Part 1. The Quantum Particle

Transcription

1 CONTACT CONTACT Introuction to Nanoelectronics Part. The Quantum Particle This class is concerne with the propagation of electrons in conuctors. Here in Part, we will begin by introucing the tools from quantum mechanics that we will nee to escribe electrons. We will introuce probabilistic escriptions of the key physical properties: position, momentum, time an energy. In the net part we will consier electrons in the simplest possible moel of a conuctor a bo i.e. we will ignore atoms an assume that the material is perfectly homogeneous. particle in a bo approimation Fig... The particle in a bo takes a comple structure like a molecule an approimates it by a homogeneous bo. All etails, such as atoms, are ignore. This moel of electrons in conuctors is known as the particle in a bo. It is surprisingly useful, an later in the class we will employ it to escribe the behavior of moern transistors. But first we will nee a way to escribe electrons. It is often convenient to imagine electrons as little projectiles pushe aroun by an electric fiel. An in many cases, this classical moel yiels a fairly accurate escription of electronic evices. But not always. In nanoscale evices especially, electrons are better escribe as waves. (a) big transistor view of an electron Electric Fiel (b) small transistor view of an electron z atom electron H C H C C H C y H H H C C C H C H H H Fig... Two representations of electrons in a soli. In (a) the electrons are represente as har little spheres, propelle by the electric fiel, an bouncing off atoms. In (b) we raw an approimate representation of the molecule,3-butaiene positione between contacts. Now the electrons are represente by probability clous. 3

2 Single slit Part. The Quantum Particle Waves in electronics Consier a beam of electrons propagating through a small hole - a very crue moel for electrons moving from a contact into a nanoscale conuctor. We might epect that the electrons woul continue in straight lines after passing through the hole. But if the hole is small enough (the imensions of a nanoscale transistor, for eample), then the electrons are observe to iffract. This clearly emonstrates that we must consier the wave properties of electrons in nanoelectronic evices. Fig..3. A simulation of electron iffraction through a single slit. This eperiment is analyze in Problem. The iffraction pattern shown above is obtaine by assuming each point insie the single slit is a source of epaning waves; see Problem. An easier eample to analyze is the ouble slit eperiment, in which we assume there are only two sources of epaning waves. Like the single slit eample, the result of a ouble slit eperiment is consistent with electrons behaving like waves. (a) Classical preiction (b) Eperimental result slits slits electrons electrons Fig..4. Classically, we woul preict that electrons passing through slits in a screen shoul continue in straight lines, forming an eact image of the slits on the rear screen. In practice, however, a series of lines is forme on the rear screen, suggesting that the electrons have been somehow eflecte by the slits. 4

3 Introuction to Nanoelectronics Review of Classical Waves A wave is a perioic oscillation. It is convenient to escribe waves using comple numbers. For eample consier the function ik e (.) where is position, an k is a constant known as the wavenumber. This function is plotte in Fig..5 on the comple plane as a function of position,. The phase of the function k (.) is the angle on the comple plane. Re Im Im Comple plane Comple plane Re Fig..5. A staning wave with its phase plotte on the comple plane. The wavelength is efine as the istance between spatial repetitions of the oscillation. This correspons to a phase change of. From Eqns. (.) an (.) we get k (.3) This wave is inepenent of time, an is known as a staning wave. But we coul efine a function whose phase varies with time: i t t e (.4) Here t is time, an is the angular frequency. We efine the perio, T, as the time between repetitions of the oscillation. (.5) T Plane waves We can combine time an spatial phase oscillations to make a traveling wave. For eample ikt, t e (.6) 5

4 Part. The Quantum Particle We efine the intensity of the wave as * (.7) Where * is the comple conjugate of. Since the intensity of this wave is uniform everywhere it is known as a plane wave. A plane wave has at least four imensions (real amplitue, imaginary amplitue,, an t) so it is not so easy to plot. Instea, in Fig..6 we plot planes of a given phase. These planes move through space at the phase velocity, v p, of the wave. For eample, consier the plane corresponing to =. k t (.8) Now, vp (.9) t k Re v p = /k Im Fig..6. In a plane wave planes of constant phase move through space at the phase velocity. The ouble slit eperiment We now have the tools to moel the ouble slit eperiment escribe above. Far from the ouble slit, the electrons emanating from each slit look like plane waves; see Fig..7, where s is the separation between the slits an L is the istance to the viewing screen. At the viewing screen we have L, t Aep ikr t Aep ikr t (.) 6

5 Introuction to Nanoelectronics The intensity at the screen is Aep i tepikr epikr Aep i tepikr epikr ep A A ep i k r k r A i k r k r A A cos k r r where A is a constant etermine by the intensity of the electron wave. Now from Fig..7: r L s y r L s y * (.) (.) Now, if y << s/ we can neglect the y term: r L s sy r L s sy (.3) (a) y (b) r r s slits (c) electrons L s/ s/ s/-y s/+y L r r y Fig..7. Far from the ouble slit, the electrons from each slit can be escribe by plane waves. Where the planes of constant phase collie, a bright line corresponing to a high intensity of electrons is observe. 7

6 Part. The Quantum Particle Then, Net, if L >> s/ sy r L s L s sy r L s L s r r L L sy L sy L (.4) (.5) sy Thus, r r, an L y A cos ks L (.6) At the screen, constructive interference between the plane waves from each slit yiels a regular array of bright lines, corresponing to a high intensity of electrons. In between each pair of bright lines, is a ark ban where the plane waves interfere estructively, i.e. the waves are raians out of phase with one another. The spacing between the bright lines at the viewing screen is y s (.7) L Rearranging, L y (.8) s 8

7 Introuction to Nanoelectronics Interpretation of the ouble slit eperiment It is notable that the fringe pattern is inepenent of intensity. Thus, the interference effect shoul be observe even if just a single electron is fire at the slits at a time. For eample, in Fig..8 we show the builup of the fringe pattern from consecutive electrons. The only conclusion is that the electron which we are use to thinking of as a particle - also has wave properties. electrons electrons 6 electrons 4 electrons 4 electrons Fig..8. The cumulative electron istribution after passage through a ouble slit. Just a single electron is present in the apparatus at any time. From A. Tanamura, et al. Am. J. Physics 57, 7 (989). 9

8 Part. The Quantum Particle The Wavefunction The wave-like properties of electrons are an eample of the wave-particle uality. Inee, in the early th century, quantum mechanics reveale that a combination of wave an particle properties is a general property of everything at the size scale of an electron. Without aressing the broaer implications of this unusual observation, we will simply note that our purposes require a suitable mathematical escription for the electron that can escribe both its particle an wave-like properties. Following the conventions of quantum mechanics, we will efine a function known as the wavefunction, (,t), to escribe the electron. It is typically a comple function an it has the important property that its magnitue square is the probability ensity of the electron at a given position an time.,, *,, P t t t t (.9) If the wavefunction is to escribe a single electron, then the sum of its probability ensity over all space must be. P, t (.) In this case we say that the wavefunction is normalize such that the probability ensity sums to unity. Frequency omain an k-space escriptions of waves Consier the wavefunction i t t ae (.) which escribes a wave with amplitue a, intensity a, an phase oscillating in time at angular frequency. This wave carries two pieces of information, its amplitue an angular frequency. Describing the wave in terms of a an is known as the frequency omain escription. In Fig..9, we plot the wavefunction in both the time an frequency omains. In the frequency omain, the wavefunction is escribe by a elta function at. Tools for the eact conversion between time an frequency omains will be presente in the net section. Note that, by convention we use a capitalize function (A instea of ) to represent the wavefunction in the frequency omain. Note also that the convention in quantum mechanics is to use a negative sign in the phase when representing the angular frequency +. This is convenient for escribing plane waves of the form e i(k-t). But it is eactly opposite to the usual convention in signal analysis (i.e. 6.3). In general, when you see i instea of j for the square root of -, use this convention in the time an frequency omains. Note the information in any constant phase offset,, as in epi t i amplitue prefactor, i.e. a epi t, where a epi. can be containe in the

9 Introuction to Nanoelectronics Re Time Frequency i t t e A Re t Im Comple plane Im Comple plane Fig..9. Representations of the angular frequency in time an frequency omains. Similarly, consier the wavefunction ae ik (.) which escribes a wave with amplitue a, intensity a, an phase oscillating in space with spatial frequency or wavenumber, k. Again, this wave carries two pieces of information, its amplitue an wavenumber. We can escribe this wave in terms of its spatial frequencies in k-space, the equivalent of the frequency omain for spatially oscillating waves. In Fig.., we plot the wavefunction in real space an k-space. Re Real space k-space ik e Ak k k Re k k Im Im Fig... Representations of wavenumber k in real space an k-space. Net, let s consier the wavefunction ae it A (.3) which escribes a wave with amplitue a, intensity a, an phase oscillating in the frequency omain with perio /t. This wave carries two pieces of information, its amplitue an the time t. In Fig.., we plot the wavefunction in both the time an frequency omains.

10 Part. The Quantum Particle Re Time Re Frequency A e it t t Im Im Fig... Representations of time t in time an frequency omains. Finally, consier the wavefunction Ak ae ik (.4) which escribes a wave with amplitue a, intensity a, an phase oscillating in k-space with perio /. This waves carries two pieces of information, its amplitue an the position. In Fig.., we plot the wavefunction in both real space an k-space. Real space k-space Re Ak e ik Re Comple plane Im Im Fig... Representations of position in real space an k-space. Comple plane k Note that, by convention we use a capitalize function (A instea of ) to represent the wavefunction in the k-space omain. Observe in Fig..9-Fig.. that a precise efinition of both the position in time an the angular frequency of a wave is impossible. A wavefunction with angular frequency of precisely is uniformly istribute over all time. Similarly, a wavefunction associate with a precise time t contains all angular frequencies. In real an k-space we also cannot precisely efine both the wavenumber an the position. A wavefunction with a wavenumber of precisely k is uniformly istribute over

11 Introuction to Nanoelectronics all space. Similarly, a wavefunction localize at a precise position contains all wavenumbers. Linear combinations of waves Net, we consier the combinations of ifferent comple eponential functions. For eample, in Fig..3 we plot a wavefunction that coul escribe an electron that equiprobable at position an position. The k-space representation is simply the superposition of two comple eponential functions corresponing to an. ik ik c c A c e c e (.5) Real space Re k-space ce ik Im Re Comple plane c c c c Fig..3. The k-space wavefunction corresponing to two positions an is simply the superposition of the k-space representations of (- ) an (- ). We can also generalize to an arbitrary istribution of positions, (). If () escribes an electron, for eample, the probability that the electron is locate at position is (). Thus, in k-space the electron is escribe by the sum of comple eponentials e -ik each oscillating in k-space an weighte by amplitue (). ik A k e (.6) You may recognize this from 6.3 as a Fourier transform. Similarly, the inverse transform is Im Im Re Comple plane Comple plane + ce ik k k Note that this wave function is not actually normalizable. 3

12 Part. The Quantum Particle ik Ak e k. (.7) To convert between time an angular frequency, use an i t A t e t (.8) it t Ae. (.9) Note that the factors of / are present each time you integrate with respect to k or. Note also that when converting between comple eponentials an elta functions, the following ientity is useful: Wave packets an uncertainty u ep iu. (.3) We now have two ways to escribe an electron. We coul escribe it as a plane wave, with precisely efine wavenumber an angular frequency: ikt, t e. (.3) But as we have seen, the intensity/probability ensity of the plane wave is uniform over all space (an all time). Thus, the position of the electron is perfectly uncertain it is probability istribution is uniform everywhere in the entire universe. Consequently, a plane wave is not usually a goo escription for an electron. On the other han, we coul escribe the electron as an iealize point particle, t, t t (.3) eisting at a precisely efine position an time. But the probability ensity of the point particle is uniform over all of k-space an the frequency omain. We will see in the net section that this means the energy an momentum of the electron is perfectly uncertain, i.e. arbitrarily large electron energies an momenta are possible. The only alternative is to accept an imprecise escription of the electron in both real space an k-space, time an the frequency omain. A localize oscillation in both representations is calle a wave packet. A common wavepacket shape is the Gaussian. For eample, instea of the elta function, we coul escribe the electron s position as aep. (.33) 4 This function was chosen such that the probability istribution of the electron is 4

13 Introuction to Nanoelectronics Real space k-space ep 8 ep k A k ψ()..5 A(k) / 8.5 k =/ / k/ k Fig..4. A normalize Gaussian. The Gaussian has the same shape in real space an k-space. Note that a Gaussian approimates a elta function in the limit. a ep (.34) where is the stanar eviation. measures the with of the Gaussian an is often thought of as the uncertainty in the position of the electron. The constant, a, is etermine by normalizing the probability ensity over all space, i.e., integrating Eq. (.34) over, we get a (.35) 4 Strictly, the uncertainty of a given quantity is efine by (.36) where signifies the average or epectation value of. Because is a constant Eq. (.36) may be simplifie: (.37) We will leave it as an eercise to show that for the Gaussian probability ensity: (.38) 5

14 Part. The Quantum Particle Thus, the Gaussian is a convenient choice to escribe a wavepacket because it has a reaily efine uncertainty. In k-space, the electron is also escribe by a Gaussian (this is another of the convenient properties of this function). Application of the Fourier transform in Eq. (.6) an some algebra gives 4 A k 8 ep k (.39) The probability istribution in k-space is 8 ep k A k (.4) Thus, the uncertainty in k-space is k (.4) The prouct of the uncertainties in real an k-space is k. (.4) The prouct k k. Thus, Eamples of wavepackets k. (.43) A typical Gaussian wavepacket is shown in Fig..5 in both its real space an k-space representations. Initially the probability istribution is centere at = an k =. If we shift the wavepacket in k-space to an average value <k> = k, this is equivalent to multiplying by a phase factor ep[ik ] in real space. Similarly, shifting the center of the wavepacket in real space to <> = is equivalent to multiplying the k-space representation by a phase factor ep[-ik ]. Real coorinates (,t) shift by Inverse coorinates (k,) ep[-ik ] ep[ik ] shift by k shift by t ep[it ] ep[-i t] shift by Table.. A summary of shift transformations in real an inverse coorinates. 6

15 Introuction to Nanoelectronics (a) Re Real space Re k-space Ak Im Comple plane Im Comple plane k (b) Re Re Akepik Im Comple plane Im Comple plane k (c) Re epik Re A k k k k Comple plane Im Comple plane Im Fig..5. Three Gaussian wavepackets. In (a) the average position an wavenumber of the packet is = an k=, respectively. In (b) the average position has been shifte to <>=. In (c) the average wavenumber has been shifte to <k>=k. Epectation values of position Given that P() is the probability ensity of the electron at position, we can etermine the average, or epectation value of from P P Of course if the wavefunction is normalize then the enominator is. We coul also write this in terms of the wavefunction (.44) 7

16 Part. The Quantum Particle Where once again if the wavefunction is normalize then the enominator is. (.45) Since *, Bra an Ket Notation * * (.46) Also known as Dirac notation, Bra an Ket notation is a convenient shorthan for the integrals above. The wavefunction is represente by a Ket: (.47) The comple conjugate is represente by a Bra: * (.48) Together, the bracket (hence Bra an Ket) symbolizes an integration over all space: * (.49) Thus, in short form the epectation value of is (.5) Parseval s Theroem It is often convenient to normalize a wavepacket in k space. To o so, we can apply Parseval s theorem. Let s consier the bracket of two functions, f() an g() with Fourier transform pairs F(k) an G(k), respectively.. * f g f g (.5) Now, replacing the functions by their Fourier transforms yiels 8

17 Introuction to Nanoelectronics * * ik ik f g F ke k G k e k Rearranging the orer of integration gives (.5) * ik ik F k e k G k e k (.53) * ik k F k G k e kk From Eq. (.3) the integration over the comple eponential yiels a elta function * i kk F k Gk e kk F k * G k k k k k Thus, * * f g F k G k k (.55) It follows that if a wavefunction is normalize in real space, it is also normalize in k- space, i.e., AA (.56) where (.54) Epectation values of k an * A A Ak Ak k (.57) The epectation value of k is obtaine by integrating the wavefunction over all k. This must be performe in k-space. Ak * kak k A k A k (.58) AA Ak * Ak k From the Inverse Fourier transform in k-space ik Ak e k (.59) note that ik i kak e k (.6) Thus, we have the following Fourier transform pair: i k Ak (.6) 9

18 Part. The Quantum Particle It follows that A k A i k (.6) AA Similarly, from the Inverse Fourier transform in the frequency omain it t Ae (.63) We can erive the Fourier transform pair: i t A t (.64) It follows that A A i t. (.65) AA We efine two operators ˆ k i (.66) an ˆ i. (.67) t Operators only act on functions to the right. To signify this ifference we mark them with a caret. We coul also efine the (somewhat trivial) position operator ˆ. (.68) The Commutator One must be careful to observe the correct orer of operators. For eample, ˆk ˆ kˆ ˆ (.69) but ˆ ˆ ˆ ˆ. (.7) In quantum mechanics we efine the commutator: qˆ, rˆ qˆ rˆ rˆ qˆ (.7) We fin that the operators ˆ an ˆ commute because ˆ, ˆ. Consiering the operators ˆ an ˆk : ˆ, ˆ k i i (.7) We have applie Parseval s theorem; see the Problem Sets. 3

19 Introuction to Nanoelectronics To simplify this further we nee to operate on some function, f(): ˆ f ˆ, k f i i f f f i if i if Thus, the operators ˆ an ˆk o not commute, i.e. ˆ, kˆ i (.73) (.74) Although we use Fourier transforms, Eq. (.43) can also be erive from the relation (.74) for the non-commuting operators operators ˆ an ˆk. It follows that all operators that o not commute are subject to a similar limit on the prouct of their uncertainties. We shall see in the net section that this limit is known as the uncertainty principle. 3

20 Electron Kinetic Energy Part. The Quantum Particle Momentum an Energy Two key eperiments revolutionize science at the turn of the th century. Both eperiments involve the interaction of light an electrons. We have alreay seen that electrons are best escribe by wavepackets. Similarly, light is carrie by a wavepacket calle a photon. The first phenomenon, the photoelectric effect, was eplaine by assuming that a photon s energy is proportional to its frequency. The secon phenomenon, the Compton effect, was eplaine by proposing that photons carry momentum. That light shoul possess particle properties such as momentum was completely unepecte prior to the avent of quantum mechanics. (i) The Photoelectric Effect It is not easy to pull electrons out of a soli. They are boun by their attraction to positive nuclei. But if we give an electron in a soli enough energy, we can overcome the bining energy an liberate an electron. The minimum energy require is known as the work function, W. e - Fig..6. The photoelectric effect: above a critical frequency, light can liberate electrons from a soli. By bombaring metal surfaces with light, it was observe that electrons coul be liberate only if the frequency of the light eceee a critical value. Above the minimum frequency, electrons were liberate with greater kinetic energy. Einstein eplaine the photoelectric effect by postulating that, in a photon, the energy was proportional to the frequency: E (.75) Where h = Js is Planck s constant, an is shorthan for h. Note the units for Planck s constant energy time. This is useful to remember when checking that your quantum calculations make sense. Thus, the kinetic energy of the emitte electrons is given by electron kinetic energy W. (.76) This technique is still use to probe the energy structure of materials W 3

21 Introuction to Nanoelectronics Note that we will typically epress the energy of electrons in electron Volts (ev). The SI unit for energy, the Joule, is typically much too large for convenient iscussion of electron energies. A more convenient unit is the energy require to move a single electron through a potential ifference of V. Thus, ev = q J, where q is the charge on an electron (q ~.6-9 C). (ii) The Compton Effect If a photon collies with an electron, the wavelength an trajectory of the photon is observe to change. After the collision the scattere photon is re shifte, i.e. its frequency is reuce an its wavelength etene. The trajectory an wavelength of the photon can be calculate by assuming that the photon carries momentum: p k, (.77) where the wavenumber k is relate to frequency by k, (.78) c where c is the spee of light. Scattere photon cky p k y yˆ p k ˆ c k k y p k ˆ k yˆ y y e - Incient photon Fig..7. The wavelength shift of light after collision with an electron is consistent with a transfer of momentum from a photon to the electron. The loss of photon energy is reflecte in a re shift of its frequency. These two relations: E an p k are strictly true only for plane waves with precisely efine values of ω an k. Otherwise we must employ operators for momentum an energy. Base on the operators we efine earlier for k an, we efine operators for momentum pˆ i (.79) 33

22 Part. The Quantum Particle an energy ˆ E i (.8) t Recall that each operator acts on the function to its right; an that ˆp is not necessarily equal to ˆp. The Uncertainty Principle Now that we see that k is simply relate to momentum, an is simply relate to energy, we can revisit the uncertainty relation of Eq. (.43) k, (.8) which after multiplication by ħ becomes Dp. D. (.8) This is the celebrate Heisenberg uncertainty relation. It states that we can never know both position an momentum eactly. For eample, we have seen from our Fourier transform pairs that to know position eactly means that in k-space the wavefunction is (k) = ep[-ik ]. Since (k) = all values of k, an hence all values of momentum are equiprobable. Thus, momentum is perfectly unefine if position is perfectly efine. Uncertainty in Energy an Time For the time/frequency omain, we take Fourier transforms an write DE. Dt. (.83) However, time is treate ifferently to position, momentum an energy. There is no operator for time. Rather it is best thought of as a parameter. But the epression of Eq. (.83) still hols when Dt is interprete as a lifetime. Application of the Uncertainty Principle The uncertainty principle is not usually significant in every ay life. For eample, if the uncertainty in momentum of a g billiar ball traveling at a velocity of m/s is %, we can in principle know its position to D = (ħ /)/(./) = 3-3 m. m ~ g v ~ m/s if Dp = %, then D = 3-3 m Fig..8. The uncertainty principle is not very relevant to everyay objects. 34

23 + - Introuction to Nanoelectronics In nanoelectronics, however, the uncertainty principle can play a role. For eample, consier a very thin wire through which electrons pass one at a time. The current in the wire is relate to the transit time of each electron by e - V DS Fig..9. A nanowire that passes one electron at a time. q I, (.84) where q is the charge of a single electron. To obtain a current of I =. ma in the wire the transit time of each electron must be q.6 fs, (.85) I The transit time is the time that electron eists within the wire. Some electrons may travel through the wire faster, an some slower, but we can approimate the uncertainty in the electron s lifetime, Dt = =.6 fs. From Eq. (.83) we fin that DE =. ev. Thus, the uncertainty in the energy of the electron is equivalent to a ranom potential of approimately. V. As we shall see, such effects funamentally limit the switching characteristics of nano transistors. Schröinger s Wave Equation The energy of our electron can be broken into two parts, kinetic an potential. We coul write this as total energy kinetic energy potential energy (.86) Now kinetic energy is relate to momentum by p kinetic energy mv (.87) m Thus, using our operators, we coul write ˆ pˆ E, t, t Vˆ, t (.88) m Another way to think about this is to consier the aition of an electron to the nanowire. If current is to flow, that electron must be able to move from the wire to the contact. The rate at which it can o this (i.e. its lifetime on the wire) limits the transit time of an electron an hence the current that can flow in the wire. Recall that moern transistors operate at voltages ~ V. So this uncertainty is substantial. 35

24 Part. The Quantum Particle Where V ˆ is the potential energy operator. Vˆ V, t. (.89) We can rewrite Eq. (.88) in even simpler form by efining the so calle Hamiltonian operator ˆ pˆ H Vˆ. (.9) m Now, Eˆ Hˆ. (.9) Or we coul rewrite the epression as i, t, t V, t, t. (.9) t m All these equations are statements of Schröinger s wave equation. We can employ whatever form is most convenient. A Summary of Operators Note there is no operator for time. Position ˆ Energy Ê Wavenumber Angular Frequency Momentum ˆk ˆ ˆp i i t i Potential Energy Kinetic Energy Hamiltonian Vˆ Tˆ Ĥ i t V pˆ m pˆ ˆ m V Table.. All the operators use in this class. The Time Inepenent Schröinger Equation When the potential energy is constant in time we can simplify the wave equation. We assume that the spatial an time epenencies of the solution can be separate, i.e., t t (.93) Substituting this into Eq. (.9) gives i t t V t (.94) t m t yiels Diviing both sies by 36

25 Introuction to Nanoelectronics i t V. (.95) t t m Now the left sie of the equation is a function only of time while the right sie is a function only of position. These are equal for all values of time an position if each sie equals a constant. That constant turns out to be the energy, E, an we get two couple equations E t i t (.96) t an E V. (.97) m The solution to Eq. (.96) is E t ep i t (.98) Thus, the complete solution is E, t ep i t (.99) By separating the wavefunction into time an spatial functions, we nee only solve the simplifie Eq. (.97). There is much more to be sai about this equation, but first let s o some eamples. Free Particles In free space, the potential, V, is constant everywhere. For simplicity we will set V =. Net we solve Eq. (.97) with V =. E m (.) Rearranging slightly gives the secon orer ifferential equation in slightly clearer form me (.) A general solution is ep ik (.) where me k (.3) Inserting the time epenence (see Eq. (.99)) gives, t ep i k t (.4) where 37

26 Part. The Quantum Particle E k. (.5) m Thus, as epecte the solution in free space is a plane wave. The Square Well Net we consier a single electron within a square potential well as shown in Fig... As mentione in the iscussion of the photoelectric effect, electrons within solis are boun by attractive nuclear forces. By moeling the bining energy within a soli as a square well, we entirely ignore fine scale structure within the soli. Hence the square well, or particle in a bo, as it is often known, is one of the cruest approimations for an electron within a soli. The simplicity of the square well approimation, however, makes it one of the most useful problems in all of quantum mechanics. V Fig... The square well. Within the soli the potential is efine to be zero. In free space, outsie the soli, the repulsive potential is V. Since the potential changes abruptly, we treat each region of constant potential separately. Subsequently, we must connect up the solutions in the ifferent regions. Tackling the problem this way is known as a piecewise solution. Now, the Schröinger Equation is a statement of the conservation of energy total energy E kinetic energy KE potential energy V (.6) In classical mechanics, one can never have a negative kinetic energy. Thus, classical mechanics requires that E > V. This is known as the classically allowe region. But in our quantum analysis, we will fin solutions for E < V. This is known as the classically forbien regime. (i) The classically allowe region V = -L/ Rearranging Eq. (.97) gives the secon orer ifferential equation: me V (.7) 38 L/

27 Introuction to Nanoelectronics In this region, E > V, solutions are of the form Ae Be (.8) or where ik ik Asin k Bcos k (.9) In the classically allowe region we have oscillating solutions. (ii) The classically forbien region m E V k (.) In this region, E < V, an solutions are of the form Ae Be (.) i.e. they are either growing or ecaying eponentials where m V E. (.) Matching piecewise solutions The Schröinger Equation is a secon orer ifferential equation. From Eq. (.7) we observe the secon erivative of the wavefunction is finite unless either E or V is infinite. Infinite energies are not physical, hence if the potential is finite we can conclue that an are continuous everywhere. That is, at the bounary ( = ) between piecewise solutions, we require that (.3) an Boun solutions (.4) Electrons with energies within the well ( < E < V ) are boun. The wavefunctions of the boun electrons are localize within the well an so they must be normalizable. Thus, the wavefunction of a boun electron in the classically forbien region (outsie the well) must ecay eponentially with istance from the well. A possible solution for the boun electrons is then 39

28 Part. The Quantum Particle where an an A, B, C an D are constants. Ce for L Acos k Bsin k for L L De for L The limit that V (the infinite square well) (.5) m V E (.6) me k, (.7) At the walls where the potential is infinite, we see from Eq. (.6) that the solutions ecay immeiately to zero since. Thus, in the classically forbien region of the infinite square well. The wavefunction is continuous, so it must approach zero at the walls. A possible solution with zeros at the left bounary is Asin k L (.8) where k is chosen such that ψ = at the right bounary also: kl n (.9) To normalize the wavefunction an etermine A, we integrate: L cos L A n L n (.) From which we etermine that A L (.) The energy is calculate from Eq. (.7) k n E. (.) m ml The first few energies an wavefunctions of electrons in the well are plotte in Fig... Two characteristics of the solutions eserve comment:. The boun states in the well are quantize only certain energy levels are allowe.. The energy levels scale inversely with L. As the bo gets smaller each energy level an the gaps between the allowe energy levels get larger. 4

29 Energy ψ() Introuction to Nanoelectronics (a) Energy solutions (b) Wavefunction solutions Fig... The lowest four states for a single electron in an infinite square well. Note that we have plotte ψ() not ψ(). The Finite Square Well When the confining potential is finite, we can no longer assume that the wavefunction is zero at the bounaries of the well. For a finite confining potential, the wavefunction penetrates into the barrier: the lower the confining potential, the greater the penetration. From Eq. (.5) the general solution for the wavefunction within the well was Acos k Bsin k, (.3) but from the solutions for the infinite well, we can see that, within the well, the wavefunction looks like Acos k (.4) or E = -L/ L/ ml Asin k -L/ (.5) The simplification is possible because the well potential is symmetric aroun =. Thus, the probability ensity ψ() shoul also be symmetric, an inee both ψ() = sin(k) an ψ() = cos(k) have symmetric probability istributions even though ψ() = sin(k) is an antisymmetric wavefunction. We ll consier the symmetric (cos(k)) an antisymmetric (sin(k)) wavefunction solutions separately. L/ After all, if the potential is the same in both left an right halves of the well, there is no reason for the electron to be more probable in one sie of the well. 4

30 Part. The Quantum Particle (i) Symmetric wavefunction We can assume a solution of the form: e for L Acos k for L L (.6) e for L where an me k (.7) m V E (.8) Note the solution as written is not normalize. We can normalize it later. The net step is to evaluate the constant A by matching the piecewise solutions at the ege of the well. We nee only consier one ege, because we have alreay fie the symmetry of the solution. At the right ege, equating the amplitue of the wavefunction gives L Acos kl ep L (.9) Equating the slope of the wavefunction gives ' L kasin kl ep L (.3) Diviing Eq. (.3) by Eq. (.9) to eliminate A gives tan kl k. (.3) But an k are both functions of energy E V E tan E (.3) L E where we have efine the infinite square well groun state energy E L. (.33) ml As in the infinite square well case, we fin that only certain, iscrete, values of energy give a solution. Once again, the energies of electron states in the well are quantize. To obtain the energies we nee to solve Eq. (.3). Unfortunately, this is a transcenental equation, an must be solve numerically or graphically. We plot the solutions in Fig... 4

31 tan(/. E/E L ), -cot(/. E/E L ) V /E - Introuction to Nanoelectronics (ii) Antisymmetric wavefunction Antisymmetric solutions are foun in a similar manner to the symmetric solutions. We first assume an antisymmetric solution of the form: e for L Asin k for L L (.34) e for L Then, we evaluate the constant A by matching the piecewise solutions at the ege of the well. Again, we nee only consier one ege, because we have alreay fie the symmetry of the solution. At the right ege, equating the amplitue of the wavefunction gives L Asin kl ep L (.35) Equating the slope of the wavefunction gives ' L kacos kl ep L (.36) Diviing Eq. (.3) by Eq. (.9) to eliminate A gives cot kl k. (.37) Epaning an k in terms of energy E V E cot E. (.38) L E In Fig.., we solve for the energy. Note that there is always at least one boun solution no matter how shallow the well. In Fig..3 we plot the solutions for a confining potential V = 5.E L V =E L V =5E L V =E L Energy (E/E L ) Fig... A graphical solution for the energy in the finite quantum well. The green an blue curves are the LHS of Eqns. (.3) an (.38), respectively. The re curves are the RHS for ifferent values of the confining potential V. Solutions correspon to the intersections between the re lines an the green or blue curves. 43

32 Part. The Quantum Particle Energy/E L Energy ψ() (a) Energy solutions 6 (b) Wavefunction solutions 5 E=4.68E L 4 3 E=.3E L E=.6E L -L/ L/ -L/ L/ Fig..3. The three boun states for electrons in a well with confining potential V = 5.E L. Note that the higher the energy, the lower the effective confining potential, an the greater the penetration into the barriers. Potential barriers an Tunneling Net we consier electrons incient on a potential barrier, as shown in Fig..4. V L Fig..4. A potential barrier. We will assume that the particle is incient on the barrier from the left. It has some probability of being reflecte by the barrier. But it also has some probability of being transmitte even though its energy may be less than the barrier height. Transmission through a barrier is known as tunneling. There is no equivalent process in classical physics the electron woul nee sufficient energy to jump over the barrier. Once again, we solve the time-inepenent Schröinger Equation. To the left an right of the barrier, the electron is in a classically allowe region. We moel the electron in these regions by a plane wave; see Eq. (.8) an the associate iscussion. On the other han, within the barrier, if the energy, E, of the electron is below the barrier potential, V, 44

33 Introuction to Nanoelectronics the barrier is a classically forbien region. The solution in this region is escribe by epaning an ecaying eponentials; see Eq. (.) an the associate iscussion. Analyzing the potential piece by piece, we assume a solution of the form ik ik e re for ae be for L ik te for L where once again an (.39) me k (.4) m V E (.4) The intensity of the incoming plane wave is unity. Hence the amplitue of the reflecte wave, r, is the reflection coefficient an the amplitue of the transmitte wave, t, is the transmission coefficient. (The reflectivity an transmissivity is r an t, respectively). Net we match the piecewise solutions at the left ege of the barrier. Equating the amplitue of the wavefunction gives (.4) r a b Equating the slope of the wavefunction gives 'ikikra b. (.43) At the right ege of the barrier, we have L ae be te (.44) an ' L L ikl L a e b e ikte L L ikl. (.45) Thus, we have four simultaneous equations. But these are a pain to solve analytically. In Fig..5 we plot solutions for energy much less than the barrier, an energy close to the barrier. It is observe that the tunneling probability is greatly enhance when the incient electron has energy close to the barrier height. Note that the wavefunction ecay within the barrier is much shallower when the energy of the electron is large. Note also that the reflection from the barrier interferes with the incient electron creating an interference pattern. When the electron energy is much less than the barrier height we can moel the wavefunction within the barrier as simply a ecaying eponential. The transmission probability is then approimately T ep L. (.46) 45

34 Energy Energy Part. The Quantum Particle E = V /5 E =.98V V E V E L Fig..5. Plots of the wavefunction for an electron incient from the left. (a) When the electron energy is substantially below the barrier height, tunneling is negligible. (b) For an electron energy 98% of the barrier height, however, note the non-zero transmission probability to the right of the barrier. L 46

35 Introuction to Nanoelectronics Problems. Suppose we fire electrons through a single slit with with. At the viewing screen behin the aperture, the electrons will form a pattern. Derive an sketch the epression for the intensity at the viewing screen. State all necessary assumptions. Fig..6. The geometry of a single slit iffraction eperiment. How oes this pattern iffer from the pattern with two slits iscusse in class?. Compare the ifferent patterns at the viewing screen with = Å an L = nm for fire electrons with wavelengths of Å, Å, an Å. Eplain. 3. Show that a shift in the position of a wavepacket by is equivalent to multiplying the k-space representation by ep[ ik ]. Also, show that a shift in the k-space representation by k is equivalent to multiplying the position of the wavepacket by ep[ ik ]. Show that similar relations hol for shifts in time an frequency. 4. Fin F( ) where F( ) is the Fourier Transform of an eponential ecay: at F( ) [ e u( t)] where u(t) is the unit step function. 5. Show that if then ( D ) D 47

36 Part. The Quantum Particle where ( ) ep 4 6. Show the following: (a) k A k A AA i (b) A A i t AA 7. A free particle is confine to move along the -ais. At time t= the wave function is given by L e L L ik ( t, ) otherwise (a) What is the most probable value of momentum? (b) What are the least probable values of momentum? (c) Make a rough sketch of the wave function in k-space, A(k, t=). 8. The commutator of two operators  an ˆB is efine as Aˆ, Bˆ AB ˆ ˆ BA ˆ ˆ Evaluate the following commutators: 48

37 Introuction to Nanoelectronics (a) ˆ, ˆ (b) pp ˆ, ˆ (c) ˆ, pˆ () p ˆˆ, p ˆ ˆ 9. Consier the wavefunction Show that ( ) ep ( ic) k C (Hint: Show that k ). For the finite square well shown below, calculate the reflection an transmission coefficients (E > ). V a a V Fig..7. A finite potential well.. Consier an electron in the groun state of an infinite square well of with L. What is the epectation value of its velocity? What is the epectation value of its kinetic energy? Is there a conflict between your results? 49

38 GaAlAs GaAs GaAlAs GaAs GaAlAs Part. The Quantum Particle. Derive the reflection an transmission coefficients for the potentialv A where A>. V A, Fig..8. A elta function potential. One metho to solve is by taking the Schröinger equation with V A A E m an integrating both sies from to for very small to get the constraint A E m ma 3. Consier two quantum wells each with with w separate by istance. V w Fig..9. The structure of the quantum wells. (a) From your unerstaning of wavefunctions for a particle in a bo, plot the approimate probability ensity for each of the two lowest energy moes for this system when the two quantum wells are isolate from each other. (b) Plot the two lowest energy moes when the two quantum wells are brought close to each other such that <<w? 5

39 Introuction to Nanoelectronics (c) Net we wish to take a thin slice of another material an insert it in the structure above (where <<w) to kill one of the two moes of part (b) but leave the other unaffecte. How woul you choose the material (compare to the materials alreay present in the structure), an where shoul this new material be place? 5

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