples" which are seen here (also seen more clearly in Figure 3.8 in a circular geometry) are visible because the STM measurement, on a specifically 2D electron system, is sensitive to the density of states [7] as well as to the height of the surface.

Electron ripples inside a rectangular 2D box can be predicted from P{x,y) from equation (4.60) (setting nz= 0 to recover a two-dimensional case). The values of nx,ny appropriate to the ripples in Figure 4.3 are not known, but will be influenced by two facts:

1) First, the energy of the electrons being seen by the STM tip is the Fermi energy of the copper, plus or minus the voltage applied to the tip.

2) Second, the energy scale, suggested by equation (4.60): En = [h2/8mL2](n2 + n2) (which needs a small modification for unequal sides Lx, Ly) of the trapped electron levels, should be offset below the Fermi energy of copper by 0.44 eV. This is known to be the location of the bottom of the band of the 2D surface electrons on the (111) face of copper.

Electrons in a 3D "Quantum Dot"

Equations (4.59) and (4.60) are applicable to the electron and hole states in semiconductor "quantum dots", which are used in biological research as color-coded fluorescent markers. Typical semiconductors for this application are CdSe and CdTe.

A "hole" (missing electron) in a full energy band behaves very much like an electron, except that it has a positive charge, and tends to float to the top of the band. That is, the energy of the hole increases oppositely to the energy of an electron.

The rules of nanophysics that have been developed so far are also applicable to holes in semiconductors. To create an electron-hole pair in a semiconductor requires an energy at least equal to the energy bandgap, Eg, of the semiconductor.

This application to semiconductor quantum dots requires L in the range of 3 -5 nm, the mass m must be interpreted as an effective mass m*, which may be as small as 0.1 me. The electron and hole particles are generated by light of energy hc/X= En> electron + ^n.hole + (4-61)

Here the first two terms depend strongly on particle size L, as L~2, which allows the color of the light to be adjusted by adjusting the particle size. The bandgap energy, Egj is the minimum energy to create an electron and a hole in a pure semiconductor. The electron and hole generated by light in a bulk semiconductor may form a bound state along the lines of the Bohr model, described above, called an exciton. However, as the size of the sample is reduced, the Bohr orbit becomes inappropriate and the states of the particle in the 3D trap, as described here, provide a correct description of the behavior of quantum dots.

2D Bands and Quantum Wires 2D Band

A second physical situation that often arises in modern semiconductor devices is a carrier confined in one dimension, say z, to a thickness d and free in two dimensions, say % and y. This is sometimes called a quantum well. In this case ipn(x,y,z) = (2/d)1/2 sin(nzKz/d) exp(ikxx) exp(ikyy), (4.62)

and the energy of the carrier in the nth band is

In this situation, the quantum number nz is called the sub-band index and for n = 1 the carrier is in the first sub-band. We discuss later how a basic change in the electron's motion in a semiconductor band is conveniently described with the introduction of an effective mass m*.

Quantum Wire

The term Quantum Wire describes a carrier confined in two dimensions, say z and y, to a small dimension d (wire cross section d2) and free to move along the length of the wire, x. (Qualitatively this situation resembles the situation of a carrier moving along a carbon nanotube, or silicon nanowire, although the details of the bound state wavefunctions are different.)

In the case of a quantum wire of square cross section,

\pnn(x,y,z) = (2/d) sin(nyjry/d) sm(nz7iz/d) exp(ikxx), (4.64)

and the energy is

It is possible to grow nanowires of a variety of semiconductors by a laser assisted catalytic process, and an example of nanowires of indium phosphide is shown in Figure 4.5 [8].

Figure 4.5 Indium phosphide nanowires[8]. InP nanowires grown by laser-assisted catalytic growth, in 10, 20, 30 and 50 nm diameters, were studied by Atomic Force Microscope image (A) and also by observation of photoluminescence (B) and (C) under illumination by light with energy he//1 > Eg. The bandgap is about 1.4 eV. In (A), the white scale bar is 5 ¡am, so the wires are up to 10|Lim in length. (B) and (C) gray scale represantation of light emitted from 20 nm diameter InP nanowire excitation in panel (B) with bandgap light linearly polarized along the axis of the wire produces a large photoluminescence, but (C) no light is emitted when excited by bandgap light linearly polarized perpendicular to wire axis. Inset shows dependence of emission intensity on polarization angle between light and wire axis.

Figure 4.5 Indium phosphide nanowires[8]. InP nanowires grown by laser-assisted catalytic growth, in 10, 20, 30 and 50 nm diameters, were studied by Atomic Force Microscope image (A) and also by observation of photoluminescence (B) and (C) under illumination by light with energy he//1 > Eg. The bandgap is about 1.4 eV. In (A), the white scale bar is 5 ¡am, so the wires are up to 10|Lim in length. (B) and (C) gray scale represantation of light emitted from 20 nm diameter InP nanowire excitation in panel (B) with bandgap light linearly polarized along the axis of the wire produces a large photoluminescence, but (C) no light is emitted when excited by bandgap light linearly polarized perpendicular to wire axis. Inset shows dependence of emission intensity on polarization angle between light and wire axis.

The InP wires in this experiment are single crystals whose lengths are hundreds to thousands of times their diameters. The diameters are in the nm range, 10-50 nm. The extremely anisotropic shape is shown in Figure 4.2 to lead to extremely polarization-dependent optical absorption.

Since these wires have one long dimension, they do not behave like quantum dots, and the light energy is not shifted from the bandgap energy. It is found that the nanowires can be doped to produce electrical conductivity of N- and P-types, and they can be used to make electron devices.

The Simple Harmonic Oscillator

The simple harmonic oscillator (SHO) represents a mass on a spring, and also the relative motions of the masses of a diatomic molecule. (It turns out that it is also relevant to many other cases, including the oscillations of the electromagnetic field between fixed mirror surfaces.) In the first two cases the mass is m, the spring constant can be taken as k, and the resonant frequency in radians per second is

Treated as a nanophysics problem, one needs to solve the Schrodinger equation (4.30) for the corresponding potential energy (to stretch a spring a distance x):

This is a more typical nanophysics problem than that of the trapped particle, in that the solutions are difficult, but, because of their relevance, have been studied and tabulated by mathematicians.

The solutions to the SHO in nanophysics are:

where the Hn(x) are well-studied polynomial functions of x, and

These wavefunctions give an oscillatory probability distribution Pn(x) = ipnipn that has n + 1 peaks, the largest occurring near the classical turning points, x =±y/2£/fc. For large n, Pn(x) approaches the classical P(x):

where v is the classical velocity. Since v goes to zero at the turning points, P(x) becomes very large at those points. This feature is approached for large n in the solutions (4.68).

For small n, these functions differ substantially from the classical expectation, especially in giving a range of positions for the mass even in the lowest energy configuration, n = 0.

An important point here is that there is an energy too¡2 (zero point energy) for this oscillator even in its lowest energy state, n = 0. Also, from equation (4.68), this oscillating mass does not reside solely at x = 0 even in its ground state. It has an unavoidable fluctuation in position, given by the Gaussian function exp(-met)

x2¡2h), which is close to that predicted by the Uncertainty Principle, equation (4.22).

Figure 4.6 Probability Pn(x) density for n =13 state of simple harmonic oscillator. Dashed line is classical P(x), with classical turning points/ and /' indicated

Schrodinger Equation in Spherical Polar Coordinates

A more substantive change in the appearance of the Schrodinger equation occurs in the case of spherical polar coordinates, which are appropriate to motion of an electron in an atom. In this case, the energy U depends only on the radius r, making the problem spherically symmetric. In standard notation, where % = rsinO cos(p, y = rsin0 sirup , z = rcosO, with 6 and (p, respectively, the polar and azimuthal angles:

-h2 1 0 (r2dip\ H2 1 0 ( . ndxp\ 1 d2ip] rr, , n m72Fr UrJ - w ^eoe{smede) + Um = Elp (471)

The Hydrogen Atom, One-electron Atoms, Excitons

The Schrodinger equation is applied to the hydrogen atom, and any one-electron atom with nuclear charge Z, by choosing U =-kZe2/r, where k is the Coulomb constant. It is found in such cases of spherical symmetry that the equation separates into three equations in the single variables r, 6, and <p, by setting tp-RMmv). (4.72)

The solutions are conventionally described as the quantum states *Pn>i>m>m , specified by quantum numbers n, I, m, ms.

The principal quantum number n is associated with the solutions Rn,i(r)= (rlao)1 exp(-r/na0)i?n,j(r/a0) of the radial equation. Here ¿£n,i(rlao) is a Laguerre polynomial in p= rjaot and the radial function has n-l-1 nodes. The parameter aQ is identical to its value in the Bohr model, but it no longer signifies the exact radius of an orbit. The energies of the electron states of the one-electron atom, En = -Z2EJn2 (where E0= 13.6 eV, and Z is the charge on nucleus) are unchanged from the Bohr model. The energy can still be expressed as En = -kZe2/2rn, where rn = n2a0/Z, and a0= 0.0529 nm is the Bohr radius.

The lowest energy wavefunctions ^Pn.i.m.m °f the one-electron atom are listed in Table 4.1 [9].

Table 4.1 One-electron wavefunctions in real form [9]

Wavefunction designation

Wavefunction name, real form

Equation for real form of wavefunction*, where p = Zr/aQ and Cn = Z3/2/v^






C2 (2-p) e~p/2

^21,COS (p


C2 p sin0 cos(p e~p/2

^21, sin cp


C2 p sin# sin <p e_p/2



C2p cosO e~p/2



C3 (27-18p +2p2) e~p/3

^31,cos q>


C3 (6p-p2) sin<9 cosq> e~p/3


C3{6p-p2) sin(9 sinq> e"p/3



C3{6p-p2) cos6 e-p/3



C4 p2 (3cos2e -1) e~p/3



C5 p2 sine cose cos(p e~p/i

^3 2,sin <p


C5 p2 sine cose sine/? e~p/3

^32,cos2 tp


C6 p2 sin2e coslcp e~p/3


C6 p2 sin2e sin2q> e~p/3

C2= Q/4V2, C3 = 2Ci/8lV3, Q=C3/2, C5 = V6C4, Q=Cs/2.

C2= Q/4V2, C3 = 2Ci/8lV3, Q=C3/2, C5 = V6C4, Q=Cs/2.

represents the ground state.

The probability of finding the electron at a radius r is given by P(r) =4jtr2^/210o, which is a smooth function easily seen to have a maximum at r = aQ/Z. This is not an orbit of radius aQ, but a spherical probability cloud in which the electron's most probable radius from the origin is aQ. There is no angular momentum in this wave-function!

This represents a great correction in concept, and in numerics, to the Bohr model. Note that ^oo is real, as opposed to complex, and therefore the electron in this state has no orbital angular momentum. Both of these features correct errors of the Bohr model.

Complex forms of p orbitals

Real forms of p orbitals in orthogonal form

Complex forms of p orbitals

Real forms of p orbitals in orthogonal form

Figure 4.7 2p wavefunctions in schematic form. Left panel, complex forms carry angular momentum. Right panel, linear combinations have the same energy, now assume aspect of bonds

The n = 2 wavefunctions start with W200t which exhibits a node in r, but is spherically symmetric like W100. The first anisotropic wavefunctions are:

These are the first two wavefunctions to exhibit orbital angular momentum, here ±h along the z-axis. Generally where m, known as the magnetic quantum number, represents the projection of the orbital angular momentum vector of the electron along the z-direction, in units of h. The orbital angular momentum L of the electron motion is described by the quantum numbers I and m.

The orbital angular momentum quantum number I has a restricted range of allowed integer values:

This rule confirms that the ground state, n = 1, has zero angular momentum. In the literature the letters s,p,d,f,g, respectively, are often used to indicate 1 = 0,1,2,3, and 4. So a 2s wavefunction has n = 2 and I = 0.

The allowed values of the magnetic quantum number m depend upon both n and I according to the scheme y21,±1 = R(r)f(d)g(cp) = c2p sine e~p/2 exp(±i<p),

V2 (2+1)

Figure 4.8 Five allowed orientations of angular momentum I = 2, length of vector and z-projections in units of h. Azimuthal angle is free to take any value

There are 2 I + 1 possibilities. For 1 = 1, for example, there three values of m: -1,0, and 1, and this is referred to as a "triplet state". In this situation the angular momentum vector has three distinct orientations with respect to the z-axis: 0=45°, 90° and 135°. In this common notation, the n = 2 state (containing 4 distinct sets of quantum numbers) separates into a "singlet" (2s) and a "triplet" (2p).

For each electron there is also a spin quantum number S with projection

These strange rules, which are known to accurately describe the behavior of electrons in atoms, enumerate the possible distinct quantum states for a given energy state, n.

Following these rules one can see that the number of distinct quantum states for a given n is 2ft2. Since the Pauli exclusion principle for electrons (and other Fermi particles) allows only one electron in each distinct quantum state, 2ft2 is also the number of electrons that can be accommodated in the ftth electron shell of an atom. For n = 3 this gives 18, which is seen to be twice the number of entries in Table 4.1 for ft = 3.

A further peculiarity of angular momentum in nanophysics is that the vector L has length I = ^1(1(1 + 1)) h and projection Lz = mh. A similar situation occurs for the spin vector S, with magnitude S = V(s(s + 1)) h and projection mji. For a single electron ms = ±l/2. In cases where an electron has both orbital and spin angular momenta (for example, the electron in the n = 1 state of the one-electron atom has only S, but no I), these two forms of angular momentum combine as J =L + S, which again has a strange rule for its magnitude: J =^(j(j + 1)) h.

The wavefunctions W2it±i = C2 p sintf e~p/2 exp(±iq?) are the first two states having angular momentum. A polar plot of ^2141 has a node along z, and looks a bit like a doughnut flat in the xy plane.

The sum and difference of these states are also solutions to Schrodinger's equation, for example

^211 + ^21-1 = C2 p sinO e~p'2 [exp(icp) + exp(-i^)] = C2 p sin<9 e~p/2 2cos<p. (4.79)

This is just twice the 2px wavefunction in Table 4.1. This linear combination is exemplary of all the real wavefunctions in Table 4.1, where linear combinations have canceled the angular momenta to provide a preferred direction for the wavefunction.

A polar plot of the 2px wavefunction (4.79), shows a node in the z-direction from the sind and a maximum along the %-direction from the coscp, so it is a bit like a dumb-bell at the origin oriented along the %-axis. Similarly the 2py resembles a dumb-bell at the origin oriented along the y-axis.

These real wavefunctions, in which the exp(im^) factors have been combined to form sin<p and cos(p, are more suitable for constructing bonds between atoms in molecules or in solids, than are the equally valid (complex) angular momentum wavefunctions. The complex wavefunctions which carry the exp(imcp) factors are essential for describing orbital magnetic moments such as occur in iron and similar atoms. The electrons that carry orbital magnetic moments usually lie in inner shells of their atoms.

10 12 14 16

Figure 4.9 Radial wavefunctions of one-electron atoms exhibit n-€-l nodes, as illustrated in sketch for (bottom to top) 1 s, 2s, 3s, 2p and 3p wavefunctions

Magnetic Moments

The magnetic properties of atoms are important in nanotechnology, and lie behind the function of the hard disk magnetic memories, for example. A magnetic moment fi= iA (4.80)

is generated by a current loop bounding an oriented area A. The magnetization M is the magnetic moment per unit volume. A magnetic moment produces a dipole magnetic field similar to that of an ordinary bar magnet. The magnetization of a bar of magnetite or of iron is caused by internal alignment of huge numbers of the atomic magnetic moments. These, as we will see, are generated by the orbital motion of electrons in each atom of these materials.

How does an orbiting electron produce a magnetic moment? The basic formula ¡i= iA for a magnetic moment can be rewritten as (dq/dt)nrz = (ev^jtrJjtr2 = (e/2m)mrv. Finally, since mru is the angular momentum L, we get

This is the basic gyro-magnetic relation accounting for atomic orbital magnetism. Recalling that the z-component of L is Lz = mh, we see that the electron orbital motion leads to piz=(eh/2me) m. The Bohr Magneton /¿B is defined as ¿¿B = eh/2me=5.79x 10-5eV/T. The magnetic moment arising from the electron spin angular momentum is described by a similar formula,

Here g, called the gyromagnetic ratio, is g = 2.002 for the electron spin and ms=± 1/2. The energy of orientation of a magnetic moment fx in a magnetic field B is where the dot product of the vectors introduces the cosine of the angle between them. Thus, the difference in energy between states ms =1/2 and ms = -l/2for an electron spin moment in a magnetic field B (taken in the z-direction) is AE =guBB.

The reasons for the alignment of atomic magnetic moments in iron and other ferromagnetic substances, which underlies the operation of the magnetic disk memory, will be taken up later in connection with basic symmetry of particles under exchange of positions.

Positronium and Excitons

In the treatment of the hydrogen and one-electron atoms, it has been assumed that the proton or nucleus of charge Z is fixed, or infinitely massive. This simplification is not hard to correct, and a strong example is that of positronium, a bound state of a positron (positive electron) and an electron. These two particles have opposite charges but equal masses, and then jointly orbit around their center of mass, half the distance between the plus and minus charges. It is an exercise to see that the Bohr radius for positronium is 2a0 and its binding energy is EJ2. All of the results for the hydrogenic atom can be transcribed to this case by interpreting r as the inter-particle distance and interpreting m to the central mass:

The classic case is positronium, but a more relevant case is the exciton.

The exciton is the bound state of a photoexcited electron and photoexcited hole in a semiconductor, produced momentarily by light illumination of energy greater than the band gap. The electron and hole particles thus produced orbit around each other, having exactly equal and opposite charges. (In a short time the charges unite, giving off a flash of light.) In the medium of the semiconductor, however, the electron and hole assume effective masses, respectively, m"e and m \, which are typically significantly different from the free electron mass.

If, as is typical, one mass, say the electron mass, is small, say 0.1 of the free electron mass, then, from equation (4.84), mcm becomes small. Because the interparticle Bohr radius is inversely dependent on mass rncm, this makes the Bohr radius large, roughly 10 aQ. In turn, because the bound state energy is -kZe2/2r, this makes the energy small, roughly 0.1 E0.

Fermions, Bosons and Occupation Rules

Schrodinger's equation tells us that, in physical situations described by potential energy functions U, there are quantum energy states available for particles to occupy. It turns out that particles in nature divide into two classes, called fermions and bosons, which differ in the way they can occupy quantum states. Fermions follow a rule that only one particle can occupy a fully described quantum state, such as a hydrogen electron state described by n, 1, mh and ms. This rule was first recognized by Pauli, and is called the Pauli exclusion principle. In a system with many states and many fermion particles to fill these states, the particles first fill the lowest energy states, increasing in energy until all particles are placed. The highest filled energy is called the Fermi energy EF.

For bosons, there is no such rule, and any number of particles can fall into exactly the same quantum state. Such a condensation of many photons into a single quantum state is what happens in the operation of a laser.

When a nanophysical system is in equilibrium with a thermal environment at temperature T, then average occupation probabilities for electron states are found to exist. In the case of fermions, the occupation function (the Fermi-Dirac distribution function, fFD), is

For photons, the corresponding Bose-Einstein distribution function,/BE, is

[2] E. Schrodinger, Annalen der Physik 79, 361 (1926).

[4] C. Davisson and L. Germer, Proc. Natl. Acad. Sei. 14, 619(1928).

[5] W. Heisenberg, Zeit. fur. Phys. 43, 172 (1927).

[6] Courtesy IBM Research, Almaden Research Center. Unauthorized use not permitted.

[7] E. L. Wolf, Principles of Electron Tunneling Spectroscopy, (Oxford, New York, 1989), page 319.

[8] Reprinted with permission from Wang, J.S. Gudiksen, X. Duan, Y. Cui, and

C.M. Lieber, Science 293, 1455 (2001). Copyright 2001AAAS.

[9] F. J. Pilar, Elementary Quantum Chemistry, (Dover, New York., 2001), p. 125.

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