Quantum Mechanics for Nanoscience andNanotechnology

Karl Hess

Beckman Institute, Department of Electrical Engineering and Departmentof Physics,University of Illinois, Urbana, Il 61801

1 Origins of the Quantum Theory

1.1 Black body radiation and Planck’s hypothesis

Black body radiation was the topic of interest in the late 1800’s and early 1900’s. It evencaptured the interest of the lay people and during tours in the physics institute of theUniversity of Vienna the question often was asked: “where is the famous absolutely blackbody?”. To their embarrassment the tour guides had to answer “it is this white box in thecorner”. The absolutely black body is actually a tiny hole in basically any box, e.g. a cubeof side length L. The box is heated to a temperature T and the radiation coming out ofthe hole is measured to give the energy content U(ν) at frequency ν in the frequency rangedν. It has been demonstrated experimentally that in equilibrium U(ν) depends only onthe temperature T of the walls and not on the material of which the walls are made. Theexperimental result for U(ν) follows approximately Wien’s law (later improved by Planck):

U(ν) ∼ ν3e−hν/kT (1)

where T is Boltzmann’s constant. From all the classical knowledge at that time, however, itwas found that it should be

U(ν) ∼ kTν2 (2)

The first solution of this puzzle was given by Planck and we will derive below a briefdescription how this solution was achieved.

We have to start by describing the electromagnetic waves in the black body.The wavesare characterized by a wave vector k with k = |k| = 2π/λ where λ is the wave length. Forall practical boundary conditions, the waves must be standing plane waves of all possiblewave lengths and amplitudes. Therefore we must have for the components of the wave vectork = (kx, ky, kz):

kx = 2πl/L , ky = 2πm/L , kz = 2πn/L (3)

with l, m, n being integers 0,±1,±2, ....

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Figure 1: Energy content U(ν) as a function of the frequency ν in arbitrary units for Planck’slaw (green solid line), Boltzman’s law (red dotted line) and Wien’s law (blue dashed line)

Figure 2: Standing waves in a rectangular box with λx = L/2,λy = L (top figure) andλx = 2L,λy = 1.5L (bottom figure)

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Figure 3: Distribution of k-states on a 2 dimensional lattice

Using the wave vector as given above one can describe for example the electrical fieldpattern in the box by a Fourier series

∑kak(t)e

ikr + cc (4)

where ak is a suitable complex amplitude, t is the time-coordinate, i is the imaginary unit,r = (x, y, z) is the space-coordinate and cc indicates the complex conjugate. Thus theelectrical field that is a function of the space-time continuum is fully characterized by the setof k vectors which is an infinite set but countable-infinite because of l, m, n being integers0,±1,±2, .... Because of this fact, one can also describe the electromagnetic field as acountable sum of oscillators, each with finite energy content and one for each separate valueof l, m, and n. The frequencies of these oscillators are between ν and ν+dν with ν = kc/(2π)where c is the velocity of light. To find the energy content of the electromagnetic field wemust therefore find the energy and the density of these oscillators.

The number of oscillators δN1 in the volume dkxdkydkz can be found by dividing k-spaceinto cubes with side-length 2π/L. Because L is very large, we have

δN1 = (L

2π)3

dkxdkydkz (5)

To express δN1 in terms of the frequencies ν it is convenient to use polar coordinates andintegrate over the solid angle which gives a factor of 4π. We therefore have

δN1 = 4π(L

2π)3

k2dk (6)

and from ν = kc/2π we obtain

δN1 = 4πL3ν2dν/(c3) (7)

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If we also permit two directions of polarization of the oscillators then we have to multiplyEq.(7) by a factor of two to obtain the total number of oscillators δN to be

δN = 8πL3ν2dν/(c3) (8)

It is now important to realize that classically the average energy of an oscillator is equalto kT . This is shown in detail in classical statistical mechanics. Therefore we have

U(ν)dν = kTδN = 8kTπL3ν2dν/(c3) (9)

which is in disagreement with experiment (approximately given by Eq.(1)).Planck’s resolution of the puzzle proceeds as follows. As in classical mechanics Planck

assumed that the probability W for an oscillator to have an energy E is proportional toe−E/kT . In addition, however, he assumed that the energy of the oscillators can only assumethe values Ej = jhν where j = 0, 1, 2, ...,∞ and h = is the famous constant of Planck. Wetherefore have for the correctly normalized probability W

W (j) =e−jhν/kT

∑∞j=0e

−jhν/kT(10)

The average energy E is then given by∑∞

j=0EjW (j) which can be found after a little algebra(HW1) to be:

E =hν

ehν/kT − 1(11)

Multiplying by δN we obtain Planck’s famous law

U(ν) = (8hπL3ν3/(c3))1

ehν/kT − 1(12)

which also is in full agreement with the experimental results. Planck presented in the year1900 the following explanation for his law: “We...consider-and this is the essential point-the energy E to be composed of a determinate number of equal finite parts, and employ intheir determination the natural constant h = 6.55x10−27 (erg sec). This constant multipliedby the frequency ν of the resonator yields the energy element hν in ergs, and dividing Eby hν we obtain the number of energy elements...”. Five years later, Einstein accepted thefull consequences of Planck’s work and stated that radiation consists of quanta of energyhν, which not only are important for emission and absorption but can have an independentexistence as particles in empty space. These particles are now called photons. Note alsothat the above treatment contained considerations of probability and statistics in particularstatistical mechanics. One can therefore say that quantum mechanics was brought forth bystatistical considerations.

The factor1

ehν/kT − 1(13)

that appears in Eq(12) is typical for photons and for all particles that are called bosons andis often called the occupation number that gives the probability to find a boson. Note thatwe will mostly use h = h/(2π) instead of Planck’s constant h.

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Figure 4: Bose Distribution for values around kTh∼ 1

1.2 Einstein’s 1917 work on stimulated emission

Einstein gave another derivation of Planck’s law and, in the course of it, derived and statedmany important facts about the interaction of matter and radiation, including the possibil-ity of stimulated emission. His work shows, how far one can get in the understanding ofmolecular problems by use of the “old” (Einstein’s, Planck’s) quantum mechanics. Einsteinalways refers to molecules but everything he says applies basically to any nano structure.We follow here Einstein’s work from 1917 closely.

According to quantum theory (that known to Einstein), a given molecule (or nano struc-ture) can exist only in a discrete series of states Z1, Z2, ..., Zn, .... The “inner” energy of thesestates is denoted by E1, E2, ..., En, .... The word state that

Einstein used is borrowed from thermodynamics where, for example, a gas of particleswas said to be in a certain state depending on temperature, pressure etc.. The word statewill be used throughout this text in various context. We will not give a precise definitionand only attempt to endow the word with more detailed properties as we go on. At theend we have to admit that we really do not exactly know what a quantum state is and canonly point to Goethe’s famous quote (in loose translation): “whenever concepts are uncleara sounding word will fast appear”.

If these molecules are part of a gas that is at temperature T , then the relative frequencyof occurrence Wn of the states Zn is given by the canonical distribution of states of statisticalmechanics:

Wn = pne−EnkT (14)

Here pn is a constant that is determined by the specific molecule (nano structure) and thestate Zn. Note that pn, En and details of the state Zn could not be determined by thequantum theory known in 1917. It takes the Schrodinger equation and its solutions to

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Figure 5: Discrete series of states in a box

Figure 6: Transition with light emission

determine all of these constants. However, if we just assume that these are somehow given,e.g. by some experiment, then we will see that great progress in the general understandingcan be made. We therefore proceed with Einstein to consider the energy exchange betweenmatter and radiation.

Consider two states Zn and Zm of a nano structure with energies

Em > En (15)

The nano structure can make a transition from Zn to Zm by absorbing electromagneticradiation (light) of energy Em − En.

The nano structure can also make a transition from Zm to Zn by emitting electromagnetic(em) radiation of the same energy Em − En. The radiation has then a frequency ν thatcorresponds to the index combination (m, n). We compare now the nano structure with a

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resonator for the em-field and consider the resonator and it’s energy changes when it interactswith with the electromagnetic field:

(i) Effects proportional to the radiation density.

Under the influence of an em-field of frequency ν and radiation density U(ν), thenano structure (resonator) can make a transition that, depending on the phase of theresonator, will either increase or decrease the energy of the resonator. (An electricmotor can be accelerated by the applied ac-voltage and gain rotational energy, orgenerate ac-voltage and current by loosing rotational energy). Correspondingly wemake the hypothesis that the nano structure can make a transition Zn → Zm byabsorbing the energy Em − En according to the probability

dW = Bmn U(ν)dt (16)

Analogously the nano structure can make a transition Zm → Zn by loosing energy tothe radiation field according to the probability

dW = BnmU(ν)dt (17)

a process that is now called stimulated emission. Bmn and Bn

m are constants now calledEinstein coefficients.

(ii) Effects independent of the radiation density.

The nano structure can radiate independent of the fact whether a radiation field ispresent or not. Einstein deduced this fact in analogy to the radioactive decay. Thiseffect is now called spontaneous emission and Einstein assigned it the probability

dW = Anmdt (18)

Anm is again a constant also called an Einstein coefficient.

We now consider thermal equilibrium and ask what the radiation density U(ν) has tobe in order not to change or disturb the probability distribution of states given by Eq.(14).For this it is necessary and sufficient that on average per unit time we have an equality ofall energy loss and gain as given by Eqs.(16) - (18). Therefore we must have for the indexcombination (m,n):

pne−EnkT Bm

n U(ν) = pme−Em

kT (BnmU(ν) + An

m) (19)

Einstein postulates now that the radiation density U(ν) grows to infinity as T approachesinfinity which gives:

pnBmn = pmBn

m (20)

and thus we obtain for U(ν):

U(ν) =An

m

Bnm

1

ehν/kT − 1(21)

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This, of course, can be considered to be a derivation of Planck’s law. Inversely, if we takePlanck’s law of Eq.(12)as given, we can deduce the ratio An

m

Bnm

of spontaneous to stimulatedemission coefficients. As we now know, this gives us the basis for the theory of lasers. Quitean achievement of Einstein considering the simplicity of his derivation. This also teaches usa lesson: one can deduce results of quantum mechanics quite simply just by counting thestates and using the probability of occurrence of a given state from statistical mechanics onecan go a long way in the derivation. This is illustrated again in the section where we derivethe quantum conductance of nano structures in the same elementary fashion.

Finally, we would like to repeat that the factor 1/(ehν/kT − 1) is often called the occu-pation factor, meaning that it relates to the probability to encounter an oscillator with theenergy hν. The −1 in the denominator arises from the fact that a difference is taken be-tween absorption and stimulated emission. We can guess that any “field” that is similar tothe em-field and interacts with matter by absorption, stimulated emission and spontaneousemission will lead to a similar occupation factor. Bose and Einstein have derived this factorlater in a different fashion and it therefore has also the name Bose-Einstein distribution.The particles that follow such a distribution are called bosons.

Fermions are particles with half-integer spin such as electrons. As we will discuss later,they follow a different statistics. The probability f(E) to find an electron with energy E isgiven by the Fermi distribution:

f(E) =1

exp[(E − EF )/kT ] + 1(22)

Here EF is the Fermi energy. As can easily be seen, for infinitely low absolute temperatureT, EF is the energy above which the probability to find an electron is zero. In spite of thealgebraic similarity of Fermi distribution and Bose occupation factor (or distribution) thetwo functions behave very differently as can easily be found by test.

1.3 The Waves of Louis de Broglie

Louis de Broglie asked the question whether one could assign a wave vector not only tophotons but also to electrons. Such assignment would not immediately mean that the elec-tron behaves like a wave. It could follow from a formal mathematical Fourier analysis ofthe physical functions that describe the electron behavior. One could just use Eq.(4) andassume that the limitations of k to a countable number corresponding to the integers l,m, nwould follow from some periodicity or boundary conditions that the functions describing theelectrons must obey for purely mathematical reasons. However, the work of Louis de Broglieand Schrodinger and much subsequent work indicates that there must be a deeper physicalsignificance in relating electron behavior to a wave vector. The electron does in many re-spects indeed behave like a wave and a particle at the same time. This wave particle dualityis, after all these years of research, still not entirely understood. The main purpose of thissection is to explain some of the major ideas that suggest wave like behavior of electronsand other entities that were previously considered as typical particles. This is, of course, theopposite of what we discussed in the first sections where we derived particle like propertiesof entities that were considered in the past to be wave like.

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Figure 7: De-Broglie frequencies for a clock (c), particle (p) and relativistic mass (m) as afunction of β

De Broglie found the roots for his derivation of wave like electron properties in the theoryof relativity. We know from Einstein’s famous work a number of important equations andfacts. For example, it is well known and experimentally established that a moving clock thatrotates with a frequency νc is slowed down relative to the clock of a stationary observer thatrotates with a frequency ν ′c. Einstein showed that

νc = ν ′c(1− β2)1/2

(23)

where β = v/c and v is the velocity of the moving clock relative to the stationary one.De Broglie also considered Einstein’s following other famous equations, first

E = m′c2 (24)

with m′ being the mass of the particle, and second Planck’s equation now also applied to aparticle to which he now assigned a frequency ν ′p

E = hν ′p (25)

De Broglie realized now that the mass m of a moving system increases with velocity as

m = m′/(1− β2)1/2

(26)

and therefore deduced that from Eq.(25) it followed that the frequency νp assigned to themoving particle must also increase in the moving system so that

νp = ν ′p/(1− β2)1/2

(27)

However, this is now opposite to the frequency of the clocks that slows down in a movingsystem. De Broglie therefore concluded the particle behavior can not be analogous to a

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clock. The only way out was then that it behaves analogous to a wave. If we introduce awave, then we have another parameter to work with, the wavelength λ or the wave vectork and by adjusting these one might get the correct results. Indeed this works out and deBroglie added the following derivations.

Consider a one dimensional model with movements in the x-direction. A wave can thenbe represented by a function sin(φ) with φ = ωt−k ·x where ω = 2πν. Using then mc2 = hνwe have

dφ = ωdt− k · dx =2πm′c2

h(1− β2)1/2dt− k · dx (28)

It is typical for the wave particle duality or dilemma that no derivation can do with wave orparticle picture only. Therefore de Broglie needed now to relate the wave also to the particleby saying in essence the following: If a clock would move at the place of the particle thenthe clock also would show a phase φ1 and the phase of the clock must be equal to the phaseof the wave i.e. φ = φ1. This is one of the famous pidooma theorems (what pidooma meanscan only be explained orally in class), and is usually called the “theorem of the harmony of

phases”. From Eq.(23) we have dφ1 = 2πν ′(1− β2)1/2

dt and from φ = φ1 we obtain

ωdt− k · dx =2πm′c2

h(1− β2)1/2dt− k · dx = 2πν ′(1− β2)

1/2dt (29)

which gives after a little algebra

hk/(2π) = mv or λ = h/(mv) (30)

This is the famous de Broglie equation which relates the momentum of a particle to awavelength, the de Broglie wavelength. For any problem in nano-science and -technologyone should always calculate the appropriate de Broglie wavelength. When the feature sizes ofa nano structure are of the order of the de Broglie wavelength, then quantum considerationsare bound to be important. Note already at this point that in solid state type of problemsthe free electron mass must often be replaced by an effective mass that is typically but notalways smaller than the free electron mass.

The work of de Broglie and the importance of his wavelength for any considerations onthe atomic scale show that quantum mechanics was not only brought forth by statisticalconsiderations as mentioned above but also by considerations involving Einstein’s relativity.I believe that the relation of statistics and relativity to quantum mechanics can not be justaccidental and is indeed very important if an understanding of quantum mechanics and it’sapplications, however incomplete, is to be achieved. There are, of course, many scientistswho believe that scientific theory should follow Ernst Mach’s suggestions who taught thefollowing: The world consists of colors, tones, warmth, pressures, spaces, times, etc., whichwe do not want to call “sensations” or “phenomena”, because in both names there liesa one-sided arbitrary theory (influence of our own eyes, ears etc.). We call them simply“elements”. The comprehension of the flux of these elements, whether directly or indirectly,is the actual aim of science. Thus Mach believed that the aim of science was the orderingof the elements in the most economical way. Modern quantum mechanics is often taught in

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Figure 8: De-Broglie wavelength

this spirit. An axiomatic system and operator algebra are presented as the basic frameworkand the claim is made that this abstract system orders the elements most economicallyand beyond this avoids contradictions. Boltzmann, on the other hand, pursued a differentapproach to physical theory. He developed minute models that underlie a physical theory.For example, he explained thermodynamics by considering very detailed properties of atomsin gases and deriving from these properties macroscopic equations that are averages of themicroscopic properties. The derivations of Planck, Einstein and Schrodinger are based onsuch considerations. Schrodinger himself considered both the approach of Mach and thatof Boltzmann viable and valid provided that neither the principles of Mach’s nor those ofBoltzmann’s approach are violated i.e. Schrodinger maintained a pragmatic balance betweenMach’s “positivism” and Boltzmann’s “realism”. We adopt this pragmatic approach herebecause it is, in our opinion, the most effective way to transmit an understanding to theapplications oriented nano structure area. We will give a more thoughtful discussion of thefoundations of quantum mechanics at the end of these lectures when talking about quantuminformation and quantum computing.

1.4 Discretization of wave vectors and conductance quantization

In this section we apply the Einstein-Planck approach to the conductance of a very thin,essentially one dimensional, wire. We start again from what is classically known. We assumethat the wire is very short and the electrons are not scattered very much by any imperfectionsof the wire material and the effect of charge accumulations (i.e. the electrostatics of the wires)can be neglected. We also assume that the wire is connected to two huge reservoirs thatare three dimensional and form ideal contacts. Then we know that the current through thewire is the thermionic emission current, well known from vacuum tubes, from both contacts.Lets assume for simplicity that there is a bias voltage Vbi that is large enough and directedsuch that current flows only from the left to the right and in z−direction corresponding to a

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Figure 9: Hess and Datta

current jLR. Assume now also that the length L of the wire is still long enough so that ourtreatment of counting wave vectors that is shown in Eq.(5) and (6) is valid. Then for thecase of one dimension we obtain:

δN1 = (L

2π)dkz (31)

Of course, we are not dealing with the oscillators of the electromagnetic field but withelectrons which we also have characterized by a wave vector and a corresponding state suchas Z1, Z2, ..., Zn, ... from above. As we know, the electron can have two spin values thatcorrespond to the two polarizations we discussed above and therefore we obtain

δN = (L

π)dkz (32)

To calculate the current we must multiply Eq.(32) by the elementary charge e and bythe velocity vz that the electron has in z−direction. There is one more problem to be takencare of: we must know that there actually is an electron in the given state with given kz

and then we must sum over all the full states kz. Because L is very large we will replacethat sum by an integral. We denote the probability that the state with wave vector kz isactually occupied by the distribution function f . f is derived in all elementary quantumtexts and we assume it here as given. The form of f is similar to that shown in Eq.(13)with an important difference in sign (the minus one in the denominator becomes a plus one).This form is typical for particles like electrons that are called Fermions. f contains also thequantity EF which is called the Fermi energy. Each state can only hold two electrons withopposite spin and this gives:

f =1

e(E−EF )/kT + 1(33)

E is the energy of an electron with wave vector kz. Using the above, the absolute value ofthe current becomes:

|jLR| = e

π

∫ ∞

0

vzfdkz (34)

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Figure 10: Quantum conductance

We now use the relation of velocity and energy to the wave vector as given by de Broglie i.e.Eq.(30) to obtain:

|jLR| = eh

mπ

∫ ∞

0

kz1

e( h2kz2

2m−EF )/kT + 1

dkz (35)

Here h = h/(2π). The integral is easily performed by introducing a new integration variableequal to k2

z and one obtains:

|jLR| = 2e

hkT ln(1 + eEF /kT ) (36)

Because the Fermi energy is raised by the applied bias voltage we use a suitable energy scaleand put EF ≈ eVbi >> kT to obtain:

|jLR| = 2e2

hVbi (37)

We have up to now only considered the counting of states in z−direction. Of course we havealso standing waves perpendicular to the z−direction in the wire. For each such additionalstanding wave or “mode” we obtain a current equal to that in Eq.(38) so that we have:

|jLR| = n2e2

hVbi (38)

where n is an integer.The quantity 2e2

his called the quantum conductance. The corresponding resistance would

be about 13kΩ. The existence of a quantum conductance was experimentally proven after1980, a very long time after the work of Einstein and de Broglie. Nevertheless, an under-standing of the quantum conductance could have been achieved by them, using their methodsof the old quantum mechanics. What they could not achieve was a calculation of the energiesof the electrons in the various quantized modes or states. This was only possible after thework of Schrodinger which is discussed next.

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2 Schrodinger’s Quantum Mechanics

2.1 Hamilton’s Classical Mechanics and Action Function

Hamilton formulated classical mechanics in terms of his function H that represents the totalenergy as function of spatial and momentum coordinates of a particle. For example, if wehave a particle with number i = 1, ..., N and it’s x-coordinate is xi while its momentumcoordinate in the x-direction is pxi. Then we have H = H(xi, pxi) and first

dpxi

dt= −∂H

∂xi(39)

and seconddxi

dt= +

∂H

∂pxi

(40)

and similar equations for the y, z space components. A trivial example would be a would beone free particle (we give it no number) moving in x-direction. Then H = p2

x/(2m) and thefirst equation gives px = constant and the second gives v = px/m. Other examples are givenas HW. We can see that in this simple case the function H does not depend on the spacecoordinate x and therefore the corresponding momentum is constant. For general problems,however, H does depend on the space coordinate which complicates things.

Hamilton-Jacobi had now an interesting idea. They suggested that one should transformthe coordinate system by a function S in such a way that the function H of the new systemthat they denoted by H would not depend on the space coordinates but only on new mo-mentum coordinates that we denote by pxi. As in the trivial example above, these pxi arethen constants and the two Hamilton equations are immediately solved. Naturally it is notalways easy to find such a function S. If it exists, then S is a function of the old spatialcoordinates and the new momentum coordinates i.e. in one dimension S = S(xi, pxi). Alittle algebra shows that we have

pxi =∂S(xi, pxi)

∂xiand xi =

∂S(xi, pxi)

∂pxi

(41)

We consider now a system with time independent constant total energy E. Then we haveH = H(xi, pxi) = E and using Eq.(41 obtain the following differential equation for S interms of the original space coordinates, called the Hamilton-Jacobi equation:

H(xi,∂S(xi, pxi)

∂xi) = E (42)

The function S is now a function of the space coordinates and of the constant momentumcoordinates of the transformed system. I am adding now a few speculations in order toexplain Schrodinger’s procedure to derive his famous equation that is presented below. Letsassume that we have a system of particles that we want to describe. Lets also assume that weknow from the origins of quantum mechanics that there are constants of the particle systemthat are essentially described by the quantum numbers l, m, n or related quantities. Lets

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speculate then that these constants are also related in a simple way (e.g. are proportional)to the constant momenta that appear in the function S. This gives us already a lot ofinformation about S and how it must look like for a particle system that we want to treatquantum mechanically. Before we discuss this further, though, lets back up and discuss thesituation with which Schrodinger was dealing before he derived his equation.

2.2 Particles, waves and Hamilton’s equations, and what Schrodingermade of it

Schrodinger had carefully studied relativity and not only immediately understood the sig-nificance of de Broglie’s work but had already entertained considerations that were basedon writings of Hermann Weyl about relativity and came to similar conclusions as de Brogliedid. From that time on, Schrodinger was trying to explain electronic properties by wave-typethought experiments and gave seminars on the topic. During one of these seminars PeterDebye made the remark that in order to deal properly with waves one needs a wave equation.And, of course as we know now, Schrodinger came up with one.

It is well known from electro-magnetics that the space dependence in three dimensionsof an amplitude ψ of a wave motion, independent of time, is given by the following partialsecond order differential equation (often called Helmholtz equation):

(∂2

∂x2+

∂2

∂y2+

∂2

∂z2)ψ + k2ψ = 0 (43)

were k is the absolute value of the wave vector. Using de Broglie’s relation from Eq.(30) andthe fact that total energy E minus potential energy V equal E − V = mv2/2 one obtainsfrom Eq.(43:

(∂2

∂x2+

∂2

∂y2+

∂2

∂z2)ψ +

2m

K2(E − V )ψ = 0 (44)

where K = h/(2π) ≡ h.Schrodinger noticed now a great similarity of this wave equation with the Hamilton-

Jacoby equation. He renamed Hamiltons function S, the action function, in terms of ψ

S = Klnψ (45)

Now, everyone who has been at Ludwig Boltzmann’s grave knows that Eq.(45) is one ofBoltzmann’s greatest achievements only Boltzmann wrote S = klnW which is the famouslaw connecting entropy S with thermodynamic probability W . Schrodinger did not use thisform just accidentally. He is telling us something and must certainly have had some thoughtsconnecting ψ with probability. Using Eq.(45) together with the Hamilton-Jacobi equationEq.(45) and replacing we get:

K2

2mi

[(∂ψ

∂xi)2

+ (∂ψ

∂yi)2

+ (∂ψ

∂zi)2

]− (E − V (xi, yi, zi))ψ2 = 0 (46)

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Here we have considered only one particle that numbered by i and have assumed that it hasthe mass mi.

Anyone who is good in the calculus of variations (and Schrodinger certainly was) knowsnow that Eq.(44) is the Euler-Lagrange equation for Eq.(46) if the following variationalproblem is considered: “We seek those functions ψ that fulfill Eq.(46) which are finite andcontinuously twice differentiable and make the integral of Eq.(46) over all space an ex-tremum”. The last sentence in quotation marks is called the Schrodinger condition. Thus,applying the Schrodinger condition to Eq.(46) we obtain:

K2

2mi

(∂2

∂(xi)2 +∂2

∂(yi)2 +∂2

∂(zi)2 )ψ + (E − V (xi, yi, zi))ψ = 0 (47)

which is identical to the wave equation derived from the Helmholtz equation except thatnow x, y, z are replaced by the particle coordinates xi, yi, zi. The remarkable point of thislatter derivation is that we have used only the Hamilton-Jacobi equation and the Schrodingercondition. We have made no reference to waves or to the de Broglie wavelength except thatwe later will use K = h.

Eq.(47) is for one particle (the i’s). Had we derived it for two particles, we would haveobtained:

K2

2m1

(∂2

∂(x1)2 +∂2

∂(y1)2 +∂2

∂(z1)2 )ψ+K2

2m2

(∂2

∂(x2)2 +∂2

∂(y2)2 +∂2

∂(z2)2 )ψ+(E−V (r1, r2))ψ = 0

(48)where r1 = (x1, y1, z1) and r2 = (x2, y2, z2). Of course, a generalization for a large numberof particles is easy only the solution becomes more complicated. With each particle oneadds three more variables that enter the differential equation. Even the most powerfulcomputers of the day take a month of cpu time to solve the Schrodinger equation exactly for10-20 particles. A small nano structure that contains about 50 atoms still typically containshundreds of electrons in addition to the 50 nuclei. It is therefore clear that approximatemethods are needed to understand nano structures.

Eq.(48) shows the basic problem of Schrodinger’s quantum mechanics. At the one side wewish to describe particles such as electrons as waves, on the other side we can not abandonthe individualization of particles and the distinction between their coordinates. In orderto proceed we need to just say that the electron is particle and wave at the same time or,perhaps, that some properties of an electron can be described by assigning a wave to theelectron while other properties, for example the mass, are best described by imagining theelectron as a point-like particle. Of course, even such a picture has it’s limitations. A crystallattice, for example, has the consequence that we need to assign an effective mass to theelectron. This point-like effective mass is, however, a consequence of the spatially extendedcrystal. Such logical difficulties may persuade the purist to not consider any pictures andjust work with operators in an abstract Hilbert space. Quantum mechanics can be donethat way. However, I believe when it comes to nano- science and technology pictures areextremely helpful and fun and no real mastership can be achieved without them. Thereforewe just accept the particle-wave schizophrenia (or nicer said “duality”). Again, a deeper

16

Figure 11: One dimensional potential well

discussions of the foundations of quantum mechanics is given at the end of the course inconnection with quantum information.

3 Important special solutions of the one particle Schrodinger

equation

In this section we do some practising by solving problems that are important for nanotechnology. What we want to obtain from the Schrodinger equation are results for thequantities that the old quantum mechanics had to assume as given. Most important in theold quantum mechanics are the energy levels E1, E2, ..., En, ... of a nano structure. We haveencountered these when we discussed Einstein’s approach of interaction of radiation withnano structures. We start with the calculation of the energy levels of a quantum well andthen proceed to a crystal lattice and finally the two-dimensional exciton which is equivalentto the exciton in a narrow quantum well. All of these cases we will first treat for one particleonly and therefore can drop the indices that number the particles, i.e. we will just use xinstead of xi etc..

3.1 The infinite potential well

Consider a one dimensional (x−direction) potential energy profile that is infinite everywhereexcept for −d ≤ x ≤ d.

The Schrodinger equation for an electron in the well then becomes:

− h2

2m

∂2

∂x2ψ = Eψ (49)

Solving this equation, it is important to note that both the energy E of the particleand its wave function ψ are unknown. This is typical for equations of the type of Eq.(49).Such equations are called eigenvalue equations and are characterized by the fact that theresult of a (differential) operator acting on the function (ψ) is proportional to the function

17

(equals Eψ). As a consequence of the structure of this type of equation, the unknown E andthe unknown ψ are determined by completely different methods or requirements as we willillustrate below. Usually it is much simpler to determine E and often that is all one needsanyway.

If we had no potential well, the electron would be free and as can be seen by inspectionthe following equation for ψ solves Eq.(49):

ψ = A+eikx (50)

Inserting Eq.(50) into Eq.(49) we obtain E = h2k2

2mwhich is the kinetic energy of an electron

with de Broglie wavelength λ and corresponding wave vector k (k = |k|). We see here alreadythat we have obtained the energy E without the necessity to know A+ and the completewave function ψ. The full ψ is determined by normalization conditions that we will discusslater.

We now return to the quantum well. Because of the boundaries the solutions that wenow can guess will be standing waves. Therefore in addition to the wave from Eq.(49) whichpropagates in one direction we will need at least a second reflected wave that propagates inthe opposite direction and therefore:

ψ = A+eikx + A−e−ikx (51)

Because the potential boundaries are infinite, ψ must be zero at the points d,−d which givesfor d:

A+eikd + A−e−ikd = 0 (52)

and for −d:A+e−ikd + A−eikd = 0 (53)

The last two equations contain the unknowns A+, A− and are homogeneous. This meansthat they have only then a solution if the determinant of the coefficients vanishes, that is

∣∣∣∣∣∣

eikd e−ikd

e−ikd eikd

∣∣∣∣∣∣= 0. (54)

This leads to the requirement that:

sin(2kd) = 0 (55)

which is equivalent to:

k =nπ

2dfor n = 1, 2, ... (56)

Using Eq.(49) one obtains

En =(nπh)2

8md2for n = 1, 2, ... (57)

As one can see, we did not need to know the wave function and we obtained the resultof discrete energies En that the electron can have. These discrete energies correspond to the

18

discrete photon energies and to the standing waves of the photons in the box that representedthe source of the black body radiation. Only now we defined standing waves for the electronby using the Schrodinger equation. The real miracle is, of course, that the discrete energies,the energy levels of the well, are in complete agreement with experiments. We note at thispoint that this astonishing agreement with experiments is limited to the discrete energies.Other results of Schrodinger’s quantum theory are more difficult to compare with appropriateexperiments. As we will see, the wave function does not correspond to a measurable quantityand the square of the wave function that will be interpreted as the probability to find theelectron is, naturally, much more difficult to measure than En.

The wave functions for the quantum well can be determined in the following way. UsingEq.(52) together with Eq.(56) we have:

A+

A−= −einπ = (−1)n+1 (58)

which givesψn(x) = A+cos(nπx/(2d)) for n = 2, 4, 6... (59)

andψn(x) = A+sin(nπx/(2d)) for n = 1, 3, 5... (60)

The coefficient A+ is determined through the interpretation that the square of the wavefunction is the probability to find the electron. Therefore, the integral of the square of thewave function from −d to +d must be 1. One can easily see that therefore A+ = 1/

√d. We

will discuss the probability interpretation later in some detail.The solutions of Eqs.(59) and (60) can be combined into one by offsetting the position

to obtain:ψn(x) = A+sin[nπ(x + d)/(2d)] for n = 1, 2, 3... (61)

Any wave function can be written as a linear combination of the functions given in Eq.(61)i.e.

ψ(x) =∑

n

cn√dsin[nπ(x + d)/(2d)] (62)

with ∑n

|cn|2 = 1 (63)

There is a general principle behind it. The solutions of the Schrodinger equation form acomplete orthogonal system (orthonormal if normalized). The system is called completebecause any ψ can be written as a linear combination of the elements of the system. Itis called orthonormal because

∫ψ∗j · ψldr = δj,l. Here ∗ denotes the complex conjugate,

r = (x, y, z) and δj,l is the Kronecker symbol.We finish this example by the following remarks. Actual quantum wells are not sand-

wiched between infinite but only between finite potentials. Furthermore the sandwiching isdone usually between two different materials. This means that the well and the boundariesare having different atomic structure. It is then actually surprising that the solutions still

19

Figure 12: Two dimensional square well

have anything to do with the idealized case. However, we will see that different materialsand different atomic structure often necessitate only the use of an effective mass for theelectron instead of its “free” electron mass and, whenever appropriate, of the dielectric con-stant of the material instead of that of the free space. The finite height of the potential ormore complicated forms of the potential necessitate either the use of perturbation theory ornumerical methods to arrive a result. If, in addition, the atomic structure plays a role, thenwe need to use more basic methods (ab initio methods) to arrive at valid solutions of theSchrodinger equation. We will discuss all of these methods.

3.2 Quantum wires and dots

Rather than considering a well, we might construct a system with vanishing boundary condi-tions for ψ in two dimensions say x, y so that the electrostatic potential is zero for 0 ≤ x ≤ Lx,0 ≤ y ≤ Ly and −∞ ≤ z ≤ +∞ which defines a slab of infinite extension. Outside the slabthe potential is infinite. For small Lx, Ly one calls such a slab a quantum wire. The energylevels of such a wire can be derived similarly to those of a well and are:

Ekz =(n1πh)2

2mL2x

+(n2πh)2

2mL2y

+(hkz)

2

2mfor n1, n2 = 1, 2, ... (64)

Such a quantum wire, connected to two three dimensional contacts, would show thequantum conductance that we had discussed before. The integers n1, n2 describe the modesperpendicular to the direction of conduction and must be inserted into Eq.(38) instead ofthe generic integer n that we have used in this equation.

If we confine the vanishing electrostatic potential also to a small distance in the z−directioni.e. we have zero potential for 0 ≤ x ≤ Lx, 0 ≤ y ≤ Ly and 0 ≤ z ≤ Lz and infinite potentialotherwise, then we have a quantum dot. The name derives, of course, from the fact that wenow have quantization, called in these cases size quantization, in all direction. Free motionof the electron is therefore not possible in any direction and we have, in this sense, a zero-dimensional structure, a dot. Of course, if the de Broglie wavelength is very large as couldbe the case in a crystal where the electron has a very small effective mass, then the quantum

20

Figure 13: A quantum wire

Figure 14: A quantum dot

dot can also be rather large. In the semiconductor GaAs, for example, we have an effectivemass that is 0.067 of the free electron mass and a quantum dot with Lx = Ly = Lz = 10nmshows still significant size quantum effects i.e. even the lowest energy level is still of theorder of 150 meV. The equation for the energy levels En of the quantum dot described aboveis easy to derive and one obtains:

En =(n1πh)2

2mL2x

+(n2πh)2

2mL2y

+(n3πh)2

2mL2z

for n1, n2, n3 = 1, 2, ... (65)

As in the case of quantum wells, wires and dots are made in nano technology by usingdifferent materials that lead to the confinement. Again, the different materials can often becharacterized just by use of an effective mass. When the structures become extremely small,however, then the nature of the atoms that constitute the material may become importantand also the approximations that lead to the possibility to use an effective mass becomeinvalid. Then one needs to use more advanced methods, e.g. ab initio methods to calculatethe energy levels.

21

Figure 15: One dimensional quantum well with different materials: AlAs-GaAs-AlAs

Figure 16: Two dimensional quantum wells and wires

22

-10 -5 0 5 10-10

-5

0

5

10

Figure 17: An exciton in two dimensions

3.3 Energy levels of an exciton in a quantum well

In the following and approximate the quantum well simply by taking the two dimensionallimit, i.e. we calculate everything in a plane instead of in three dimensions. Therefore thespace coordinate are r1 = (x1, y1) and r2 = (x2, y2).

An exciton is a combination of an electron and a hole in a solid.We will discuss electrons and holes in connection with crystal lattices. For now we just

can assume that an electron is a negatively charged particle with mass m1 while a hole ispositively charged with mass m2. The electron and hole attract each other and form anentity that is similar to an atom. As one does for atoms we assume for the moment that thehole mass is very large, basically infinite. We then need only consider one moving particleand therefore can again drop for simplicity the indices (1, 2) and only consider the electronwho’s mass we denote by m∗ where the asterisk reminds us of the fact that we are in a crystaland the mass is an effective mass not the free electron mass. The hole just forms the zeroof the coordinate system. If it moves i.e. performs a translation in space, the electron justtranslates with it. Therefore the Schrodinger equation is given by:

h2

2m∗ (∂2

∂(x)2 +∂2

∂(y)2 +)ψ + (E +e2

4πεε0r)ψ = 0 (66)

where e is the absolute value of the elementary charge, r = |r|, εε0 is the dielectric constantof the crystal in question and we have used Coulomb’s law to express the potential energy.Now we introduce cylinder coordinates r = (r, φ) and write for the wave function:

ψ(r) = R(r)eilφ (67)

where i is the imaginary unit and l is an integer. We therefore have in cylinder coordinates:

[− h2

2m∗1

r

d

dr(r

d

dr)− l2

r2 − e2

4πεε0r]R(r) = ER(r) (68)

A differential equation, a so called eigen-value equation, such as Eq.(68) is, of course difficultto solve. One therefore has only two possibilities: either one finds a differential equation

23

2 4 6 8 10x

-1

1

2

3

4

5

ΨHxL State Z3, with n=2, l=1

2 4 6 8 10x

0.5

1

1.5

2

2.5

3

ΨHxL State Z2, with n=2, l=0

2 4 6 8 10x

0.5

1

1.5

2

ΨHxL State Z1, with n=1, l=0

Figure 18: The radial wavefunction Rn,l for (n,l) = (1,0),(2,0) and (2,1)

in the mathematical literature that is equivalent after some substitution or one solves theequation numerically. We will discuss numerical solutions as special projects. Fortunately,however, Eq.(68) is equivalent to Laguerre’s differential equation after substituting R =be−arr|l|L. One can see then that L represents a Laguerre polynomial (HW problem).

It is now important to note that we are often not interested in the functions L. These canbe identified as describing the states that were denoted by Einstein as Z1, Z2, ..., Zn, .... How-ever, as stated at the beginning what we want is mostly the discrete energies E1, E2, ..., En, ....Characteristically for all these type of problems a solution of the differential equation existsonly for certain discrete energies when En < 0 which means that the particle is confined orbound by the form of the electrostatic potential of the other particle or the quantum welletc. These discrete energies for which solutions exist are usually given by simple expressions.In our case of Laguerre’s differential equation we have:

En = − m∗e4

8h2π2ε2ε20(n + 1/2)2

(69)

It is interesting to note that this solution is distinctly different to the solution for thehydrogen atom in three dimensions that can be found in all quantum mechanics texts. Thereis a 1/2-number in the denominator, and the En’s are almost a factor of two larger thanthose of the hydrogen atom in 3 dimensions which are given by:

En = − m∗e4

32h2π2ε2ε20n

2(70)

The formula for hydrogen atoms in free space does, of course, not contain the dielectricconstant ε. However, a donor atom in a semiconductor such as arsenic in silicon would bedescribed exactly by Eq.(70)

This larger value of En is indeed observed for excitons in quantum wells. Why is thisinteresting? Well, the classical problem of planets going around the sun has actually solutionsin a plane and one obtains the same solutions whether or not one starts the calculation in twoor three dimensions. Not so in quantum mechanics! The Schrodinger condition apparently

24

makes a difference for the cases when solutions are sought in spaces of different dimensionalityand experiments on nano structures, e.g. the exciton in a quantum well, have confirmed this!Here we have one important example why nano structure research is even important for veryfundamental questions in quantum mechanics. It gives us the unique possibility to investigatequantum solutions as a function of dimensionality. One can sandwich an atom in a quantumwell and make it almost two dimensional. One can go further and create quantum wiresthat are almost one dimensional and finally one can create entities that are very small in allthree dimensions, quantum dots.

3.4 Crystals, energy band structures and effective masses

We have repeatedly referred to the fact that in semiconductor crystals the mass of the electronmust be replaced by an effective mass. In this section, we present justification for this andbeyond that show how an electron propagates in a crystal. In general, this propagationcan not be calculated from a one electron treatment of the Schrodinger equation becausea crystal contains, of course, many electrons that interact with each other. However, thereexists a very powerful approximation that applies for most of the interesting problems ofnano science and technology. Within this approximation one can consider one electron, theelectron whose propagation we want to calculate, and describe all the other electrons and thenuclei of the crystal by a potential V (r) that is periodic with respect to the space coordinater. Of course, it is extremely complicated to actually compute this potential. To do this,one would have to solve the Schrodinger equation for the many electron system. It turnsout, however, that one can deduce the most important properties of the crystal electron inquestion, if one only knows a few Fourier-components of V (r). We will discuss below howone can obtain these Fourier components. For now we just assume that somehow we knowthem and we derive a solution of the Schrodinger equation from this knowledge.

3.4.1 Energy bands derived by Fourier analysis

The crystal is a set of atoms or molecules whose location is described by points Rl that aregenerated by three vectors a1, a2, a3 from the relation:

Rl = l1a1 + l2a2 + l3a3 (71)

In the simplest case, a1, a2, a3 are the orthogonal vectors of three dimensional Euclideanspace (a, 0, 0), (0, a, 0), (0, 0, a).

The corresponding crystal lattice is called the cubic simple lattice with distance a betweenthe lattice points. Because the crystal is periodic, the crystal potential is also periodic i.e.

V (r + Rl) = V (r) (72)

It is natural then to assume that the physical phenomena in the crystal are also of thesame periodicity as shown in Eq.(72) and that therefore Fourier analysis is the appropriate

25

Figure 19: Cubic simple crystal in 2 dimensions

mathematical tool to solve the Schrodinger equation for this case. Fourier analysis teachesus that if we have a function f that obeys:

f(r + Rl) = f(r) (73)

then we can expand f into the Fourier series

f(r) =∑

h

AheiKh·r (74)

with

Ah =1

Ω

∫

Ω

f(r)e−iKh·rdr (75)

Here h = 0,±1,±2, ... is an integer (not to be mixed up with Planck’s constant), Ω is thevolume of the cell that when repeated over and over fills the whole crystal which is for thecubic simple crystal Ω = a3. The vectors Kh are called reciprocal lattice vectors and, for thecubic simple crystal, are given by:

Kh = h1b1 + h2b2 + h3b3 (76)

where b1 = (2πa

, 0, 0), b2 = (0, 2πa

, 0) and b3 = (0, 0, 2πa

).We can now guess the wave function from the following considerations. For a free electron

ψ is proportional to eik·r. Because of the periodicity of the crystal the wave function willalso be modified to reflect the periodicity so we might have:

ψ(k, r) = eik·ruk(r) (77)

26

where uk has the periodicity of the crystal i.e. uk(r + Rl) = uk(r). This “guess” of thewave function can in fact be proven mathematically and Eq.(77) is called Bloch’s theorem.Because uk is periodic, we can write according to Eq.(74):

uk(r) =∑

h

AheiKh·r (78)

Inserting Eq.(78) into the Schrodinger equation we obtain:

h2

2m

∑

h

|k + Kh|2Bh − eV (r)∑

h

Bh = E∑

h

Bh (79)

where Bh = Ahe−i(k+Kh)·r.

The termsEh(k) = |k + Kh|2 (80)

appearing in Eq.(79)are called the empty lattice energy terms or empty lattice energy bandsbecause they form the solution for the energy when V (r) = 0. Although these terms arejust “stroboscopic” replicas of the free electron parabola energy E vs k relation, they areactually very useful for the following reasons.

Eq.(79) contains all the information on the crystal structure through the vectors Kh. Allthe rest of the information on the crystal material and atomic properties is buried in thecrystal potential V (r). Assume for a moment that the crystal potential is very smooth andpresents only a small perturbation. Then one can deduce from perturbation theory, thatwe will treat later, that significant changes of the solution due to the crystal potential areonly introduced around certain values of k which correspond to the so called Brillouin zoneboundary. The solution of the Schrodinger equation is therefore identical to the empty latticebands except around these boundary points. This means that one can get a pretty goodidea about the solution just by considering the empty lattice bands. To gain this insight isparticularly important when the crystal lattice is more complicated than just cubic simple.Then also the Kh are more complicated and it becomes very valuable to obtain first a plotof the empty lattice bands. Examples for simple one dimensional lattices are shown in class(Figures).

Of course, for any precise calculation, the crystal potential can not be neglected. In-cluding the potential, one can solve the Schrodinger equation in the well known way that isprescribed by Fourier analysis. We multiply Eq.(79) by the factor e−i(k+Kl)·r and integratethe resulting equation over all the crystal volume Vol and normalize (i.e. divide) by thevolume. Using this procedure, we encounter the following integrals:

1

Vol

∫ei(k+Kh)·re−i(k+Kl)·rdr = δh,l (81)

The simple equality to the Kronecker δ follows from the fact that the integrand equals 1if h = l. For h 6= l the integrand oscillates rapidly over the large volume and the integral

27

Figure 20: Empty lattice bands for one dimensional lattice

Figure 21: Empty lattice band for two dimensional cubic simple crystal

28

Figure 22: Empty lattice band for two dimensional cubic face centered crystal

therefore equals 0. For the term that contains the crystal potential, however, things becomemore complicated. Then we have integrals of the form

< Kl|V (r)|Kh >=1

Vol

∫e−i(k+Kl)·rV (r)ei(k+Kh)·rdr (82)

Here we have used the customary shorthand notation < Kl|V (r)|Kh > for the integral.There is a deeper mathematical sense in this shorthand and we refer the interested reader tomore detailed descriptions of the “Dirac notation” in standard texts on quantum mechanics.We also will give a brief outline of the general features of this notation below. For now weonly need to know that Eq.(82) represents nothing but the Fourier transform of the crystalpotential with respect to Kh −Kl i.e.

< Kl|V (r)|Kh >=1

Vol

∫ei(Kh−Kl)·rV (r)dr (83)

Using Eqs.(82) and (83) we obtain then from (79) for each given index −∞ < l < +∞the following algebraic equation:

[El(k)− E(k)]Al +∑

h

Ah < Kl|V (r)|Kh >= 0 (84)

If we still set aside the fact that we do not know the crystal potential, then Eq.(84) containsthe unknown E(k) and Al, Ah while El(k) is given by Eq.(80). Thus we have an infinitesystem of homogeneous linear algebraic equations. We can, of course, not solve such aninfinite system. One therefore usually attempts a numerical solution by limiting l, h and bythen obtaining a finite number of equations. A computer can easily deal with about one

29

Figure 23: Empty lattice bands with a ”gap”

hundred values of l and h. Note, however, that the situation is exactly the same as it wasin the case of the quantum well. We have a system of homogeneous equations that has onlythen a solution if it’s determinant vanishes. This condition gives us then a result for E(k)even if we do not know the coefficients Al, Ah and, in fact, E(k) is all we want to know. Toshow the procedure explicitly we seek now a solution for the values h, l = 0,−1. To makethings simpler we choose an energy scale so that < 0|V (r)|0 >=< −1|V (r)| − 1 >= 0 andthen obtain the following determinant.

∣∣∣∣∣∣

E0(k)− E(k) < 0|V (r)| − 1 >

< −1|V (r)|0 > E−1(k)− E(k)

∣∣∣∣∣∣= 0. (85)

With the notation S(k) = E0(k) + E−1(k) and using the fact that < −1|V (r)|0 > is thecomplex conjugate of < 0|V (r)| − 1 > one obtains from Eq.(85):

E(k) =1

2S(k)± 1

2

√S(k)2 + 4| < 0|V (r)| − 1 > |2 (86)

The curve corresponding to this result looks exactly like the corresponding empty latticecurves except that for k = ±π

a, which is the Brillouin zone boundary in one dimension, we

obtain a splitting of the curves with an energy gap EG = 2| < 0|V (r)| − 1 > |.This is the well known energy gap of band structure theory that separates, for exam-

ple, valence and conduction bands in semiconductors. Such a band gap can also easily bemeasured by investigating the optical properties of the semiconductor. Roughly speaking,the semiconductor is transparent for light energies hω that are below the gap energy i.e.hω ≤ EG. From the experimentally known energy gap we can therefore deduce the Fouriertransform | < 0|V (r)| − 1 > | of the unknown crystal potential. In practice it turns out that

30

Figure 24: Illustration of tight binding Hamiltonian

even if we use a much larger number of integers h, l, one needs only a relatively small numberof Fourier components of the crystal potential which can be determined from a number ofexperimental results. For various historical reasons, this method of calculation of E(k) iscalled the empirical pseudo potential method.

3.4.2 Energy bands by nearest neighbor interactions

Energy bands have been calculated by a variety of methods that, on the surface, look com-pletely different. Accordingly, these methods have also received very different names. Inprinciple, however, the methods are often mathematically very similar if not identical. Toillustrate this we describe here a so called “tight binding” nearest neighbor method and takethe opportunity to explain the Dirac notation a little better.

We can formally write the Schrodinger equation as

H0ψn = Enψn (87)

where H0 is called the Hamiltonian operator, which can be a differential operator as givenin the above examples for the Schrodinger equation. We know then that the solutions ψn

form a complete orthogonal system and if they are properly normalized we have:∫

ψ∗j · ψldr = δj,l (88)

It is known from mathematics that such a system of functions also forms a vector space. Wecan therefore denote the functions as vectors e.g.

ψl → |l > (89)

31

and can define a scalar product of vectors such that

< j|l >=

∫ψ∗j · ψldr = δj,l (90)

Because this basic system of vectors (functions) |n > is complete, any vector |λ > can bewritten as a sum of these vectors with coefficients that are the projections of |λ > on |n >i.e.

|λ >=∑

n|n >< n|λ > (91)

From this equation we can see that we can take the symbol |n >< n| as a projectionoperation (or projection operator) that projects any vector |λ > onto |n >. One can alsosee that

∑n|n >< n| is the identity operator because it turns any |λ > into itself. It is also

very useful and practical to represent other operators in terms of such projection operators.For example, the Hamiltonian H0 can be written as:

H0 =∑

n|n > En < n| (92)

because then we have:

H0|n >=∑

n|n > En < n|n >= En|n > (93)

We turn now to describe a simple way to write the Hamiltonian of a crystal. If oneconsiders a crystal and likes to solve band structure problems, one could think of H0 asbeing the Hamiltonian describing totality of all the single separated atoms. The atoms areat this point assumed to be independent. We can attach to each atom a wave function |i >.In contrast to the examples above, the index i labels now simply the different atoms andnot the different eigenvalues of the energy for a given atom. We assume, for simplicity, thateach atom has a single Eigenvalue E0 and a single corresponding atomic wave function |i >.Addition of more energy eigenvalues and wave functions for each single atom complicatesthe notation but needs to be done for realistic calculations. If we now think of any wavefunction |λ > defined by a certain amplitude ci of finding an electron at a given site, we canrepresent this wave function by:

|λ >=∑

ici|i > (94)

or|λ >=

∑i|i >< i|λ > (95)

where we have used ci =< i|λ > as above. Thus the localized atomic wave functions |i >form the complete ortho normal system generated by H0 exactly as we had it for the caseof e.g. one quantum dot and its energy eigenvalues as well as corresponding wave functions.This illustrates the great flexibility of the Dirac notation and illustrates the fact that thegeneral mathematical formalism of eigenvalue equations is very flexible and can be adjustedto and adopted for many different physical situations.

To make a crystal out of these unconnected atoms and to permit propagation of elec-trons between them i.e. to be “delocalized”, one needs to define the contributions to the

32

Figure 25: Nearest neighbour interaction

Hamiltonian due to the overlap of the atomic potentials of the different sites. In general,this overlap can be described by a term

∑lm

Vlm|l >< m| (96)

which then gives in total the so called tight binding Hamiltonian

H =∑

i|i > Ei < i|+

∑lm

Vlm|l >< m| (97)

Vlm represents, of course, a potential energy not a potential. We are sometimes using the sym-bol V for potential and at other instances for potential energy because it is often convenientnot to include the symbol e for the elementary charge as it can conflict with the exponent e.g.ex. The physical meaning of these two terms can be further explained as follows. The totalHamiltonian of an electron in a crystal is the sum of kinetic energy, T , and potential energyV (r). The potential energy is to first approximation given by the sum of the potential v(ri

of the single isolated atoms. Therefore we have H0 = T +∑

i v(ri =∑

i|i > Ei < i|. Thecontributions of all atoms to the potential energy at site l are δVl =

∑m6=l v(rm and these

give rise to the off diagonal terms in Eq.(97. The contributions of these non diagonal termslead to a connection of the originally separated states of H0 and to a nonzero probability foran electron to escape from one distinct state to another.

A good approximation to this Hamiltonian is often the nearest neighbor interaction thatis we have

Vlm = V if l, m are nearest neighbors (98)

andVlm = 0 if l,m are not nearest neighbors (99)

33

We assume now a one dimensional chain of N atoms with lattice spacing a and use the trialwave function:

|k >=1√N

∑j

eikaj|j > (100)

Inserting this |k > into the nearest neighbor tight binding Hamiltonian and using Ei = E0

for all i = 1, 2, ... we obtain for the energy Eigenvalues E(k):

E(k) = E0 + V (eika + e−ika) = E0 + 2V cos(ka) (101)

3.5 The effective mass

See reference [1].

3.6 Counting and filling states

See reference [2].

3.7 Tunnelling

We know that electromagnetic waves are not necessarily propagating anywhere but undercertain circumstance decay exponentially within fractions of a wavelength and “die out”. Onecalls this decaying portion of the wave an “evanescent” wave. A well known case is that ofan em-wave propagating in a medium of high dielectric constant with a neighboring mediumof lower dielectric constant. When a light ray falls at a shallow angle on the interface it isreflected. However, this does not mean that the electromagnetic field is zero in the mediumof smaller dielectric constant. There exists still the evanescent wave. One could check thisin the following way, If one sandwiches a medium of small dielectric constant between twolayers with large dielectric constant then there will still be propagation of the light beamdepending how thin the medium with the small dielectric constant is.

A very similar phenomenon is known for the Schrodinger waves. Consider instead of aquantum well a thin quantum barrier that is designed as follows.

The static electrical potential is zero everywhere except for a region −d ≤ x ≤ +dwhere it assumes a (negative) value V0 so that we have a positive potential energy eV0. ASchrodinger-De Broglie wave impinging upon this barrier will decay and become evanescent.However, if the barrier is thin enough, it will “leak” through and propagate on the otherside. With the probability interpretation for the wave function, this means that an electronwill have a certain non zero probability of going through the barrier. W can now representthe propagating free electron that has a kinetic energy E by a wave vector k whose absolutevalue is as usually given by k = (

√2mE)/h. The exponentially decaying wave in the barrier

can be characterized by a wave vector γ∗ = iγ that is imaginary (γ is real), just as we knowit from the Helmholtz equation and from electromagnetic waves. We have

γ = [√

2m(V0 − E)]/h (102)

34

Figure 26: Schematic picture of Tunneling through a rectangular barrier

The wave function has then the general form:

ψ(x) = Aeikx + Be−ikx for x < −d

ψ(x) = Ceγx + De−γx for |x| < +d

andψ(x) = Eeikx + Fe−ikx for x > +d (103)

We must now match the wave functions and their derivatives at the boundaries x = ±d.This is a laborious algebraic exercise and we perform the procedure therefore with an ap-proximation. We assume that left of the barrier we have an incoming and a reflected waveand therefore both A,B 6= 0. At the left barrier we assume we have only a wave decayingfrom left to right and therefore C = 0. At the right barrier we need to admit a reflection andtherefore both C, D 6= 0. Finally we have only one wave outgoing to the right and thereforeF = 0. This approximation is obviously good for thicker barriers. It certainly simplifies thealgebra. The condition of continuous wave function and derivative gives for x = −d:

Ae−ikd + Be+ikd = Deγd

andik[Ae−ikd −Be+ikd] = −γDeγd (104)

For the barrier at x = d we obtain:

Ceγd + De−γd = Eeikd

andγ[Ceγd −De−γd] = ikEeikd (105)

35

Figure 27: Exact transmission coefficient for tunneling

Eliminating C from the last two equations gives us D in terms of E and eliminating B fromEqs.(104) gives us therefore A in terms of E i.e. the transmitted amplitude in terms of theincoming amplitude. We know that the transmission probability T is equal to the square of

the amplitude ration T = (EA)2

which is:

T ≈ 4γ2

k2 + γ2e−4γd (106)

which reflects the exponential decay of the evanescent mode. For thinner barriers this ap-proximation does not hold and one obtains T ≈ 1/[1 + (kd)2]. The transmission coefficientT above the barrier also can only be calculated by the full matching procedure without theabove approximation. It shows an oscillating behavior as can be seen in the figure.

3.7.1 Arbitrary forms of potentials, WKB

The wave function matching method outlined above, works only for simple rectangular po-tential shapes. If one deals with arbitrary potentials, a numerical approach is necessary.There also exists an approximation due to Wentzel, Kramers and Brillouin (WKB) that isoften very good and that we will describe here.

Assume that we have a one dimensional barrier of potential energy V (x) > 0 for −d ≤x ≤ d and zero potential energy outside this region exactly as we had it above except thatV is now not constant. The Schrodinger equation reads then:

(d2

dx2− γ2)ψ = 0 (107)

36

Choosing an arbitrary point x = b left of −d we try now the solution

ψ =1√γe

∫ xb γ(x′)dx′ (108)

Inserting Eq.(108) into Eq.(107)we see that Eq.(108) indeed solves Eq.(107) provided thatone can neglect the term proportional to

d2

dx2(

1√γ

) (109)

One can indeed neglect this term if (i) the potential energy varies relatively smoothly and if(ii)

√γ 6= 0 (the latter because

√γ is in the denominator of Eq.(109)). These two conditions

are unfortunately a lot to ask for because of the following reason.√

γ vanishes exactly forE = V . If a particle approaches the potential barrier from the left having a kinetic energyE, and if the barrier rises smoothly, then we have E = V just at the point where the particlehits the barrier. If, on the other hand, the barrier rises abruptly at −d, then the potentialenergy does not change smoothly and we need to suspect that this part of the approximationfails. Therefore it appears that we have no possibility for any practical case that Eq.(109) isa great approximation. Lets therefore just apply the WKB approximation to the rectangularbarrier from above by calculating the transmission coefficient T exactly as we did above. Wefind T by dividing the wave function slightly to the right of d by the wave function slightlyto the left of −d and squaring:

T = e−4γd (110)

which reproduces the important exponential factor of Eq.(106). It is the fact, that WKBgets the exponential factor correctly that makes it often a reasonable approximation forrelatively thick barriers. One can then fit most experimental data by just slightly modifyingthe thickness 2d of the barrier that is often not even well known. Because of this fact,the WKB approximation takes an honorable place among the approximations to tunnellingproblems.

3.7.2 Coulomb blockade

Tunnelling as described above is based on the solution of the Schrodinger equation for asingle electron and does not consider any effects of the electrons charge. For a long timethis deficiency was not noticed and experiments could be satisfactorily explained. Theseexperiments typically involved a thin insulator sandwiched between two metal plates usuallyhaving a rather large area. The insulator represents a tunnel barrier because the conductionband of the insulator is above the energy of the conduction electrons in the metals, the Fermisea, as we have pointed out in the section on band filling. As the technology related to thetunnelling experiments was developed further, a strange phenomenon was observed. Thetunnelling current should start flowing as soon as a voltage, no matter how small, is appliedto the tunnelling barrier because the probability of tunnelling is nonzero and can be largefor thin barriers (with a thickness of the order of a few Angstrom). However, the tunnelling

37

Figure 28: Coulomb Blockade

38

current was observed to be zero as long as the applied voltage was below a certain thresholdvoltage Vth.

This lack of current for small (close to zero) voltages was called the zero point anomalyand the literature of the time was full of speculations where this anomaly could arise from.The explanation turned out to be simple. The two metal contacts with an insulator inbetween form a capacitor with capacity C. This capacity can be calculated from the parallelplate formula to be approximately C = Arεε0/d where A is the area of the metal plates andd the thickness of the insulator. Elementary courses in electrical engineering tell us thatthe energy Eth that is necessary to transfer a single electron from one capacitor plate to theother is given by

Eth =e2

2C(111)

which requires a voltage Vth = eC. This means that no tunnelling current can flow unless the

applied bias is above this voltage threshold.It becomes now clear why for large contact areas and a correspondingly large capacitance

this threshold was not noticed. In that case and for relatively large temperatures T thethreshold voltage is smaller than the always “available” thermal voltage kT/e. Only if thecapacitance is very small, of the order of Ato-Farad, becomes the threshold noticeable atroom temperature because then we have:

|eVth| >> kT (112)

The existence of such a threshold, or the visibility of the zero point anomaly, is now calledthe coulomb blockade effect. This effect permits the control of tunnelling down to singleelectrons and so called single electron transistors have been proposed.

These devices would have the advantage to deal with the absolute smallest amount ofcharge that any electron device can deal with. It has the disadvantage that any imperfectionthat carries charge will significantly disturb its operation. In addition, the feature sizes ofsuch devices must not exceed the length of about one nanometer if one wants to work at roomtemperature because otherwise Eq.(112) is not fulfilled. As a consequence it is generally seenas very difficult to develop integrated circuits based on single electron transistors. One needsa whole circuit of nano structures with very few charge carrying imperfections that woulduncontrollably shift the threshold of the single devices.

3.7.3 Resonant tunnelling

3.7.4 Inter band tunnelling

4 Stationary perturbation theory

We have seen above that the various explicit method of solving the Schrodinger equationdepend sensitively on the details of the problem and that an explicit solution can usually befound only for the simplest geometries. For more complicated structures there are mainly twomethods available: (i) a numerical solution of the problem and (ii) a solution by perturbationtheory. We will first describe perturbative methods and deal with numerical approaches later.

39

Figure 29: IEEE spectrum paper

40

4.0.5 The perturbation series

The idea of perturbation theory is the following. We have given a Hamiltonian H in theSchrodinger equation that defies any attempt of explicit solution. However, there is a portionof H that we denote by H0 and for which we know how to obtain the complete orthogonal andnormalized system of Eigenfunctions |n > and the eigenvalues En. The remaining portion ofH we denote by εH1. The factor epsilon is thought to be a small number, indicating that thisterm is not the main term and represents just a correction. The idea is then to constructseries expansions in powers of ε that are smaller and smaller as the power of ε increases.Thus we have

H = H0 + εH1 (113)

andH0|n >= En|n > n = 0, 1, 2, ... (114)

and we wish to find eigenvalue E and wave function |ψ > of the equation

H|ψ >= E|ψ > (115)

To achieve this goal we consider expansion around the lowest zero order solution |0 >,E0

using the following power series expansions:

|ψ >= |0 > +ε|ψ1 > +ε2|ψ2 > +... (116)

andE = E0 + εEψ1 + ε2Eψ2 + ... (117)

As usual with such power series expansions, we now insert Eqs.(116) and (117) int Eq.(115)and group all terms with the same coefficient εn together since all these terms need to vanishseparately. Thus we obtain for the three lowest equations:

ε0 : H0|0 >= E0|0 > (118)

ε1 : H0|ψ1 > +H1|0 >= E0|ψ1 > +Eψ1|0 > (119)

ε2 : H0|ψ2 > +H1|ψ1 >= E0|ψ2 > +Eψ1|ψ1 > +Eψ1|0 > (120)

Eq.(118) is just the zero order equation. Eq.(119) is the basis for first order perturbationtheory Expanding ψ1 in terms of the complete ortonormal system of H0 we have

|ψ1 >=∑

n

an|n > (121)

We put a0 = 0 because |0 > appears separately in Eq.(116). Then multiplying Eq.(119) by< 0| gives immediately:

Eψ1 =< 0|H1|0 > (122)

This is the important result for the perturbed energy level. Multiplying Eq.(119) with< i| 6=< 0| gives

ai =< i|H1|0 >

E0 − Ei

(123)

41

Figure 30: Scattering of an electron by a small object

from which we get the perturbed wave function:

|ψ >= |0 > +∞∑1

< n|H1|0 >

E0 − En

(124)

Of course, we would have been able to expand around |j > instead of |0 > and would havethen obtained similar results just with 0 > replaced by |j > in the last three equations.

Higher order perturbation terms require a little more algebra and we just list the per-turbed energy arising from E0 to second order:

E = E0+ < 0|H1|0 > +∞∑1

< 0|H1|n >< n|H1|0 >

E0 − En

(125)

Perturbation theory is extremely useful for a host of problems including scattering ofparticles by small obstacles, corrections to well known problems due to small additions ofarbitrary potentials etc...

We will return to several applications of perturbation theory in the course of these lec-tures, here we give only two examples, that of the effects of an electric field (called starkeffect) in a quantum well and scattering by a periodic potential.

Consider the infinite quantum well problem but add a constant electric field F for −d ≤x ≤ +d. Note that the addition

42

Figure 31: Quantum well with an additional constant electric field

43

Figure 32: Electron ”impinging” on periodic crystal

of the field is not enough to specify the problem because the Schrodinger equation dependson the potential. We could, for example specify a potential energy that is symmetric withrespect to the middle x = 0 of the quantum well, negative to the left and positive to theright e.g. V = eFx. Then we obtain the first order energy correction from Eq.(122) andEq.(61):

Eψ1 =< 0|eFx|0 >=eF

d

∫ +d

−d

xsin[π(x + d)/(2d)]2dx = 0 (126)

which tells us that for such a symmetric potential there is no first order Stark effect. Hadwe chosen a potential energy eFd + eFx then we would have obtained the non-zero answereFd.

For the second example we consider a free electron with a wave function proportional tieik·r impinging on a periodic crystal with periodic potential energy

V (r) =∑

h

BheiKh·r (127)

where the Kh are reciprocal lattice vectors as discussed in connection with band structurecalculations. We use now the fact that the perturbation series involves matrix elements ofthe form

< k′|V (r)|k >=

∫e−ik′·r ∑

h

BheiKh·reik·rdr (128)

44

The integrand in this equation varies rapidly and the integral therefore vanishes except if wehave

k + Kh = k′ (129)

Only when this condition is fulfilled do we have a non zero contribution to the perturbationseries. The collision of a single electron with the huge crystal must be elastic and therefore

|k| = |k′| (130)

Eq.(129) and (130) together represent the condition for Bragg diffraction, in other words theelectron is scattered by the crystal if this condition is fulfilled. In one dimension the Braggconditions are equivalent to k = ±Kh

2. This is exactly the Brillouin zone boundary where

we had the opening of the band gap in the band structure calculation i.e. the point wherethe empty lattice bands crossed. The same is true in two or three dimensions.

4.0.6 Green’s functions and perturbation theory

Given a Schrodinger equation of the usual form (H−E)ψ = 0 with a Hamiltonian differentialoperator H one defines the Green’s function as the solution of the equation

(H − E)G(r, r′, E) = −δ(r− r′) (131)

If one has the Green’s function, what is the use of it? Most commonly, it is used to ob-tain the solution of the inhomogeneous differential equation if we have the solution of thehomogeneous one. For, if we wish to solve the equation

(H − E)ψ = f(r) (132)

then

ψ(r) = −∫

f(r′)G(r, r′, E)dr′ (133)

represents one solution as can be immediately found out by inserting Eq.(133) into (132).This fact can now also be used in a special way to actually obtain a Green’s function G byperturbation theory if the Green’s function G0 is known for the unperturbed problem. Letsassume that H = H0 +V (r) where V (r) is just any given small perturbation of the potentialenergy and lets further assume that we know the solution G0 of

(H − E)G0(r, r′, E) = −δ(r− r′) (134)

We now like to obtain the solution G of

(H + V (r)− E)G(r, r′, E) = −δ(r− r′) (135)

To obtain this solution, we first observe that the perturbed problem can also be writtenas:

(H0 − E)ψ = −V (r)ψ(r) (136)

45

which is just the inhomogeneous equation corresponding to the homogeneous equation of theunperturbed problem. Therefore substituting −V (r)ψ(r) for f(r in Eq.(133) we obtain

ψ(r) =

∫V (r′)ψ(r′)G0(r, r

′, E)dr′ (137)

Adding to this solution of the inhomogeneous equation the homogeneous solution we havethe general solution of the perturbed problem:

ψ(r) = ψ0(r) +

∫V (r′)ψ(r′)G0(r, r

′, E)dr′ (138)

The only problem is now that Eq.(138) is actually an integral equation because the unknownψ appears on both sides. However, we can now try to get a perturbation series by iteration.We start by approximating ψ by ψ0 on the right hand side of Eq.(138). This gives

ψ(r) = ψ0(r) +

∫V (r′)ψ0(r

′)G0(r, r′, E)dr′ (139)

Inserting now this ψ into Eq.(138) gives

ψ(r) = ψ0(r)+

∫V (r′)ψ0(r

′)G0(r, r′, E)dr′+

∫G0(r, r

′, E)V (r′)G0(r, r′′, E)V (r′′)ψ0(r

′′)dr′dr′′

(140)This process can be continued with more and more elaborate integrals. Of course, theformalism can also be put into matrix and vector form by the following discretization. Wewrite the potential energy as a vector V with

V = (V(r1),V(r2),V(r1), ...) (141)

The Green’s function as a function of two variables becomes then a matrix which we denote byG and G0. For the integral we can then substitute a Riemann-sum which is just a product ofmatrix and vector multiplied by a constant characteristic for the distance between dicretizedpoints. We then have ∫

V (r′)G(r, r′, E)dr′ = GV (142)

With these changes, we can write the whole series expansion as

|ψ >= |ψ0 > +G0V|ψ0 > +G0VG0V|ψ0 > +... (143)

It is easy to see that this is equivalent to

|ψ >= |ψ0 > +G0V|ψ > (144)

which replaces the original integral equation Eq.(137). |ψ > and |ψ0 > are here, of course,also vectors obtained by the same discretization procedure. Remember now that the Green’sfunction is by its definition closely related to the wave function. We therefore can go through

46

the whole procedure that we just went through using the Green’s function instead of thewave function. This gives

G = G0 + G0VG (145)

This equation is usually called a Dyson equation. The Dyson equation is useful in numerousways. Here we just show its power by calculating the transmission coefficient for a T-shapednanostructure as shown in the Figure.

We treat the T-shaped nanostructure in the tight binding formulation for a one dimen-sional infinite chain of atoms numbered 0,±1,±2, .... At the atom with number 0 we attach asidearm of finite length with atom numbers 1′, 2′, 3′, .... as shown in the figure. The tight bind-ing wave functions we denote then by |0 >, | ± 1 >, ... and for the sidearm by |1′ >, |2′ >, ....As in all such problems, it is now important to choose the unperturbed problem in the bestpossible way which mostly means as close to the complete perturbed problem but still simpleenough so that we can either solve it explicitly ourselves or at least find a solution in theliterature. There exists a solution in the literature for the infinite chain and also one fora finite chain (sidearm) [3]. Therefore we take these two still unconnected chains as theunperturbed problem. The perturbed problem just includes the coupling of the chains. Thiscoupling can be described in the tight binding method by the nearest neighbor interactionthrough:

V01′ = V (|0 >< 1′|+ |1′ >< 0| (146)

Now we use the Dyson equation

G = G0 + G0V01′G (147)

We do not need to solve this equation because all we want is the transmission coefficientTs of an electron propagating from the left of the sidearm to the right. Therefore we needonly certain matrix elements of the Green’s functions G (we know those of G0). As in thecase of tunnelling, the transmission coefficient Ts is given by the square of the transmissionamplitude ts which is

ts =< −d|G|+ d >

< −d|G0|+ d >(148)

Here < +d| denotes a given site to the left of the sidearm and |−d > a corresponding one tothe right of the sidearm. We can create now equations for the unknown matrix elements ofG0 the following way. We multiplying the Dyson equation from the left by < −d| and fromthe right by | + d > as well as remembering that G0 and therefore all its matrix elementsare known from reference [3], we obtain:

< −d|G|+ d > = < −d|G0|+ d > + < −d|G0|0 >V < 1′|G|+ d > (149)

Here we have also used the fact that < −d|G0|1′ > = 0 because the unperturbed Green’sfunction G0 can not lead to any coupling to the sidearm. We now have an equation for thematrix element < −d|G|+ d > that we are seeking but we have introduced another unknownmatrix element < 1′|G|+ d > because the Dyson equation contains G twice. The procedureis therefore the following. We use the Dyson equation again and multiply it now by the two

47

appropriate vectors (one from the left one from the right) to obtain the new unknown matrixelement. We continue the procedure until we have as many equations as we have unknownsand then solve these equations e.g. numerically. For the particular case of the T -shapednanostructure we need to repeat this procedure only two more times. First we multiply theDyson equation by < 1′| from the left and by | − d > from the right which gives:

< 1′|G|+ d > = < 1′|G0|1′ >V < 0|G|+ d > (150)

Second we need < 0|G|+ d > which we obtain by another use of the Dyson equation:

< 0|G|+ d > = < 0|G0|+ d > + < 0|G0|0 >V < 1′|G|+ d > (151)

We have now the three equations (149)-(151) for three unknown matrix elements involvingG and we can solve this system of linear equations to obtain for the exact transmissionamplitude:

ts = 1 +< 0|G0|+ d >V < 1′|G0|1′ >V < 0|G0|+ d >

(1− V 2< 0|G0|0 >< 1′|G0|1′ >)/< −d|G0|+ d >(152)

This amplitude is exact because we obtained the matrix element for G which is the exactGreen’s function. In general one is not that lucky and needs either a much larger number ofequations or some assumptions to neglect certain higher order terms.

There is still quite a bit of algebraic work left to compute all the matrix elements involvingG0 that are in principle known. This work was done for us by the basic results derived byEconomou [3] and, for the T -shaped nanostructure by Guinea and Verges [4] who devisedthe whole calculation that we just have performed. We only give their result. The actualcalculation could be done as a (pretty elaborate) homework problem. The result is:

ts = [1− isin(Nθ)

2sin(θ)sin((N + 1)θ)]−1

(153)

Here i is the imaginary unit and θ equals ka as defined by the one dimensional E(k)relation for the tight binding band structure as given by Eq.(101). In other words, θ isdetermined by the energy of the particle and by the equation cosθ = (E(k) − E0)/(2V ).As a homework problem one can easily calculate that the transmission coefficient dependssensitively on the length of the sidearm i.e. on the number N of atoms in the sidearm. Thisnumber can actally be controlled by a gate electrode in a real nanostructure because wecan put a gate over the sidearm and repel electrons by biasing this gate negatively. Thetransmission then changes with the gate voltage due to quantum interference and we havea new transistor that switches because of quantum interference. The effect has actuallybeen seen experimentally. Again, however, as in the case of Coulomb blockade, the effect isextremely sensitive to imperfections and it is therefore a question whether such a device haspractical use. Nevertheless, it represents a new type of device that is similar in its actionsto waveguide devices with sidearms as known in the electro-magnetics of waveguides.

48

-0.75 -0.5 -0.25 0.25 0.5 0.75J

-0.4

-0.2

0.2

0.4

Im@TsD

Figure 33: Imaginary part of the Transmission amplitude from equation (152)

49

4.1 Numerical solutions of the one particle Schrodinger equation

D. K. Ferry, pp 64-67Once the Schrodinger equation is discretized and such brought into matrix form, the nu-

merical packages LAPACK/ARPACK should be used to diagonalize the matrices. LAPACKis for full matrices ARPACK for sparse matrices (most elements of entry zero). Help withthis packages can be obtained from the web. See e.g.

http://www-heller.harvard.edu/ shaw/programs/lapack.html

5 The many particle Schrodinger equation, identical

particles

5.1 General form and probability interpretation

We have already derives the Schrodinger equation for many identical particles and havewritten it out for two particles in Eq.(48) which we repeat:

K2

2m1

(∂2

∂(x1)2 +∂2

∂(y1)2 +∂2

∂(z1)2 )ψ+K2

2m2

(∂2

∂(x2)2 +∂2

∂(y2)2 +∂2

∂(z2)2 )ψ+(E−V (r1, r2))ψ = 0

(154)As discussed, we now can not use the ordinary space coordinates x, y, z as we did for oneparticle. We now need to use the particle coordinates r1 = (x1, y1, z1) and r2 = (x2, y2, z2).One says that the Schrodinger equation is an equation in configuration space, i.e. in thespace involving the coordinates of the particles. This brings home the difficulties in theinterpretation of what the wave function really represents. As long as we were dealing withthe space coordinates, we could, without logical contradiction, regard the wave function assomething corresponding to a field as in the electromagnetic field. The similarity of the waveequation with the Helmholtz equation enforced this picture. However, now we suddenly haveparticle coordinates i.e. we talk about an entity which is located at a point in space andat the same time calculate a wave function and use the picture of waves. Max Born solvedthis puzzle of interpretation. This solution resembles somewhat the solution that Alexanderthe great found for the Gordian knot. The knot was untangled but not without expense.Born simply postulated that the square of the wave function |ψ(r)|2 be the probability tofind a particle. We have already mentioned that for the one particle case that leads to thenormalization condition: ∫

|ψ(r)|2dr = 1 (155)

The wave function thus becomes a “guiding” wave for the probability to find a particle.This is somewhat consistent with the derivation of the Schrodinger equation that uses theHamilton-Jacobi formalism which deals with constants of the motion, such as the constantangular velocity of a circular motion around a central point. This interpretation, which isnow generally adopted, can also by generalized for many particles. One just has to modifythe normalization condition in the obvious way that the integral over all space over the two

50

particle coordinates must now be one because one certainly will find the two particles if allspace is checked out. Thus ∫

|ψ(r1, r2)|2dr1dr2 = 1 (156)

and similarly for N identical particles.This probabilistic interpretations of Schrodinger’s wave mechanics has been tested in

numerous experiments, for example the famous two-slit-type experiments, and is generallyfound in agreement with experiments. Nanostructure technology offers a great playgroundfor further and more sophisticated testing such as observing the scattering of electrons byassembles of carbon nanotubes. Here we simply continue by assuming that the probabilityinterpretation is correct and investigate the consequences and the necessary generalizations.

5.2 Bosons and Fermions

Consider free particles in different states i.e with different wave vectors k1 and k2. Assumefurther that the particles do not exert any force on each other. Electrons would, of coursehave a Coulomb interaction, so this example is somewhat esoteric. As can be seen fromEq.(154), the wave function is then a simple product

ψ(r1, r2) = Aψk1(r1)ψk2(r2) (157)

For our example of free plane waves in a given volume we have ψk1(r1) = ek1·r1 etc. whilefor electrons in a quantum well the appropriate wave functions need to be used. However,there is a basic problem with this simple solution. The particles of quantum mechanics arenot distinguishable and we have no way of marking an electron or a proton to designate itas number 1 or 2. Therefore we need to somehow accommodate the fact that the particlesare indistinguishable otherwise we get the counting of possible different states wrong. TheSchrodinger formalism can neatly accommodate the existence of indistinguishable particles.We just need to construct a wave function that is noncommittal as to which particle is inwhich state. There are exactly two ways to do this. We can have

ψ(r1, r2) = A[ψk1(r1)ψk2(r2) + ψk2(r1)ψk1(r2)] (158)

where A is just the normalization. If particles have such a wave function they are calledbosons. Or we can have

ψ(r1, r2) = A[ψk1(r1)ψk2(r2)− ψk2(r1)ψk1(r2)] (159)

and such particles are called fermions. We have not learned at this point much about spinand polarization and simply note that all particles with integer spin are bosons all withhalf-integer spin are fermions. Generally we have

ψ(r1, r2) = ±ψ(r2, r1) (160)

where the plus holds for bosons and the minus for fermions.

51

The normalization constant A can easily be found by performing the integral of Eq.(156)and using the fact that

|ψ(r1, r2)|2 = ψ(r1, r2)ψ∗(r1, r2) (161)

For bosons confined to a volume Vol we obtain A = 1Vol

√2

For distinguishable particles and

the simple product wave function of Eq.(156) the normalization would have just equaled 1Vol

.The necessity of including the fact that particles can not be distinguished thus has con-

sequences for the normalization and form of the wave function. One of the most importantconsequences is that the wave function for fermions given by Eq.(159) vanishes if k1 = k2.This is the famous Pauli exclusion principle that appears here as a consequence of the rulesfor constructing wave functions of indistinguishable particles. Therefore we can not have twoidentical fermions in one and the same given state. We can have two with opposite spin asdescribed below. There is another consequence of the “anti symmetry” (the minus sign) ofthe wave functions of fermions. Identical fermions in different states of a given volume willtend to be farther apart from each other than distinguishable particles would be because thewave function tends to zero when they come close. The probability to find them close to eachother is therefore diminished. It is as if there were a force of repulsion between fermions andwe call this geometric consequence of the requirements on the wave function the exchangeforce. There is a further factor that one needs to take into account if one considers theexchange force. We have been up to now ignoring the spin. But the description of the wavefunction needs also to include the spin. We will show later how exactly this can be done andremark here only the following. To describe the spin we need to multiply the wave functionwith a spin dependent portion. This spin dependent portion contributes also a sign to thewave function. For a singlet spin pair (when measured and one is found up the other isfound down) a minus sign is contributed such that the spatial function products of ψ appearactually with a plus sign as in the case of bosons. This is the reason why we always saidthat each quantum state can accommodate two electrons with opposite spin.

Bosons like to be closer to each other as compared to distinguishable particles independentof their polarization (integer spin) because of the plus sign in Eq.(158). For them the sayingapplies “the closer we are, the merrier we will be”.

5.3 Interacting Particles and the Hartree Theory

We have already mentioned in the section on band structure that it is difficult to determinethe crystal potential because all the electrons and all the nuclei contribute to it. The sameis true for any nanostructure whether or not it is crystalline. We would like to considernow the precise many body Hamiltonian of a nanostructure made out of atoms or ions. Wefollow in essence the notation and approach of Inkson [5]. We immediately make a veryreasonable approximation. Because the nuclei are so massive compared to the electrons theymove much slower than the electrons and can be considered stationary. This means that thedifferentiation in the Schrodinger equation that relates to the kinetic energy concerns the

52

electrons only. We denote the differential operator for the electron numbered i by ∇2i with:

∇2i =

∂2

∂(xi)2 +∂2

∂(yi)2 +∂2

∂(zi)2 (162)

We place the positively charged nuclei of the nanostructure at locations Rl which may ormay not be regular crystal lattice sites and we denote the location of the electrons by ri andrs. Two different indices are necessary for the electrons if we consider the interaction of twoelectrons. The Hamiltonian H of the Schrodinger equation for the whole nanostructure thenbecomes:

H = − h2

2m

∑i

∇2i +

1

2

∑i,s

′ e2

|ri − rs| −∑

i,l

e2

|ri −Rl| (163)

The summation that describes the potential energy due to the Coulomb interaction of elec-trons has a factor 1

2because we can not count the interaction of the i’s and the s’s electron

twice. The sum has also a ′ to indicate that s 6= i so that the electron does not exert aCoulomb repulsion on itself. In the sum that describes ion-electron interactions no such pre-cautions are necessary because the space coordinate of ions is R while that for electrons is r.This Hamiltonian forms the staring point for many body investigations of nanostructures.The real problem is the second term which contains the electron interactions. The thirdterm can be dealt with in various ways. We could identify the third term with a crystalpotential as we have done in the section on band structure. In a further approximation wecould use the effective mass approximation and drop the third term just using an effectivemass instead of m. The only problem that would remain now is that our nanostructure,containing a number of electrons, would not be neutral. However, we could include posi-tively charged donors to make it neutral or even simpler introduce a homogeneous positivelycharged background to make it neutral. The latter model is called a jellium model in solidstate theory. This leaves essentially for now the second term as a novelty in finding solutionsof the Schrodinger equation.

The geometrical exchange force discussed above complicates things greatly. As men-tioned, any numerical approach needs to store all the particle coordinates and find a solutionof the Schrodinger equation for each particle configuration. This can be done exactly for afew particles but any larger number requires some approximations. The first good modelapproximation of this kind is due to Hartree and to Sommerfeld.

The object is, of course, to simplify the “interacting” many body problem into a problemthat depends only on the individual positions of the particles. Then the problem can bedecomposed into single particle equations that we know how to solve. What we thereforneed to have is an approximation that lets us rewrite the interaction term as follows:

1

2

∑i,s

′ e2

|ri − rs| =∑

i

Vi(ri) (164)

Electrostatics suggests a physically satisfying way to do this. If we take the i’s particleat the position ri, then the potential “seen” by it will be given by the contributions of all

53

the other electrons that we also could consider as fixed i.e.

Vi(ri) ≈∑

s

′ e2

|ri − rs| (165)

and therefore we will have a Schrodinger equation for the i’s particle given by:

(H0(ri) + Vi(ri)− Ei)ψi(ri) = 0 (166)

where H0 represents the electron kinetic energy and the electron-ion (nuclei) interactions. Inthis way we can obtain a wave function for all of the electrons corresponding to particularfixed positions of particles. However, with moving particles these wave functions change andwe can not assess how good our procedure actually is. If we postulate, however, that weare only talking about wave functions that are occupied and multiply the potential energyby the probability to find and electron at point s then the procedure is at least consistent.Thus the potential energy becomes:

Vi(ri) ≈∑

s,occupied

′ e2|ψs(rs)|2|ri − rs| (167)

This is the Hartree approximation which postulates that each electrons moves in a field dueto the distributed charge given by the |ψ|2 of all the others. As a further approximation, weassume that the wave function is that of distinguishable particles i.e. a product wave functionwith each factor of the product determined by Eq.(166). Of course, we have then neglected alleffects of the exchange force in addition to the approximations that are involved in Eq.(167)and described above in connection with this equation. In fact, Sommerfeld showed that inthe jellium approximation the positive background cancels almost all the effects and thatthe Hartree term of Eq.(167) vanishes as the fraction of the area S divided by the volumeVol of the system goes to zero i.e. S

Vol→ 0. It turns out that this is a good approximation

for simple metals such as sodium and potassium when the volume is large. The electronscan then be treated as free and non-interacting plane waves. However, the fraction S

Voldoes

not go to zero for nanostructures. In fact as the volume goes to zero, SVol

becomes larger andlarger.

The Hartree approximation describes therefore a large number of effects that are impor-tant in nanotechnology. For example, the Coulomb blockade effect, due to the quantizationof charge of the electrons is described well by the Hartree approximation. If we have aquantum dot and add electrons from the infinite point (the vacuum level) then, the energythat is necessary to add an electron is also described well by the Hartree approximation. Inthis way one can calculate the capacitance of a quantum dot. A reasonable approximationcan in this way also be obtained for the switching energy of cellular automatas (Figures,description). We will return to some of these topics later.

In the next section we describe the physics of how the Hartree approximation can beimproved, the physics of exchange and correlation and subsequently we describe densityfunctional theory as a method to include the physics of exchange and correlation.

54

5.4 Exchange, Correlation and the Thomas-Fermi Model

In this section we give a qualitative discussion of the effects of exchange and Coulombinteraction named exchange correlation effects. Consider a jellium solid and think of justone “test” electron that we wish to study. What effect does it have on all other charges.Neglecting for the moment quantum effects, the other electrons will just move away fromour test electron due to Coulomb repulsion until an equal and opposite positive backgroundcharge has been induced so that the electric field at large distances is removes. The field ofthe test electron is screened out. The electron is left surrounded by a hole in the electrondensity which contains an equal and opposite charge. This is the simplest example of whatone calls a correlation hole.

We now change our thinking towards a quantum system. This change will actually notalter the situation all that much. For, if there were no coulomb interaction the exchangeeffect alone would also lead to a repulsion of all the other electrons having the same spinstate. Thus exchange and Coulomb effects have much the same net result: the productionof a hole in the electron density surrounding the test electron.

Which effect is more important, the Coulomb repulsion or the Pauli exclusion? Thisdepends on the situation. For typical solid state densities, the Coulomb repulsion oftenprevents electrons coming close enough so that Pauli exclusion would play a role. Then onecan introduce a simple model that is still consistent with quantum ideas, the Thomas-Fermimodel.

The local electron density in a solid is determined by the Fermi level. If we apply alocal electrostatic force (Figure), there will be a tendency for the electrons to rearrange untilthe density is again consistent with the Fermi level. Provided we have a slowly varyingpotential, we can assume that each part of the system acts like the bulk solid but with adifferent potential energy. Suppose the applied potential is Uapp and the total potential is Uand that due to the rearrangement of charge is Ure. We then have:

U(r) = Uapp(r) + Ure(r) (168)

Ure(r) satisfies Poisson’s equation that describes the rearranged charge density ρre(r) i.e.

∇2Ure(r) = ρre(r)/(εε0) (169)

For ρ one can often write in an extended metallic system

ρ(r) = eU(r)N(EF ) (170)

Here NF is the density of states at the Fermi energy. Poisson’s equation then becomes:

∇2[U(r)− Uapp(r)] = eU(r)N(EF )/(εε0) (171)

which can be rewritten as

(∇2 − 1/L2TF )U(r) = ∇2Uappl(r) (172)

55

with 1/L2TF = eN(EF )/(εε0). For Uappl = e/r one obtains the well known screened solution

U(r) = ee−r/LTF

r(173)

We can now take into account the coulomb correlation in the Hartree model and writefor the interaction term:

1

2

∑i,s

′ e2

|ri − rs| →1

2

∑i,s

′ e2

|ri − rs|e−|ri−rs|/LTF (174)

This screened potential is often a better starting point to include exchange effects. Wehave now, however, tied up some energy in the correlations of the electrons that lead to thescreening. This correlation energy is defined as the difference between Coulomb energy andscreened Coulomb energy at small distances i.e.

ECORR = e(1− e−r/LTF )/r ≈ −e2/(2LTF ) (175)

We have thus discussed the concepts of exchange and correlation in a qualitative way.The equations above give us a possibility to estimate exchange correlation effects. If forexample the Thomas-Fermi length is such that electrons are not getting close enough forthe exchange interaction to be important, then one can hopefully neglect it or in some casesinclude it more easily by using Eq.(174) as a starting point. There is, however, a methodemerging that permits to include exchange and correlation effects more directly and in a fairlystraightforward fashion. This method is due to Hohenberg, Kohn and Sham (HKS) and iscalled density functional theory. This theory has become useful for physicists and chemistsand is very useful in nanoscience and technology. A rough description of the method is givennext.

5.5 Density Functional Theory

HKS showed in their well known theorem [5] that the ground state properties of a systemcan be derived in terms of the electron density which is then considered a variable. For theHartree interaction term it is clear, or at least plausible, that in the ground state we canreplace it in the following way

∑

s,occupied

′ e2|ψs(rs)|2|ri − rs| → e2

∫n(r′)|r− r′|dr

′ (176)

Here we put the test electron simply at the place of the variable r and have transformed thediscrete rs into the continuous variable r′ and the summation into an integral. The squareof the wave function of the full states s was naturally transformed into the electron densityn(r′). This does, of course, not bring us in any way beyond the Hartree approximation.The important part of the proof of HKS is that exchange and correlation effects and theircontribution to the Hamiltonian, i.e. to the effective potential energy, can be represented as

56

a functional HKS[n(r)] of the electron density n(r). A functional is, of course, a much moredifficult entity than a function. HKS depends on the functional form of n(r) at all r and doesnot as a function would map one number of n(r) into just another number. A considerableamount of research has shown, however, that HKS can be approximated by local functions(not functionals) of the electron density and (in higher order approximations) in addition ofthe gradient of the electron density. The local density functional theory (LDFT) has beenquite successful even with the lowest order approximation which is:

HKS[n(r)] ≈ d[n(r)εxcn(r)]

d[n(r)](177)

where n(r)εxc is the exchange correlation energy. This exchange correlation energy has beencalculated in detailed research projects for various interesting cases e.g. for two dimensionalelectron densities, with and without spin effects included and has been tabulated e.g. interms of polynomial approximations. We give here only the result for a three dimensionalelectron density (in the so called Hartree-Fock approximation):

HKS[n(r)] ≈ −(3π2n(r))1/3

/π (178)

The complete Schrodinger equation now is transformed into a set of one electron Schrodingerequations for the lowest eigenstates i = 1, 2, ..., N . The Hamiltonian is:

H = − h2

2m∇2 + e2

∫n(r′)|r− r′|dr

′ + HKS[n(r)] + Uapp(r) + Vnu (179)

Here we have added a possible applied potential Uapp which could be due to a donor oracceptor in a semiconductor or some other “external” potential. We have also added Vnu

which represents for example the crystal potential, of the positive jellium background or, forany nanostructure, the positively charged nuclei or ions.

The Schrodinger equation that we need to solve is then

Hψi = Eiψi (180)

and we must use

n =N∑

i=1

|ψi|2 (181)

with n and ψ being, of course, functions of r.The density functional theory as presented here has been found remarkably accurate and

numerically efficient to handle. We believe that it is and will continue to be the method ofchoice for many body nanostructure calculations. Numerical codes such as Gaussian, VASPand Abinit are often available can be used.

57

6 Nanostructures and Quantum Treatments

6.1 Electronic Nanostructures

6.1.1 Schrodinger-Poisson

The many body Hamiltonial in the local density functional approximation as given inEq.(179) is usually further simplifies when quantum wells, wires or dots made our of semi-conductor heterolayers are considered. One then is not interested in the effects of all theelectrons of the nanostructure but only, for example, in those populating the conductionband. The action of the other electrons can then often be approximated by introducing adielectric constant. Then we can replace the interaction term in the following way:

e2

∫n(r′)|r− r′|dr

′ → e2

4πεε0

∫nc(r

′)|r− r′|dr

′ (182)

where nc is the space dependent electron density in the conduction band.Instead of taking the potential as obtained from the Coulomb-integral (or summation)

of Eq.(182) we may also, particularly if we have relatively many electrons and other chargesuse the equation of Poisson to find that potential and at the same time include the appliedpotential and corresponding energy Uapp. If, for example the applied potential arises frompositively charged donors that have a density N+

D , then we can obtain the potential energydue to nc and N+

D from the following Poisson equation:

∆ΦPoi =e

εε0

[nc(r)−N+D (r)] (183)

The many body Hamiltonian reads then:

H = − h2

2m∇2 − eΦPoi + HKS[n(r)] + Vnu (184)

If the nanostructure is still containing many atoms and we can use the effective mass ap-proximation, then we can remove the potential energy Vnu of the nuclei or ions and justintroduce the effective mass instead. We then have:

H = − h2

2m∗∇2 − eΦPoi + HKS[n(r)] (185)

where we have introduced the effective mass m∗. Note, however, that for some semicon-ductors the effective mass is anisotropic and need to be introduced that way as we havedescribed it in the section on effective mass.

Eq.(185) needs to be solved then together with the equation of Poisson Eq.(183). To dothis we need to express the electron density by the wave-function in the spirit of the Hartreetype of treatment. Thus we have:

nc =N∑

i=1

|ψi|2 (186)

58

where the summation is over all full states. If we deal with a quantum dot, we just need toknow how many states are full i.e. how many dot states we have and how many electrons wehave. This means, we need to know essentially the Fermi energy or the corresponding electro-chemical potential. (If we have just a few electrons in a quantum dot, we may want to go backto the Coulomb integration instead of the solution of the Poisson equation.) The so calledself-consistent calculation (mostly numerical) that solves Eqs.(183), (185) and (186) givesus then the many body energy levels of a quantum dot and describes coulombic (includingCoulomb blockade) and quantum effects. A detailed example (reading assignment) is workedout in references [8] and [9].

If quantum wells or wires are considered, then we need to modify Eqs.(186) because wenow have a propagating wave function in one direction for the quantum wire and in twodirections for the quantum well. Consider therefore a quantum well and denote the directionof propagation by the symbol|| (for parallel to the quantum well heterojunction). The wavefunction of the electrons is therefore described in the propagating direction by

ψ|| =1√A

eik||·r|| (187)

where A is the area of the quantum well and r|| is the space coordinate parallel to theheterolayer boundaries of the well. Perpendicular to the well (z-direction), we have, ofcourse, the discrete quantization of the well energy levels i = 1, ..., N with normalized wavefunction ζi(z) so that the total wave function ψ becomes:

ψ =1√A

eik||·r||ζi(z) (188)

We obtain therefore for nc which now depends only on the z-coordinate and is homogeneousin the r|| direction.

nc(z) =∑

k||

1

A

N∑i=1

|ζi(z)|2f(k||) (189)

where we have multiplied the expression by the Fermi-distribution of the electrons to guar-antee that the sums are performed over full states. The sum over k|| can be replaced by theintegral that results in the two dimensional density of states as we have shown before.

The procedure for the quantum well is, of course, very similar to that one needs to do tocalculate quantum effects in inversion layers of MOS transistors. this procedure is describedin detain in reference [10].

6.1.2 Scattering in Nanostructures and Modulation Doping

See reference [?]

6.2 Nanostructure Mechanics

See lecture notes of Gang Li.

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6.3 Einstein-Podolsky-Rosen, Quantum Measurement and Quan-tum Information

6.3.1 Quantum Information

Classical information has been given mathematical definition by Claude Shannon [11]. Quan-tum information is not yet as well defined. Both types of information are thought to deal with“static resources” that represent the substance of information and with dynamical processesthat relate to the flux of information. In the language of Ernst Mach, the static resourceswould be the “elements” and information theory relates to the flux of these elements. Themost important element of classical information theory is the bit which is mathematicallyrepresented by either a zero or a one and physically by the on or off state of a current,light etc.. Correspondingly one defines in quantum information the “qubit”. The qubit canassume values of minus one and one. We could also say zero instead of the minus one for thequbit. However, because we link the qubit below to spin measurements with the outcomesspin up or down, we let the qubit assume values of one (for spin up) and minus one (for spindown). These values materialize in actuality only with a certain probability and only afterthe physical act of measurement. Before the measurement the qubit is neither minus onenor one but in a so called superposition of both. The mathematical definition follows thequantum mechanical formalism as outlined in section 3.4.2. We define a two dimensionalspace of quantum states using the orthonormal basis | − 1 >, |1 > and a corresponding wavefunction

|ψ >= c−1| − 1 > +c1|1 > (190)

with c−1 and c1 being complex numbers, |c−1|2 representing the probability that the quantumparticle is found in state | − 1 > and |c1|2 representing the probability that the quantumparticle is found in state |1 >. The qubit is thus a much more complicated entity than the bit.Nevertheless, the definition appears mathematical enough to permit the use of qubits withina mathematical framework of quantum information theory. One could simply view the qubitas a combination of digital and analog information in the following way. The outcome of ameasurement gives the digital result minus one or one depending on the analog informationcontained in c−1, c1. However, this definition contains the mathematically “unclean” wordmeasurement which also presents the following complication: the qubit is not just a “staticresource” but contains in its definition the dynamics of the measurement. It further containsan epistemological element of quantum theory. After Bohr, the value minus one or one isassumed through and by the act of measurement. Before that act, no “element” in the senseof Mach exists that determines the actual outcome of zero or one. One can also follow Bohrto say that therefore before the act of measurement no static resource exists, no basis forinformation exists. Einstein did not like Bohr’s view and concocted together with Podolskyand Rosen (EPR) a situation or Gedanken Experiment that could be described in termsof qubits as follows. Consider two qubits that have “met” and therefore are correlated toeach other. For example, two particles with spin. If these particles are in a so called singletstate then, if we measure the spin of each and if one particle is measured to have spinup (characterized e.g. by the state |1 > and by the measurement outcome one), then we

60

know with probability one that the measurement of the other particle will give a minus onecorresponding to the state |−1 >. We can separate the qubits over large distances after theyhave met. The measurement outcome still will be the same. One calls the qubits entangled.In contrast to the teachings of Bohr, Einstein therefore maintained that the particles withspin (corresponding to our qubits) must carry some elements (Einstein called them elementsof reality) that will determine in some way the outcome of the measurement. This is becauseof the finite maximum velocity (the velocity of light) of information transfer, the separatedparticles can not communicate. If they are still correlated, then according to Einstein theymust carry information that correlates them. The debate of Einstein and Bohr might seemto be theoretical only. However, entanglement is currently claimed to be a cornerstone ofquantum information and is seen as a very important, if not the most important, extensionbeyond classical information. Why beyond classical information? Can one not describe theentanglement by some analog addition to the digital outcome for the spins? Most quantuminformation researchers would say that entanglement can not be described in a classical wayusing classical information theory. They would say that entanglement and therefore quantuminformation go beyond what can be described by Shannon’s information theory. The basisfor this widespread opinion is the Theorem of Bell which we therefore describe next.

6.3.2 The EPR Experiment and the Theorem of Bell

Bell considered the general outcome of EPR type experiments. If we measure the spin of oneparticle with a magnet oriented along direction a, where a is a unit vector, then we will havea certain outcome Aa = ±1 for the spin. If we measure the second particle of the singletpair with another magnet oriented along direction b then we will have a certain outcomeBb = ±1 for that particle. We know already that if Aa = 1 then Ba = −1. Therefore weobtain for the average product < AaBb >= −1. In general, quantum theory gives the result:

< AaBb >av= −a · b (191)

Bell maintains that no classical correlation between the particles based on local informationor elements of reality for a given particle can lead to this result. Bell’s proof is based on thesimple observation that

AaBb + AaBc + AdBb − AdBc = ±2 (192)

This is easily proven by inspecting all possible cases. One can also easily convince oneselfthat for certain directions of the magnet Eq.(191) gives for the average

< AaBb + AaBc + AdBb − AdBc >av > 2 (193)

But how can that be in view of Eq.(192)? Obviously Eq.(192) contradicts Eq.(193). HenceEq.(192) contradicts Eq.(191), the result predicted by QM. This is the central result thatproves Bell’s theorem which says in essence that no reasonable theory based on Einsteinlocality can yield the quantum result.

Actual experiments were made using quantum optics i.e. polarized photon pairs. Andthe actual experiments by Aspect and coworkers [12] agreed with the theoretical result of

61

quantum mechanics. This is the basis for the claim that quantum information can not bedescribed in classical terms.

6.3.3 Critique of the Theorem of Bell

There is a problem with the theorem of Bell. It is assumed in Eq.(192) that all quantities withthe same index are the same. In the Aspect experiment, however, only one term of Eq.(192)can be measured at a given time and for a given pair of photons. The other terms withdifferent setting have to be measured at a different time and involve necessarily a differentphoton pair. Therefore the quantities with the same index are not guaranteed to be the same.This leads to grave problems in the comparison of Eq.(193) with any real experiment andparticularly with the Aspect experiment. This can be seen most clearly in certain versionsof Bell’s proof [13]. It is doubtful whether or not Bell-type proofs can be repaired and itis therefore doubtful whether entanglement can or can not be described by the means ofclassical information theory. However, it is not doubtful that quantum information as awhole represents something new and useful. The use of quantum optics in cryptography, forexample, has been shown beyond any doubt. One simply can not “listen” to a communicationbased on single photon transmission without perturbing the communication in such a waythat it can be detected. The workings of quantum computing also are in no doubt, whetheror not the Theorem of Bell can be proven using flawless probability theory. However, thenimbus of quantum information would suffer greatly if it can not.

6.4 Student Projects

Posted on class web-site.

References

[1] K. Hess, Advanced Theory of Semiconductor Devices, IEEE Press, New York, pp 42-45,2000.

[2] K. Hess, Advanced Theory of Semiconductor Devices, IEEE Press, New York, pp 67-77,2000.

[3] E. N. Economou, Green’s Functions in Quantum Physics, Springer Verlag, Berlin, Newyork, Tokyo, p 80, 1983

[4] F. Guinea and J. A. Verges, Phys. Rev. B, Vol 35, pp 979-986, 1987

[5] John C. Inkson, Many-Body Theory of Solids, Plenum Press, New York and London,pp 5- 10, 1984

[6] John C. Inkson, Many-Body Theory of Solids, Plenum Press, New York and London,pp 265-270, 1984

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[7] K. Hess, Advanced Theory of Semiconductor Devices, IEEE Press, New York, pp 157-158, 2000.

[8] M. Macucci, Karl Hess and G.J. Iafrate, Electronic energy spectrum and the concept ofcapacitance in quantum dots, Phys. Rev. B, Vol.48, pp 17 354-17 366, 1993

[9] M. Macucci, Karl Hess and G.J. Iafrate, Numerical simulation of shell-filling effects incircular quantum dots, Phys. Rev. B, Vol.55, pp 4879-4882, 1997

[10] K. Hess, Advanced Theory of Semiconductor Devices, IEEE Press, New York, pp 157-159, 2000.

[11] Shannon, C. E. and Weaver, W. ,The Mathematical Theory of Communication, Uni-versity of Illinois Press, Urbana and Chicago, 1998

[12] Aspect, A., Dalibard, J. and Roger, G., Phys. Rev. Letters 49, 1804-1807, 1982

[13] Karl Hess and Walter Philipp, Breakdown of Bell’s theorem for certain objective localparameter spaces, PNAS Vol. 101, pp 1799-1805, 2004

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