Quantum Transport: Atom to Transistor

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Including some of the most advanced concepts of non-equilibrium quantum statistical mechanics, this book presents the conceptual framework underlying the atomistic theory of matter. No prior acquaintance with quantum mechanics is assumed. Many numerical examples provide concrete illustrations, and the corresponding MATLAB codes can be downloaded from the web.

Videostreamed lectures linked to specific sections of the book are also available through web access. Specifications Publisher Cambridge University Press. Customer Reviews. Write a review. See any care plans, options and policies that may be associated with this product. Email address. Please enter a valid email address. Walmart Services. Get to Know Us. Customer Service. In The Spotlight. Shop Our Brands. All Rights Reserved. Kamakhya Prasad Ghatak. The Thermoballistic Transport Model.

Reinhard Lipperheide. Applications of Chalcogenides: S, Se, and Te. Gurinder Kaur Ahluwalia. Bogoliubov-de Gennes Method and Its Applications. Jian-Xin Zhu. Sergio C. Mircea Dragoman. Spin Physics in Semiconductors. Mikhail I. Control of Magnetotransport in Quantum Billiards. Peter Schmelcher. Physics of Graphene. Hideo Aoki. Seiji Samukawa. Robson Ferreira. Low Dimensional Semiconductor Structures.

Excitonic and Photonic Processes in Materials. Jai Singh. Umar Ibrahim Gaya. Nanostructured Lead, Cadmium, and Silver Sulfides. Stanislav I. Bismuth-Containing Compounds. Handong Li. Handbook of Bioelectronics. Sandro Carrara. Magnetic Resonance of Semiconductors and Their Nanostructures. Pavel G. FIB Nanostructures. Chips Bernd Hoefflinger. The Physics of Semiconductors.

Marius Grundmann. Lessons from Nanoelectronics. The total capacitance is denoted CE , where E stands for electrostatic. Iterative procedure for self-consistent solution: For a small device, the effect of the potential U is to raise the DOS in energy and can be included in our expressions for the number of electrons N Eq. We start with an initial guess for U, calculate N from Eq. Discovering an appropriate function U N if there is one! However, there is one aspect that I would like to mention right away, since it can affect our picture of current flow even for a simple one-level device and put it in the so-called Coulomb blockade or single-electron charging regime.

Let me explain what this means. Energy levels come in pairs, one up-spin and one down-spin, which are degenerate, that is they have the same energy. Usually this simply means that all our results have to be multiplied by two. The expressions for the number of electrons and the current should all be multiplied by two.

However, there is a less trivial consequence that I would like to explain. Consider a channel with two spin-degenerate levels Fig. We expect the broadened DOS to be twice our previous result see Eq. Since the available states are only half filled for a neutral Prologue: an atomistic view of electrical resistance 0. However, under certain conditions the DOS looks like one of the two possibilities shown in Fig.

It is hard to understand why the two peaks should separate based on the simple SCF picture. The point is that no electron feels any potential due to itself. Suppose the up-spin level gets filled first, causing the down-spin level to float up by U0. But the up-spin level does not float up because it does not feel any self-interaction, leading to the picture shown on the left in Fig. Of course, it is just as likely that the down-spin will fill up first leading to the picture on the right. Describing the flow of current in this Coulomb blockade regime requires a very different point of view that we will not discuss in this book, except briefly in Section 3.

But when do we have to worry about Coulomb blockade effects? Otherwise, the SCF method will give results that look much like those obtained from the correct treatment see Fig. Answer: the extent of the electronic wavefunction. Levels with well-delocalized wavefunctions large R have a very small U0 and the SCF method provides an acceptable description even at the lowest temperatures of interest.

But if R is small, then the charging energy U0 can exceed kB T and one could be in a regime dominated by single-electron charging effects that is not described well by the SCF method. In describing electronic conduction through small conductors we can identify the following three regimes. More correctly, one could use if practicable the multielectron master equation that we will discuss in Section 3.

It is generally recognized that the intermediate regime can lead to novel physics that requires advanced concepts, even for the small conductors that we have been discussing. With large conductors too we can envision three regimes of transport that evolve out of these three regimes. We could view a large conductor as an array of unit cells as shown in Fig.

If t U0 , it will be in the CB regime and can in principle be treated using the multi-electron master equation to be discussed in Section 3. If the conductor is extended in the transverse plane, we should view each unit cell as representing an array of unit cells in the transverse direction. Indeed many believe that high-Tc superconductors whose microscopic theory is still controversial consist of unit cells whose coupling is delicately balanced on the borderline of the SCF and the CB regimes.

The more delocalized the electronic wavefunctions large t , the more accurate the SCF description becomes and in this book I will focus on this regime. Basically I will try to explain how the simple one-level description from Section 1. Note that in Eqs. Let me explain why. If we make the rather cavalier assumption that all levels conduct independently, then we could use exactly the same equations as for the one-level device, replacing the 24 Prologue: an atomistic view of electrical resistance m1 m2 Drain Source Fig.

With this in mind, I will refer to Eqs. Nanotransistor — a simple model: As an example of this independent level model, let us model the nanotransistor shown in Fig. Note that the gate capacitance CG is much larger than the other capacitances, which helps to hold the channel potential fixed relative to the source as the drain voltage is increased see Eq. The essential feature of a well-designed transistor is that the gate is much closer to the channel than L, allowing it to hold the channel potential constant despite the voltage VD on the drain.

I should mention that our present model ignores the profile of the potential along the length of the channel, treating it as a little box with a single potential U given by Eq. Nonetheless the results Fig. Luckily we do not need to pin down the precise value of this fraction, since the present model gives the same current independent of L. However, the coupling to the contacts decreases inversely with the length L of the conductor, since the longer a conductor is, the smaller is its coupling to the contact.

The reason is that we are really modeling a ballistic conductor, where electrons propagate freely, the only resistance arising from the contacts.

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The conductance of such a conductor is indeed independent of its length. The ohmic length dependence of the conductance comes from scattering processes within the conductor that are not yet included in our thinking. For example, in a uniform channel the electronic wavefunction is spread out uniformly.

But a scatterer in the middle of the channel could split up the wavefunctions into two, one on the left and one on the right with different energies. In our model we do not explicitly discuss these processes; we simply legislate that the contacts are maintained at equilibrium with the assumed electrochemical potentials. The full story requires us to include phase-breaking scattering processes that cause a change in the state of an external object. For example, if an electron gets deflected by a rigid that is unchangeable defect in the lattice, the scattering is said to be coherent.

But if the electron transfers some energy to the atomic lattice causing it to start vibrating, that would constitute a phase-breaking or incoherent process. Such incoherent scatterers are also needed to remove energy from the electrons and cause dissipation. For example, in this chapter we have developed a simple model that allows us to calculate the resistance R, but none of the associated Joule heat I 2 R is dissipated in the channel; it is all dissipated in the contacts.

This is evident if we consider what happens when an electron goes from the source to the drain Fig. In the real world too there is experimental evidence that in nanoscale conductors, most of the heating occurs in the contacts outside the channel, allowing experimentalists to pump a lot more current through a small conductor without burning it up. But long conductors have significant incoherent scattering inside the channel and it is important to include it in our model.

All quantities of interest can be calculated from these matrices. For example, in Section 1. Figure 1. The complete set of equations is summarized in Chapter The reader might wonder why the numbers become matrices, rather than just column vectors. It seems reasonable 29 1. But why do we need a matrix H m, n? This is a question that goes to the heart of quantum mechanics whereby all physical quantities are represented by matrices. In general, no single representation will diagonalize all the matrices and a full quantum treatment is needed.

In order to make this approach accessible to readers unfamiliar with advanced many-body physics, I will derive these results using elementary arguments. What we have derived in this chapter Fig. Indeed if there is a representation that diagonalizes all the matrices, then the matrix model without the s-contact would follow quite simply from Fig. We could write down separate equations for the current through each diagonal element or level for this special representation, add them up and write the sum as a trace. The resulting equations would then be valid in any representation, since the trace is invariant under a change in basis.

In general, however, the matrix model cannot be derived quite so simply since no single representation will diagonalize all the matrices. In Chapters 8—10, I have derived the full matrix model Fig. I should mention that the picture in Fig. In our elementary model Fig. In the matrix model Fig. However, this can be true only for very short channels. We then need additional equations to determine both [A E ] and [G n E ]. Let us now start where quantum mechanics started, namely, with the hydrogen atom.

Calculate the current vs. Thermoelectric effect: In this chapter we have discussed the current that flows when a voltage is applied between the two contacts. To see this, calculate the current from Eq. This problem is motivated by Paulsson and Datta Negative differential resistance: Figure 1. This problem is motivated by Rakshit et al. We start in this chapter with 1 the hydrogen atom Section 2. It was apparent that a radical departure from classical physics was called for. Bohr postulated that electrons could be described by stable orbits around the nucleus at specific distances from the nucleus corresponding to specific values of angular momenta.

If we heat up the atom, the electron is excited to higher stationary orbits Fig. This striking agreement with experiment suggested that there was some truth to this simple picture, generally known as the Bohr model. The energy E of the electron plays a role similar to that played by the frequency of the acoustic wave. It is wellknown that a sound box resonates at specific frequencies determined by the size and shape of the box.

Let us elaborate on this point a little further. Real boxes are seldom in this form but this idealization is often used since it simplifies the mathematics. The justification for this assumption is that if we are interested in the properties in the interior of the box, then what we assume at the ends or surfaces should 37 2. Anyway, for a periodic box the eigenfunctions are given by cf. But for each value of k there is a sine and a cosine function which have the same eigenvalue, so that the eigenvalues now come in pairs.

An important point to note is that whenever we have degenerate eigenstates, that is, two or more eigenfunctions with the same eigenvalue, any linear combination of these eigenfunctions is also an eigenfunction with the same eigenvalue. What is the corresponding quantity we should sum to obtain the probability current density J x, t? This ensures that the continuity equation is satisfied regardless of the detailed dynamics of the wavefunction. However, this is true only for the plane wave functions in Eq. The cosine and sine states in Eq.

Indeed Eq. To see how this is done let us for simplicity consider just one dimension and discretize the position variable x into a lattice as shown in Fig. We can write Eq. Since Eq. It can be shown that this form, Eq.

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The wavefunction at subsequent times t is then given by Eq. Later we will discuss how we can figure out the coefficients. For the moment we are just trying to make the point that the dynamics of the system are easy to visualize or describe in terms of the eigenvalues which are the energy levels that we talked about earlier and the corresponding eigenvectors which are the wavefunctions associated with those levels of [H]. That is why the first step in discussing any system is to write down the matrix [H] and to find its eigenvalues and eigenvectors.

This is easily done using any standard mathematical package like M at l a b as we will discuss in the next section. These examples are all simple enough to permit analytical solutions that we can use to compare and evaluate our numerical solutions. The advantage of the numerical procedure is that it can handle more complicated problems just as easily, even when no analytical solutions are available.

It is straightforward to set up this matrix and use any standard mathematical package like M at l a b to find the eigenvalues and the corresponding eigenvectors. We obtained eigenvalues, which are plotted in Fig. Boundary conditions: One more point: Strictly speaking, the matrix [H] is infinitely large, but in practice we always truncate it to a finite number, say N, of points. This means that at the two ends we are replacing see Eq. The actual value of the potential at the end points will not affect the results as long as the wavefunctions are essentially zero at these points anyway.

Another boundary condition that is often used is the periodic boundary condition where we assume that the last point is connected back to the first point so that there are no ends Fig. As we mentioned earlier Fig. This does change the resulting eigenvalues and eigenvectors, but the change is imperceptible if the number of points is large. Both forms are equally correct though one may be more convenient than the other for certain calculations.

Number of eigenvalues: Another important point to note about the numerical solution is that it yields a finite number of eigenvalues unlike the analytical solution for which the number is infinite. This means that in practice we are limited to very small problems. However, if the coordinates are separable then we can deal with three separate one-dimensional problems as opposed to one giant three-dimensional problem.

Resistance of a Molecule

Spherically symmetric potential: Some problems may not be separable in Cartesian coordinates but could be separable in cylindrical or spherical coordinates. Equation 2. The dots show the analytical result Eqs.

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Numerical results with a lattice spacing of a should be compared with the analytical values of f r 2 a. The 1s level agrees well, but the 2s level is considerably off. The reason is easy to see if we plot the corresponding f r 2 and compare with the analytical results. It is evident from Fig. Solid line shows the analytical result Eqs. This simple example illustrates the essential issues one has to consider in setting up the lattice for a numerical calculation. The lattice constant a has to be small enough to provide adequate sampling of the wavefunction while the size of the lattice has to be big enough to cover the entire range of the wavefunction.

If it were essential to describe all the eigenstates accurately, our problem would be a hopeless one. Luckily, however, we usually need an accurate description of the eigenstates that lie within a certain range of energies and it is possible to optimize our matrix [H] so as to provide an accurate description over a desired range. Use Eq. Finally, a supplementary section elaborates on the concepts of Section 3.

The energy levels look something like Fig. The elements of the periodic table are arranged in order as the number of electrons increases by one from one atom to the next. Their electronic structure can be written as: hydrogen, 1s1 ; helium, 1s2 ; lithium, 1s2 2s1 ; beryllium, 1s2 2s2 ; boron, 1s2 2s2 2p1 , etc. How do we calculate the energy levels for a multi-electron atom? The nuclear potential Unuc , like UL , is fixed, while USCF depends on the electronic wavefunctions and has to be calculated from a self-consistent iterative procedure. In this chapter we will describe this procedure and the associated conceptual issues.

What will the energy levels looks like? However, this does not compare well with experiment at all. How could we be off by over 30 eV? It is because we did not account for the other electron in helium. The reason is that an electron in a helium atom feels a repulsive force from the other electron, which effectively raises its energy by 30 eV and makes it easier for it to escape Fig.

This means that the calculation has to be done self-consistently as follows. Step 1. Step 2. Step 3. Step 4. The latter has to be calculated self-consistently. Step 5. For Step 2 we can use essentially the same method as we used for the hydrogen atom, although an analytical solution is usually not possible.

However, the dependence on r is quite complicated so that no analytical solution is possible. This preserves the spherical symmetry of the charge distribution and the difference is usually not significant. The radial probability distributions for hydrogen 1s level and silicon 1s level and 3p level are shown. Also shown for comparison is the 1s level of the hydrogen atom, discussed in the last chapter.

Silicon atom: Figure 3. Also shown for comparison is the 1s level of the hydrogen atom. Note that the silicon 1s state is very tightly confined relative to the 3p state or the hydrogen 1s state. This is typical of core states and explains why such states remain well-localized in solids, while the outer electrons like 3p are delocalized. However, this approach quickly gets out of hand as we go to bigger atoms with many electrons and so is seldom implemented directly.

But suppose we could actually calculate the energy levels of multi-electron atoms. The former called the ionization levels are the filled states from which an electron can be removed while the latter the affinity levels are the empty states to which an electron can be added. The answer depends on what we want our one-electron energy levels to tell us. Ionization levels and affinity levels: Our interest is primarily in describing the flow of current, which involves inserting an electron and then taking it out or vice versa, as we discussed in Chapter 1.

So we would want the one-electron energy levels to represent either the energies needed to take an electron out of the atom ionization levels or the energies involved in inserting an electron into the atom affinity levels Fig. Such photoemission experiments are very useful for probing the occupied energy levels of atoms, molecules, and solids. However, they only provide information about the occupied levels, like the 1s level of a helium atom or the valence band of a semiconductor. To probe unoccupied levels such as the 2s level of a helium atom or the conduction band of a semiconductor we need an inverse photoemission IPE experiment see Fig.

In large solids without significant excitonic effects we are accustomed to assuming that the optical gap is equal to the gap between the valence and conduction bands, but this need not be true for small nanostructures. How do we choose this effective potential? From Eqs. But to calculate the affinity levels we should use the potential due to Z electrons two electrons for helium.

The energy levels we obtain from the first calculation are lower in energy than those obtained from the second calculation by the single-electron charging energy U0. As we discussed in Section 1. Even in nanostructures that are say 10 nm or less in dimension, it can be quite significant that is, comparable to kB T. One important consequence of this is that even if an SCF calculation gives energy levels that are very closely spaced compared to kB T see Fig. Of course, this is a significant effect only if the single-electron charging energy U0 is larger than kB T.

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Device problems often require us to incorporate complicated boundary conditions including different materials with different dielectric constants. Correlation energy: The actual interaction energy is less than that predicted by Eq. The corresponding selfconsistent potential is also reduced cf. The potential in Eq. But it is really quite surprising that the one-electron picture with a suitable SCF often provides a reasonably accurate description of multielectron systems.

The fact that it works so well is not something that can be proved mathematically in any convincing way. Our confidence in the SCF method stems from the excellent agreement that has been obtained with experiment for virtually every atom in the periodic table see Fig. Almost all the work on the theory of electronic structure of atoms, molecules, and solids is based on this method and that is what we will be using.

The results are in excellent agreement with experiment adapted from Herman and Skillman For a hydrogen atom, the s and p levels are degenerate that is, they have the same energy. But this is not true of the self-consistent potential due to the electrons and, for multi-electron atoms, the s state has a lower energy than the p state. In Section 3.

We will then discuss two bonding mechanisms ionic Section 3. The numbers are taken from Harrison and Mann The energies of these valence electrons exhibit a periodic variation as shown in Fig. The main point to notice is that the energies tend to go down as we go across a row of the periodic table from lithium Li to neon Ne , increase abruptly as we step into the next row with sodium Na and then decrease as we go down the row to argon Ar.

This trend is shown by both the s and p levels and continues onto the higher rows. Indeed this periodic variation in the energy levels is at the heart of the periodic table of the elements. The energy levels of Na and Cl look roughly as shown in Fig. This is only part of the story, since the overall energetics also includes the electrostatic energy stored in the microscopic capacitor formed by the two ions as explained in the text.

But this argument is incomplete because we also need to consider the change in the electrostatic energy due to the bonding. The correct binding energy is more like 4 eV. The point is that the energy levels we have drawn here are all ionization levels. The numerical details of this specific problem are not particularly important or even accurate. EB Formation of H2 from individual H atoms with a bonding level EB and an anti-bonding a lower energy level, the detailed energetics of the process require a more careful discussion.

In general, care is needed when using one-electron energy level diagrams to discuss electron transfer on an atomic scale. The atoms on the right of the periodic table have lower electronic energy levels and are said to be more electronegative than those on the left. We would expect electrons to transfer from the higher energy levels in the former to the lower energy levels in the latter to form an ionic bond.

However, this argument does not explain covalent bonds which involve atoms with roughly the same electronegativity. The process is a little more subtle. For example, it is hard to see why two identical hydrogen atoms would want to form a H2 molecule, since no lowering of energy is achieved by transferring an electron from one atom to the other.

What happens is that when the two atoms come close together the resulting energy levels split into a bonding level EB and an anti-bonding level EA as shown in Fig. The lowest energy solution to Eq. Let us now try to understand the competing forces that lead to covalent bonding. The dashed curve in Fig. Experimentally, the bond length R for a H2 molecule is 0. To answer this question we need to calculate the overall energy which should include the electron—electron repulsion note that USCF r 69 3.

To understand the overall energetics let us consider the difference in energy between a hydrogen molecule H2 and two isolated hydrogen atoms 2H. The solid curve in Fig.

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It has a minimum around 0. Despite the crudeness of the approximations used, the basic physics of bonding is illustrated fairly well by this example. Vibrational frequency: The shape of the binding energy vs. R curve suggests that we can visualize a hydrogen molecule as two masses connected by a spring Fig. The binding energy of the hydrogen molecule see Fig.

Ionization levels: As we have discussed, the energy levels of a multi-electron system usually denote the ionization levels, that is the energy it takes to strip an electron from the system. However, it is easy to see from Eqs. However, it is possible to solve the multi-electron problem exactly if we are dealing with a small channel weakly coupled to its surroundings, like the one-level system discussed in Section 1.

It is instructive to recalculate this one-level problem in the multi-electron picture and compare with the results obtained from the SCF method. One-electron vs. Since each one-electron energy level can either be empty 0 or occupied 1 , multi-electron states can be labeled in the form of binary numbers with a number of digits equal to the number of one-particle states. N one-electron states thus give rise to 2N multi-electron states, which quickly diverges as N increases, making a direct treatment impractical.

That is why SCF methods are so widely used, even though they are only approximate. Consider a system with two degenerate one-electron states up-spin and down-spin that can either be filled or empty. All other one-electron states are assumed not to change their occupation: those below remain filled while those above remain empty.

We have four available multi-electron states which we can designate as 00, 01, 10, and Equations involving probabilities of different states are called master equations. We could call Eq. For example, if we assume that the interaction only involves the entry and exit of individual electrons from the source and drain contacts 73 3.

The entry rate is proportional to the available electrons, while the exit rate is proportional to the available empty states. The same picture applies to the flow between the 00 and the 10 states, assuming that up- and down-spin states are described by the same Fermi function in the contacts, as we would expect if each contact is locally in equilibrium. Using these rate constants it is straightforward to show from Eq.

Figure 3. Relation between the multi-electron picture and the one-electron levels: As I have emphasized in Section 3. Transitions involving the addition of one electron are called affinity levels while those corresponding to the removal of one electron are called ionization levels. For example see Fig. This is a very important general concept regarding the interpretation of the one-electron energy levels when dealing with complicated interacting objects.

The occupied or ionization levels tell us the energy levels for removing an electron while the unoccupied or affinity levels tell us the energy levels for adding an extra electron. Indeed that is exactly how these levels are measured experimentally, the occupied levels by photoemission PE and the unoccupied levels by inverse photoemission IPE as mentioned in Section 1.

Law of equilibrium: Figure 3. Equilibrium problems do not really require the use of a master equation like Eq. It is then justifiable to single out one level and treat it independently, ignoring the occupation of the other levels. The SCF method uses the Fermi function assuming that the energy of each level depends on the occupation of the other levels.

But this is only approximate. The exact method is to abandon the Fermi function altogether and use Eq. This result, known to every device engineer, could thus be viewed as a special case of the general result in Eq. Equation 3. Our primary interest is in calculating the current under non-equilibrium conditions and that is one reason we have emphasized the master equation approach based on Eq.

For equilibrium problems, it gives the same answer. However, it also helps to bring out an important conceptual point. One often hears concerns that the law of equilibrium is a statistical one that can only be applied to large systems. But it is apparent from the master equation approach that the law of equilibrium Eq. The result is compared with a calculation based on the restricted SCF method described in Section 1. The SCF current—voltage characteristics look different from Fig. We cannot use this method more generally for two reasons. Secondly, it is not clear how to incorporate broadening into this picture and apply it to the transport regime where the broadening is comparable to the other energy scales.

A lot of work has gone into trying to discover a suitable SCF within the one-electron picture that will capture the essential physics of correlation. Use the SCF method only the Hartree term to calculate the energy of the 1s level in a helium atom. Use the SCF method only the Hartree term to calculate the energies of the 3s and 3p levels in a silicon atom. Plot the wavefunction for the 1s and 3p levels in silicon and compare with that for the 1s level in hydrogen cf. Plot the approximate binding energy for a hydrogen molecule as a function of the hydrogen—hydrogen bond length, making use of Eqs.

Show from Eqs. Molecules, on the other hand, do not have this spherical symmetry and a more efficient approach is needed to make the problem numerically tractable. At the same time it is also a very important conceptual tool that is fundamental to the quantum mechanical viewpoint. In this chapter we attempt to convey both these aspects. But the concept of basis functions is far more general. One can view them as the coordinate axes in an abstract Hilbert space as described in Section 4. A specific example: To understand the underlying physics and how this works in practice let us look at a specific example.

We will now use the concept of basis functions to show how this result is obtained from Eq. Since we have used only two functions u L and u R to express our wavefunction, the matrices [S] and [H] in Eqs. The integrals can be performed analytically to yield the results stated earlier in Eq. Figure 4. How can we get accurate results using just two basis functions? What do we lose by using only one basis function instead of ? The answer is that our results are accurate only over a limited range of energies. To see this, suppose we were to use not just the 1s orbital as we did previously, but also the 2s, 2px , 2py , 2pz , 3s, 3px , 3py and 3pz orbitals see Fig.

We argue that the lowest eigenstates will still be essentially made up of 1s wavefunctions and will involve negligible amounts of the other wavefunctions, so that fairly accurate results can be obtained with just one basis function per atom. The diagonal elements are roughly equal to the energy levels 87 4. All these states could be used as basis functions for a more accurate treatment of the hydrogen molecule. But a proper treatment of the higher energy levels would require more basis functions to be included. For large molecules or solids such calculations can be computationally quite intensive due to the large number of basis functions involved and the integrals that have to be evaluated to obtain the matrix elements.

The integrals arising from the self-consistent field are particularly time consuming. For this reason, semiempirical approaches are widely used where the matrix elements are adjusted through a combination of theory and experiment. For example, we could calculate suitable parameters to fit the known energy levels of an infinite solid and then use these parameters to calculate the energy levels in a finite nanostructure carved out of that solid. However, the concept of basis functions is more than a computational tool.

It represents an important conceptual tool for visualizing the physics and developing an intuition for what to expect. Indeed the concept of a wavefunction as a superposition of basis functions is central to the entire structure of quantum mechanics as we will try to explain next.

Vector space vs. Hilbert space: It is useful to compare Eq. In principle, N is infinite, but in practice we can often get accurate results with a manageably finite value of N. We have tried to depict this analogy in Fig. In this notation the expansion in terms of basis functions see Eq. This is a property of orthogonal basis functions which makes them conceptually easier to deal with. Even if we start with a non-orthogonal basis, it is often convenient to orthogonalize it. What we might lose in the process is the local nature of the original basis which makes it convenient to visualize the physics.

As a rule, it is difficult to find basis functions that are both local and orthogonal. From here on we will generally assume that the basis functions we use are orthogonal. How do we write down the matrix [R] corresponding to an operator Rop? In general, the answer is 91 4. Like the rotation operator in vector space, any differential operator in Hilbert space has a matrix representation. An easier approach is to use the discrete lattice representation that we discussed in Chapter 2. Also, it can be shown that a matrix that is Hermitian in one representation will remain Hermitian in any other representation.

An important requirement of quantum mechanics is that the eigenvalues corresponding to any operator Aop representing any observable must be real. This is ensured by requiring all such operators Aop to be Hermitian not just the Hamiltonian operator Hop which represents the energy since the eigenvalues of a Hermitian matrix are real. The reason I am bringing it up in this chapter is that it provides an instructive example of the concept of basis functions. Let me start by briefly explaining what it means. But this is not quite right.

If we wish to write the multi-electron wavefunction in the form shown in Eq. This is seen by noting that the transformation matrix [V] is obtained by writing each of the eigenstates the old basis as a 95 4. But the point is that the relation given in Eq. The density matrix is then evaluated as described above and its diagonal elements give us the electron density n x times the lattice constant, a.