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The nanoscale and quantum mechanics At the macroscopic scale of meters, classical mechanics can be used to describe motion. At the microscopic scale of atoms, classical mechanics fails to describe motion properly and quantum mechanics must be used. Quantum mechanics is valid at all length scales. It is possible to describe the motion of macroscopic objects with quantum mechanics; it is just mathematically difficult to do so. Classical mechanics and quantum mechanics give the same predictions for macroscopic objects so we usually use the simpler classical mechanics to describe large objects. Nanometer scale objects lie near the boundary between classical mechanics and quantum mechanics and sometimes it is necessary to use quantum mechanics to describe phenomena on the scale of nanometers. Richard Feynmann had this to say about quantum mechanics, "Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that you have any direct experience about. They do not behave like waves, they do not behave like particles, they do not behave like clouds, or billiard balls, or weights on springs, or like anything you have ever seen." Feynmann also said, "I think I can safely say that nobody understands quantum mechanics." Quantum mechanics is spectacularly successful in describing phenomena on a small scale. It is also mathematically difficult. In quantum mechanics, everything moves as a wave but exchanges energy and momentum as a particle. When an electron moves, it must be treated as a wave that can interfere. The wavefunction that describes an electron has peaks and valleys that move around and reflect off walls much like water waves. You can see the electron waves in the left image of Fig. 1.1. When a peak meets a valley there is destructive interference and when a peak meets a peak there is constructive interference. While an electron is moving, don't think of it as a particle that follows a particular path through space. A wave follows many paths simultaneously. Although an electron was used as an example here, the same could be said about other particles like protons, neutrons, or photons. It is even possible to observe the wave nature of larger objects such as atoms and molecules. All of the information that can be known about a particle is contained in its wavefunction. For instance, the square of the amplitude of the wavefuction describes the probability that an particle will be observed at a particular place. Other quantities such as the velocity of the particle can also be determined from the wavefunction. The time evolution of the wavefunction is described by the Schrödinger equation, Here i2 = -1, m is the mass of the particle, and V(x,y,z,t) is a space and time dependent potential that confines the particle. Ψ is the wavefunction. It is a three dimensional, time dependent, complex field. To understand this, think about a temperature map for a minute. Temperature is a real quantity (it has no imaginary part) but it can be different at every position in space and it changes in time. A wavefunction is a complex number at every position in space that changes as a function of time. 10 If two particles interact with each other (like an electron and a proton in a hydrogen atom) then there are not two wavefunctions (one for the electron and one for the proton) there is just one wavefunction, Ψ(xe,ye,ze,xp,yp,zp,t). This wavefunction describes the joint probability of finding an electron at position xe,ye,ze, and a proton at position xp,yp,zp. This is a complex, time dependent field in six dimensions. In a typical nanostructure, there are often millions of interacting particles. The wavefunction in this case would be a complex, time-dependent field in millions of dimensions. You could write down the Schrödinger equation for this case and try to solve using a computer. However, it turns out that the Schrödinger equation is intractable for more than about ten interacting particles. Intractable means that even though we know exactly the equation that needs to be solved and we know exactly how to solve it, it would take longer than the age of the universe to find the solution numerically on even the fastest computer. For more than ten interacting particles, analytic solutions are known to the Schrödinger equation for only a few special cases. For all other cases some kind of approximation has to be used. The trick is finding an approximation that is simple enough to solve but complex enough to describe the phenomena of interest. Albert Einstein expressed this as, "Everything should be made as simple as possible -- but no simpler." To understand why the Schrödinger equation is intractable, consider how it would be solved numerically. For a numerical solution, the wavefunction could be approximated by the value of the wavefunction on a grid of points. The number of points needed depends on the number of particles and the desired precision of the solution. A wavefunction for N particles moving in three dimensions would be a complex function of 3N spatial variables plus time. Ψ(x1, y1, z1, x2, y2, z2, ... ,xN, yN, zN, t) For a numerical solution, assume that it is sufficient to consider 100 values for x1 and 100 values for y1 etc. Then the number of points of the grid that would be used to approximate the wavefunction would be 1003N. Because N is in the exponent, the number of points becomes very large for large N. For 10 interacting particles, the number of points would be 1060. This is more than the number of atoms on the earth. There is no way to numerically solve the Schrödinger equation like this for a general problem involving 10 interacting particles on a conventional computer. Simplifications always need to be made to make the problem tractable. This is a sobering result. The Schrödinger equation, the most important equation in physics, is intractable. It is impossible in general to calculate the quantum dynamics of many interacting particles on a conventional computer. Approximations must be made in the calculations and then the validity of the approximations must be checked by carefully preparing a quantum system of interacting particles and measuring what happens. Quantum computing It is sometimes possible to map one intractable onto another so that if you find the solution to one of the intractable problems the solution to the other intractable problem will be known. It has been shown that calculating the quantum dynamics of certain systems can be mapped onto the problem of finding the factors of a large integer. Finding the factors of a large integer is known to be an intractable problem. If the right quantum system is prepared and then the evolution of that system is measured, the factors of the large integer can be determined. When a quantum system is used to perform a calculation like this it is called a quantum computer. These quantum computers can in principle solve certain intractable problems that no conventional computer could. However, up until now only very simple quantum computers have been made and the most powerful calculation that 11 has been performed was the factoring of 15 into 3 and 5. Research is proceeding on mapping intractable problems onto quantum dynamics (this is called finding quantum algorithms) and building more complex quantum computers. Uncertainty principle: The uncertainty principle applies to all waves: water waves, light, sound, and the matter waves described by quantum mechanics. It states that product of the variance in the width of a pulse times the variance in the wave numbers in the pulse must be greater than or equal to 0.5. ΔxΔk ≥ 0.5 The wave number is k = 2π/λ. Any wave pulse can be written as a superposition of sinusoidal waves with a certain distribution of wavelengths. The narrower the wave pulse is in position, the more wavelengths are needed to describe it. For instance, a wavepulse with a Gaussian form can be written as a superposition of cosine waveforms with different wavenumbers, Here the width of the wavepulse, Δx = a/2 and the width of the distribution of wavenumbers is Δk = 1/a and the product is ΔxΔk = 0.5. Any wavepulse can be written as a superposition of sine and cosine waveforms like this and the product of the uncertainties is always greater or equal to 0.5. A Gaussian wavepulse is a minimum uncertainty wavepulse. The uncertainty relation is often applied to classical waves in telecommunications, seismology, ultrasound imaging, or other disciplines where waves are measured. In quantum mechanics it is common to multiply both sides of this equation by h/(2π). Since p = hk/(2π), ΔxΔp ≥ h/(4π). This relation is often loosely interpreted as saying that it is impossible to know the position and the momentum of a particle simultaneously. Quantum dots: Quantum dots are man-made structures that can be used to store a small amount of charge. 12 Typically the structure is a piece of a semiconductor with dimensions of a few tens of nanometers to a few microns. Quantum dots typically contain a charge somewhere between a single electron and a few thousand electrons. Fig. 4.1. A lateral quantum dot. In this case, gold gates are deposited on a layered semiconductor. Mobile electrons exist at the interface between two kinds of semiconductor about 100 nm beneath the surface. By apply a negative voltage to the gold gates, the regions underneath the gates can be depleted of electrons. By choosing appropriate gate voltages for all of the gates, isolated puddles of electrons can be formed between the gates. These puddles are the quantum dots. Fig. 4.2. A schematic of a vertical quantum dot. 13 Fig. 4.3. Quantum dots sorted by size emitting light of differnt colors. Reference: www.physik.uni-muenchen.de/sektion/feldmann/fieldsoi/qdot/qdhome_b.htm A crude model for the quantum states in a quantum dot is the infinite potential well. This problem is discussed in most physics textbooks and the results are summarized below. 1-d infinite square well For a particle of mass m in an infinite square well potential, the potential energy of the particle is infinite for x < 0 and x > L. For 0 < x < L, the potential energy of the particle is zero. The wavefunctions for a one-dimensional potential well are: The energies that correspond to these wavefunctions can be determined by substituting the wavefunctions into the time-independent Schrödinger equation. The energies are: En = n²h² = E1n² 8mL² n = 1,2,3,... 14 The energy states are plotted as a function of n. Density of states Sometimes it is useful to know the number of states in a certain energy range. If the 1-d potential is very long (large L) then the states will be closely spaced and the number of states in a certain energy range can be approximated as a continuous function. By looking at the figure above it is clear that there are no states with energies below E1 = h²/(8mL²), there are many states just above E1 at the bottom of the parabola, and there are then fewer and fewer states as the energy increases. This can be stated more mathematically by defining a function called the density of states g(E). The number of states between energies EA and EB is, EB N = ∫ g(E)dE EA Using the relationship for the energy, E = E1n², the density of states for a one-dimensional potential well can be determined to be, Here a factor of 2 has been included to account for the two spins allowed for every value of n. 15 The density of states for a one-dimensional potential. 2-d infinite square well For a particle of mass m in an infinite square well potential, the potential energy of the particle is infinite for x < 0, y < 0, and x > Lx, or y > Ly. For 0 < x < Lx and 0 < y < Ly, the potential energy of the particle is zero. The wavefunctions for a two-dimensional infinite square well potential are: The energies that correspond to these wavefunctions can be determined by substituting the wavefunctions into the time independent Schrödinger equation. The energies are: Enxny = h² nx² ny² ( + ) 8m Lx² Ly² nx,ny = 1,2,3,... Density of states The density of states for a two-dimensional potential square well is zero for energies below, E1 = h² 1 1 ( + ) 8m Lx² Ly² and is constant for energies above this energy. Including spin degeneracy the density of states is, 0 g(E) = π 2E1 for E < E1 for E > E1 16 The density of states for a two-dimensional potential. 3-d infinite square well For a particle of mass m in an infinite 3-d square well potential, the potential energy of the particle is infinite for x < 0, y < 0, z < 0 and x > Lx, or y > Ly, z > Lz. For 0 < x < Lx, 0 < y < Ly, and 0 < z < Lz, the potential energy of the particle is zero. The wavefunctions for a three-dimensional infinite square well potential are: The energies that correspond to these wavefunctions can be determined by substituting the wavefunctions into the time independent Schrödinger equation. The energies are: Enxnynz = h² nx² ny² nz² ( + + ) 8m Lx² Ly² Lz² nx,ny,nz = 1,2,3,... Density of states The density of states for a two-dimensional potential square well is zero for energies below, E1 = h² 1 1 1 ( + + ) 8m Lx² Ly² Lz² and is increases as a square root above this energy. Including spin degeneracy the density of states is, 17 The density of states for a three-dimensional potential. Reading Review the discussion of quantum mechanics in sections 13.6-13.10 in Understanding Physics. Review "The infinite square potential well", section 13.13 in Understanding Physics. Review chapter 19, "Atomic physics" in Understanding Physics. References Modeling and Prospects for a Solid-State Quantum Computer, H. E. Ruda and B. Qiao, Proceddings of the IEEE, vol 91, p. 1874 (2003). Britney's guide to semiconductor physics: Density of states Problems 1. In an electron microscope, electrons are accelerated through a potential of 70 kV. In the normal operation mode of an electron microscope, it is not possible to image features smaller than the wavelength of the electrons. What is the wavelength of these electrons? 2. A C60 is bound to a surface by van der Waals forces. The wavefunction that describes the position of the C60 is ψ = (2/π)1/4exp(-(x-x0)2). What is the probability that the C60 molecule will be found between 0.8x0 and 1.2x0? 3. In practically every physics textbook that discusses wave-particle duality, there is a calculation of the wavelength of the matter wave of a macroscopic object. In Understanding Physics this discussion is on page 370. The relationship between the momentum of an object and the wavelength of the object is p = h/λ. Since p is typically on the order of 1 kg m/s for a macroscopic object and h is a very small number, the wavelength of a macroscopic object is very small. The conclusion of this discussion is always that it is impossible to observe quantum interference of a macroscopic object. Waves can be scattered from a lattice. The smallest lattice that we can make is an atomic crystal. Look at the description of Bragg scattering in Understanding Physics on page 335. Then read about electron diffraction in the Davidson-Germer experiment on page 371. It is 18 the Bragg scattering of electron waves from a crystal that was observed in the DavidsonGermer experiment. Imagine scattering a C60 molecule from a crystal in the same way. Estimate the largest angle of diffraction that can be achieved (Consider only the first order maximum n = 1). Choose a reasonable spacing between the atoms in the crystal. You will need to know the minimum momentum that a particle can be given. Assume that thermal fluctuations limit the resolution of the momentum i.e. 0.5kBT = 0.5mv2. Consider temperatures down to 1 K. 19