Collective excitations in one-dimensional quantum dot arrays
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(2) Special thanks to Professor Angela Camacho for her support and for being the inspiration that brought this work to life; to Elena, Andres and Pedro for their help and encouragement; to all my Family and Friends for being there showing their support and hearing me even if they had no clue of what I was talking about..
(3) Contents Introduction. v. Chapter 1. Collective Excitation In Semiconductor Chains 1. Systems Couple By Coulomb Force 2. Systems Where Tunneling is Allowed. 1 2 4. Chapter 2. Applications 1. Systems Couple only by Coulomb Force 2. Systems Couple by Coulomb Force and Tunneling 3. Conclusions. 7 7 11 17. Appendix A. Mathematical Steps from Chapter 1 1. Fourier Transformation of The Potential 2. The Self-Consistent Approximation by Ehrenreich and Cohen 3. Calculation of the Matrix Elements. 19 19 20 24. Appendix B. Mathematical Steps from Chapter 2 1. Wave Functions 2. Solution of Equation 2.15. 27 27 27. Appendix. Bibliography. 29. iii.
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(5) Introduction On the last few decades it has been done a lot of research on the area of quantum dot systems, due to the broad field of applications thanks to their versatility. A few nano-scale applications based on the Plasmons are biological devices, perfect lenses and invisibility veil. One of the most important applications is in the growing need in electronics for a device that can transmit as much information as an optic fiber and can be construct in a nano-scale [1]. Most of the studies have been done in 3D, 2D and 1D systems[2][3][4][5]. In this work we focus on 0D systems (quantum dots)arranged in a line and coupled through Coulomb interaction. We also studied 1D arrays made of these quantum dots coupled through tunneling, which in the limit of zero lattice constant forms a quantum wire. Starting with a chain of cylindrical quantum dots completely uncoupled, but charged with a given density that permits a fast response to an external electric field parallel to the chain; we propose to analyze the propagation of plasmons through the chain due to pure Coulomb interaction and also study the effect of tunneling on the collective excitations. In addition we calculate the changes in the plasmon frequency as the lattice constant increases and the quantum dots become more and more uncoupled.. v.
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(7) CHAPTER 1. Collective Excitation In Semiconductor Chains This chapter is devoted to the study of collective excitations 1D semiconductor quantum dot arrays. The first part develops a theory of collective excitations based on The Self-Consistent Approximation by Ehrenreich and Cohen[6] (Appendix A), then is applied to systems where only Coulomb coupling is allowed; Subsequently, this theory is extended to systems where tunneling is present. In both cases the dots are longitudinally arranged and periodically spaced in the x direction, as can be seen in Figure1. Our system consists of a chain of quantum dots of InxGa1-xAs embedded in a matrix of AlxGa1-xAs [2], as shown in Chapter 3.. Figure 1. Schematic representation of the system, for quantum dots of disk geometry. In the x direction the dots are periodically spaced. We start with the integral form of the 3D Poisson equation Z d~r0 e2 n(~r0 ) V (~r) = |~r − ~r0 |. (1.1). in which the source of the potential is a charge distribution given by n(~r0 ), and = 4πo b where b is the background dielectric constant, making the Fourier transformation of it, in the x direction(Appendix A), we obtain Z Z h p i p 4e2 V (qx , y, z) = y 2 + z 2 − y 02 + z 02 dy 0 dz 0 (1.2) n(qx , y 0 , z 0 )K0 qx Where K0 is the Bessel funtion. In this study we base are modeling of the excitations in the Self-Consistent Approximation (developed in detail in the Appendix A). This method describes the behavior of an electron charge distribution under the effect of an electromagnetic perturbation, when the distribution is subject to a self consistent potential. Furthermore, Ehrenreich and Cohen show that the for the a polarized electromagnetic perturbation, in a 3D system, the dielectric function is going to 1.
(8) 2. 1. COLLECTIVE EXCITATION IN SEMICONDUCTOR CHAINS. depend in the potential, the energy levels and the occupation of the levels in the system, as well as in the frequency of the charge oscillation (ω, t) = 1 − vg. X [f (Ek+q ) − f (Ek )] k. 2. (1.3). (Ek+q − Ek − ~ω). is the 3D potential, f (Ek+q ) and Ek+q are the corresponding where vg = 4πe q2 Fermi function and energy of the k+q state. Ecuation 1.3 has to be equal to cero in order to have Plasmon oscilations[7], so that the frequency of the charge oscillation can be found by means of the energy levels, the occupation of the energy levels of the system and the potential. By using the self consistent approximation[6] we get an expression for the density response in terms of the potential X X 2 2 hkx , i| V (x) |kx0 , i0 i hkx0 , i0 | eiqx x |kx , ii n(qx , y 0 , z 0 ) = 2 |ψy0 | |ψz0 | i,i0 kx ,kx0. ×. f (i0 ) − f (i) E(i0 ) − E(i) + ~ω. (1.4). Here f(i) and E(i) are the corresponding Fermi function and energy of the i th state; and ψy and ψz are going to be the wave functions for the lowest state in the y and z directions. After replacing 1.4 into 1.2, and integrating over the y and z degrees of freedom, the potential takes the form V (qx ) = I(qx ). X X i,i0. hkx , i| V (x) |kx0 , i0 i hkx0 , i0 | eiqx x |kx , ii. kx ,kx0. × where V (qx ) = and I(qx ) =. 8e2 . Z Z Z Z. Z Z. 2. f (i0 ) − f (i) E(i0 ) − E(i) + ~ω. (1.5). 2. V (qx , y, z) |ψy | |ψz | dydz. i h p p y 2 + z 2 − y 02 + z 02 dy 0 dz 0 dydzK0 qx 2. 2. 2. 2. × |ψy0 | |ψz0 | |ψy | |ψz |. Equation 1.5 is a general expression that can be use as the starting point for studies of collective excitations in any system that is longitudinally arranged. In the following sections, equation 1.5 is going to be applied to systems coupled first by only Coulomb force and then to systems where tunneling is allowed.. 1. Systems Couple By Coulomb Force In the theory exposed it is assumed that the dots are electrically isolated, which means that the electrons do not move from one dot to another, in other words tunneling is not allowed. However, the dots are coupled by Coulomb’s force, which leads to collective excitations of the system. The wave functions of the dots for the y and z directions depend on the confinement, the geometry. As mentioned.
(9) 1. SYSTEMS COUPLE BY COULOMB FORCE. 3. before, on the x direction the dots are periodically spaced and the wave functions for different dots do not overlap, so the tight-binding wave functions are perfectly suitable for the system.[2]Hence, the wave function in the x direction is |kx , ii =. X. eikx ld ϕi (x − ld). (1.6). l. where d is the period of the arrange and i labels the energy state. To begin, recall equation 1.5, from now on referred as equation 1.7 V (qx ) = I(qx ). X X. hkx , i| V (x) |kx0 , i0 i hkx0 , i0 | eiqx x |kx , ii. i,i0 kx ,kx0. ×. f (i0 ) − f (i) E(i0 ) − E(i) + ~ω. (1.7). The calculation of the matrix products in 1.7 are done in Appendix A, for now the results are hkx0 , i0 | eiqx x |kx , ii = δkx0 −kx −qx ,nG Aii0 (qx ) Z =⇒ Aii0 (qx ) = ϕ∗i0 (x)eiqx x ϕi (x)dx. (1.8). and hkx , i| V (x) |kx0 , i0 i =. X. V (qx + n0 G)A∗ii0 (qx + n0 G). (1.9). n0. Where G = 2π d and n takes in the multipoles contributions. Substituting 1.8 and 1.9 in 1.7 gives an expression for the potential in terms of the wave equations, the polarization and the potential itself. V (qx ) = I(qx ). X X X i,i0. 0 kx ,kx. V (qx + n0 G)A∗ii0 (qx + n0 G)δkx0 −kx −qx ,nG Aii0 (qx ). n0. × V (qx ) = I(qx ). X. Πii0 (ω)Aii0 (qx + nG). i,i0. X. f (i0 ) − f (i) (1.10) E(i0 ) − E(i) + ~ω. V (qx + n0 G)A∗ii0 (qx + n0 G). (1.11). n0. and the polarization is going to be Πii0 (ω) =. X kx. f (i0 ) − f (i) E(i0 ) − E(i) + ~ω. For sake of simplicity, the potential is denoted as V (qx + nG) = Vn where (qx + nG) = qn . Finally, the potential can be expressed as. (1.12).
(10) 4. 1. COLLECTIVE EXCITATION IN SEMICONDUCTOR CHAINS. Vn = I(qn ). X. Πii0 (ω)Aii0 (qn ). i,i0. X. Vn0 A∗ii0 (qn0 ). (1.13). n0. Equation 1.13 is a general self-consistent expression that can be used in modelling collective excitation in any system that obeys the conditions and approximations expressed in the beginning of this chapter. In the first section of Chapter 2 the reader will see an application of equation 1.13 to a system with parabolic confinement on the three directions.. 2. Systems Where Tunneling is Allowed We extend in this section our model introducing tunnelling. The wave functions are going to be the same |kx , ii =. X. eikx ld ϕi (x − ld). (1.14). l. and the electronic band structure taking into account only first nearest neighbors is Ei = i − W i Cos(kx dx ). (1.15). where i is the energy of the electron in the i th orbital(on site). As done before, it begins, recalling equation 1.5, from now on referred as equation 1.16 V (qx ) = I(qx ). X X. hkx , i| V (x) |kx0 , i0 i hkx0 , i0 | eiqx x |kx , ii. i,i0 kx ,kx0. ×. f (i0 ) − f (i) E(i0 ) − E(i) + ~ω. (1.16). the calculation of the matrix products is done in Appendix A, for now the results are hkx0 , i0 | eiqx x |kx , ii =. X. 0. eid(kx l−kx L) δkx0 −kx −qx ,nG Alii0 (qx ). (1.17). l. =⇒ Alii0 (qx ) =. hkx , i| V (x) |kx0 , i0 i =. X. Z. ϕ∗i0 (x)eiqx x ϕi (x + ld)dx. 0. e−id(kx l−kx L). L. replacing this into equation 1.16. X n0. 0 V (qx + n0 G)AL∗ ii0 (qx + n G). (1.18).
(11) 2. SYSTEMS WHERE TUNNELING IS ALLOWED. V (qx ) =. X XX. 0 kx kx. ii0. I(qx ). X. 0 id(kx l−kx L). 0 V (qx + n0 G)AL∗ ii0 (qx + n G)e. n0. lL. 0. e−id(kx l−kx L) δkx0 −kx −qx ,nG Alii0 (qx ) V (qx ) =. XX ii0. I(qx + nG). X. !. ×. f (i0 ) − f (i) E(i0 , qx ) − E(i, qx ) + ~ω !. 0 l V (qx + n0 G)AL∗ ii0 (qx + n G) Aii0 (qx + nG)Πii0 (ω, qx ). n0. lL. 5. where Πii0 (ω, qx ) is the polatization. Πii0 (ω, qx ) =. X kx. f (i0 ) − f (i) − E(i, qx ) + ~ω. E(i0 , qx ). For sake of simplicity qx + nG = qn V (qx + nG) = Vn consequently the self-consistent equation for the potential y going to be V (qn ) =. XX ii0. lL. I(qn )Alii0 (qn ). X. V (qn0 )AL∗ ii0 (qn0 )Πii0 (ω, qx ). (1.19). n0. The above equation is the analog of equation 1.13. It is a general expression useful to model collective excitation in 1D quantum dot arrays, longitudinal and periodically arranged couple by Coulomb force allowing tunneling..
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(13) CHAPTER 2. Applications In this chapter the reader will find applications of the theories developed in the first chapter of this work, for a two level system of uncoupled quantum dots forming a 1D array, first by analyzing only the effect of the Coulomb interaction and then introducing a weak coupling between the dots due to tunneling. As well as a comparison of the results with previous works about collective excitations. The applications are done with systems of two energy levels, and parabolic confinement in all three directions.. 1. Systems Couple only by Coulomb Force Following reference [2], it will be assumed that the energy difference between the first two levels is going to be of 25meV. The dots have a radius of 7.5 nm and a width of 5 nm. For a two level system equation 1.13 of Chapter 1 becomes. Figure 1. Schematic representation of the system.. Vn = I(qn ) Π01 (ω)A01 (qn ). X. Vn0 A∗01 (qn0 ) + Π10 (ω)A10 (qn ). n0. X n0. Vn0 A∗10 (qn0 ). !. (2.1). where it is considered that the confinement in y and z directions is parabolic of the " E 2 2 10 form Vc = m (y + z 2 ) at all times. So in cylindrical coordinates 2 ~ 7.
(14) 8. 2. APPLICATIONS. I(qn ) =. 8e2 . Z. 2π. 0. Z. 2π. 0. Z. 7.5. 0. Z. 7.5. In [qx (ρ − ρ0 )]. 0. ×e−. mE10 ~2. ρ2 −. e. mE10 ~2. ρ02. ρρ0 dθdθ0 dρdρ0. (2.2). Here we make an aproximation, K0 [qx (ρ − ρ0 )] → In [qx (ρ − ρ0 )], wich is valid only " E 2 2 10 for very small qx [4][5][8]. For parabolic confinement of the form Vc = m x 2 ~ in the x direction A01 (qn ) =. . 4 π2. 1/4. mE10 ~2. Z. 2.5. (x + ld)eiqn x e−. mE10 ~2. x2. dx = A10 (qn ). (2.3). −2.5. A01 (qn ) = A10 (qn ) because the wave functions of the single dots are real, therefore equation 2.1 becomes Vn = I(qn ) (Π01 (ω) + Π10 (ω)) A01 (qn ). X. Vn0 A∗01 (qn ). n0. |. {z. g(qn0 ). Vn = I(qn ) (Π01 (ω) + Π10 (ω)) A01 (qn )g(qn ). }. (2.4). Wave functions for each confinement and energy are calculated in the Appendix B. Multiplying on the right an left sides of equation 2.4 by A∗01 and summing over n X n. Vn A∗01 (qn ) =. X. I(qn ) (Π01 (ω) + Π10 (ω)) A01 (qn )A∗01 (qn )g(qn ). n. g(qn ) = (Π01 (ω) + Π10 (ω)). X. 2. I(qn ) |A01 (qn )| g(qn ). n. 1 = (Π01 (ω) + Π10 (ω)). X. 2. I(qn ) |A01 (qn )|. (2.5). n. This obeys the condition of equation A.28 of chapter one being equal to cero in order to have a Plasmon oscilations[7]. The polarization can be rewritten as[2] (Π01 (ω) + Π10 (ω)) =. (nso − ns1 /2)E10 2 (~ω)2 − E10. (2.6). where nso and ns1 are the average linear electron densities and E10 is the energy difference between the ground and the first exited level, substituting this in equation 2.5 (nso − ns1 /2)E10 X 2 1= I(qn ) |A01 (qn )| 2 (~ω)2 − E10 n to finally obtain the dispersion relation.
(15) 1. SYSTEMS COUPLE ONLY BY COULOMB FORCE. 2 (~ω)2 = E10 + (ns0 − ns1 /2)E10. X. 9. 2. I(qn ) |A01 (qn )|. (2.7). n. as mentioned in the beginning of this chapter E10 = 25meV , (ns0 − ns1 )/2 = 0.02nm−1 [2],m = 0.41me and = 4π0 b where b = 6.5. The dispersion relation for collective excitations due only to Coulomb interaction, in semiconductor chains set up of quantum dots is based on the magnitude I(qn ) from equation 2.2. As can be seen in Figure 2 this interaction between charges decreases as the distance between dots (lattice constant) increases. This is expected because of the dependence of two interacting charges, follows the 1/r law. Figure 2 shows the screening effect due to the charge density, which is strong enough to balance the Coulomb coupling for distances larger than 45 nm. In Figure 3 we observe the dependence of I(qn ) on the wave vector qn for a given value of the lattice constant (10nm), the Coulomb interaction follows the equation 2.2 in the long wave limit.. π Figure 2. I(qn ) vs d for qx = 2000nm . Here is clear that the strength of the Coulomb interaction diminishes with distance.. In Figure 4 we present the dispersion relation measured in units of the energy difference between the chosen quantum dot levels, we can see the dependence of the dispersion on the lattice constant showing clearly much electronic structure as the distance between dots decreases and therefore the interaction becomes stronger. The positive growing behavior of the excitation (Figure 4 and Figure 5) agrees with experimental data[9]. In Figure 6 we observe that the mean distance between dots (lattice constant) , for which the plasmon vanishes is 50 nm. Also, Comparing Figure 2, Figure 3, Figure 5 and Figure 6 we conclude that I(qn ) rules the magnitude and behavior of the excitation, and the form factor A01 (qn ) modulates it, as expected from the theory. In this section we show the dispersion relation for.
(16) 10. 2. APPLICATIONS. Figure 3. Graphic of I(qn ) vs qx for a distance of 10 nm between dots. This shows the behavior of the Coulomb interaction for the long wave limit.. Figure 4. Dispersion relations for different distances between dots. This results come from evaluating equation2.7, where we neglect the multipoles contributions. This curves agree with the reported by Ting-Ting Kang.[5]. chains of semiconductor quantum dots in the long-wave limit as function of the lattice constant. We confirm that the Coulomb interaction causes the collective excitations in these systems..
(17) 2. SYSTEMS COUPLE BY COULOMB FORCE AND TUNNELING. 11. Figure 5. Dispersion relation for a system coupled only by Coulomb force. With a distance of 10 nm between dots. Figure 6. dispersion relation vs distance between dots, for qx = π 2000 . The influence if the Coulomb force over the Dispersion relation has been establish. 2. Systems Couple by Coulomb Force and Tunneling We study the role played by tunneling in chains of quantum dots. We assume that the tunneling is weak so that we use the tight binding method for calculating the energies of the one dimensional array of quantum dots,although not too weak, because in this case we have to take into account the electron-electron interaction within each quantum dot. However, the wavefunctions overlap is neglected. The appearing of the energy bands of the chains introduces simultaneously intra-band transitions as well as inter-band transitions. We analyze first the intra-band dispersion relation. 2.1. Intraband Excitations. For the system we described earlier, and taking into account the weak tunneling aproximation, equation 1.19 from Chapter 1 takes the form.
(18) 12. 2. APPLICATIONS. Vn = I(qn )(A00 (qn )Π00 (ω). X. Vn0 A∗00 (qn0 ). n0. + A10 (qn )Π10 (ω, qx ). X. Vn0 A∗10 (qn0 ) + A01 (qn )Π01 (ω, qx ). n0. X. Vn0 A∗01 (qn0 )). n0. (2.8). however, as mentioned before A01 (qn ) = A10 (qn ) because the wave functions of the single dots are real so Vn = I(qn )(A00 (qn )Π00 (ω). X. Vn0 A∗00 (qn0 ). n0. + A10 (qn ). X. Vn0 A∗10 (qn0 )(Π10 (ω, qx ) + Π01 (ω, qx ))). (2.9). n0. where the polarization can be rewritten as[2] ns0 (0) − ns0 (qx ) E00 (qx ) + ~ω ((ns0 − ns1 )/2)E10 (qx ) = Π(ω, qx ) (Π01 (ω, qx ) + Π10 (ω, qx )) = 2 (q ) (~ω)2 − E10 x Π00 (ω, qx ) =. (2.10). where E10 = (Ess − Wss ∗ Cos(qx ∗ d)) − (Exx − Wxx ∗ Cos(qx ∗ d)) E00 = Wss (1 − Cos(qx ∗ d)). (2.11). for simplicity X. Vn0 A∗00 (qn0 ) = g1. n0. X. Vn0 A∗10 (qn0 ) = g2. n0. therefore, the potential is now written as Vn = I(qn )(A00 (qn )Π00 (ω, qx )g1 + A10 (qn )Π(ω, qx )g2 ). (2.12). Multiplying both sides of equation 2.12 by A∗00 (qn ) and summing over n g1 =. X. 2. I(qn ) |A00 (qn )| Π00 (ω, qx )g1 +. n. |. X. I(qn )A10 (qn )A∗00 (qn ) Π(ω, qx )g2. n. {z. C11. }. |. {z. C12. }. (2.13). and subsequently, multiplying both sides of equation 2.12 by A∗10 (qn ) and summing over n.
(19) 2. SYSTEMS COUPLE BY COULOMB FORCE AND TUNNELING. g2 =. X. I(qn )A00 (qn )A∗10 (qn ) Π00 (ω, qx )g1 +. n. |. X. 13. 2. I(qn ) |A10 (qn )| Π(ω, qx )g2. n. {z. }. C21. |. {z. }. C22. (2.14). gives a system of equations that can be summarized as C12 Π(ω, qx ) C11 Π00 (ω, qx ) − 1 C22 Π(ω, qx ) − 1 C21 Π00 (ω, qx ). =0. (2.15). Solving equation 2.15 gives a third degree equation(Appendix B) 2 0 = (~ω)3 + (~ω)2 (E00 − C11 ns0 ) − ~ω(C22 (ns0 − ns1 /2)E10 + E10 ) 2 2 + E10 C11 (ns0 (0) − ns0 (qx )) − E10 E00 − E00 C22 (ns0 − ns1 /2)E10 (2.16). And its roots are the dispersion relations. We calculate this ruts using Mathematica, and only one has physical meaning. Next, is the calculation of the energies in eV .[10] E0 = 0.317151 − 2 × 0.002719Cos(qx d) E1 = 0.192731 − 2 × 0.009698Cos(qx d) E00 = 2 × 0.002719(1 − Cos(qx d)) E10 of equation 2.16 is the differences between E0 and E1 . The factor I(qn ) is going to be the same as in the systems without tunneling, equation 2.2. for parabolic " E 2 2 10 confinement of the form Vc = m x in the x direction 2 ~ A00 (qn ) = A10 (qn ) =. . . 4 π2. mE10 π~2 1/4. 1/2 Z. 2.5. eiqn x e−. mE10 ~2. x2. dx. (2.17). x2. (2.18). −2.5. mE10 ~2. Z. 2.5. eiqn x xe−. mE10 ~2. dx. −2.5. The dots have a radius of 5 nm and a width of 5nm(Figure 7). The constants used in the calculation of the results exposed below are ns0 = 0.02nm−1 , ns0 − ns1 /2 = 0.02nm−1 ,m = 0.41me and = 4π0 b where b = 6.5. First of all, let’s begin by studying the behavior of the energy difference with respect to qx . The fact that charge can tunnel introduces a new dependence of the energy difference as function of qx , which affects the possible collective excitations. In Figure 9 is shown the dispersion relation taken into account tunneling in the long wave limit. We compare it with the dispersion relation without tunneling and find out that the energy of the collective excitation is higher as a consequence of tunneling. The behavior of the relation dispersion as function of lattice constant can be observed in Figure 10, where we corroborate the same trend as in the pure.
(20) 14. 2. APPLICATIONS. Figure 7. Schematic representation of the system. Coulomb case. Although, the range in which the action of the excitation takes place is enlarge by the tunneling, in factor of 40.(all the plots of plasmon frequency are presented in units of the transition energy). Figure 8. The energy difference for the intra-band transition exhibits a clear dependence on qx . The plot is done in eV. 2.2. Interband Excitations. For a system with inter band tunneling and taking into account the weak tunneling aproximation, equation 1.19 from Chapter 1 takes the form.
(21) 2. SYSTEMS COUPLE BY COULOMB FORCE AND TUNNELING. 15. Figure 9. Dispersion relation vs qx for a distance of 10 nm between dots. this plot shows that even with weak tunneling, the magnitude of the excitation is higher than the one only due to Coulomb force.. Figure 10. Dispersion relation vs distance between dots, for qx = π 2000 . This figure shows that the Coulomb interaction is the most important factor on the collective excitation.. Vn = I(qn )(A10 (qn )Π10 (ω, qx ). X. Vn0 A∗10 (qn0 ) + A01 (qn )Π01 (ω, qx ). n0. X. Vn0 A∗01 (qn0 )). n0. (2.19). since A01 (qn ) = A10 (qn ), the potential takes de form Vn = I(qn )A10 (qn )Π(ω, qx ). X n0. Vn0 A∗10 (qn0 ). (2.20).
(22) 16. 2. APPLICATIONS. where (Π01 (ω, qx ) + Π10 (ω, qx )) =. ((ns0 − ns1 )/2)E10 (qx ) = Π(ω, qx ) 2 (q ) (~ω)2 − E10 x. (2.21). for simplicity lets denote X. Vn0 A∗10 (qn0 ) = g. n0. so Vn = I(qn )A10 (qn )Π(ω, qx )g. (2.22). multiplying both sides of equation 2.22 by A∗10 (qn ) and summing over n gives 2. 2. 1 = I(qn ) |A10 (qn )| Π(ω, qx ) = I(qn ) |A10 (qn )|. ((ns0 − ns1 )/2)E10 (qx ) 2 (q ) (~ω)2 − E10 x. (2.23). finally, the dispersion relation is 2 (~ω)2 = E10 + ((ns0 − ns1 )/2)E10. X. 2. I(qn ) |A01 (qn )|. (2.24). n. Equation 2.24 looks like equation 2.7, but the difference is that the energies on this case, depend on the wave vector. Again, The factor I(qn ) is going to be the same as in the systems without tunneling. The energies are calculate by diagonallyzing the tight-binding matrix, equation 2.25. The parameter used to calculate the results are the same ones that were used on the calculation for intraband tunneling.[10] 2iW10 Sin(qx d) e0 − 2W0 Cos(qx d) (2.25) −2iW10 Sin(qx d) e1 − 2W1 Cos(qx d) . 0.317151 − 2 × 0.002719Cos(qx d) −2i × 0.006020Sin(qx d). 2i × 0.006020Sin(qx d) 0.192731 − 2 × 0.009698Cos(qx d). . Equation 2.25 gives two Eigen values, the ground energy of the system E0 , and the first exited level E1 , E10 of equation 2.24 is the differences between those two energies. This difference, as can be seen in equation 2.24, plays an important role on the dispersion relation, because it has an stronger dependency on qx than A10 (qn ) and I(qn ). Contrary to the intraband plasmon the frequency of the interband excitation is smaller than in uncoupled chains (pure Coulomb). This means, that when a collective excitation of charge occurs on the systems is more likely to be because of the Coulomb interaction, or moreover because an intraband tunneling..
(23) 3. CONCLUSIONS. 17. Figure 11. E10 is the difference of the tight-binding energies, which exhibits a clear dependence on qx .. Figure 12. Dispersion relation vs distance between dots, for qx = π 2000 . This excitation has a lower frequency than the pure Coulomb interaction and the intraband tunneling. 3. Conclusions This study reports very interesting results, on one hand the Coulomb coupling causes very strong collective excitations with energies comparable with the ones found in two dimensional arrays[11]; however, in the case of intra-band tunneling the energies of the excitations are much larger, but for the inter-band case, the excitation shows lower energies that in the pure Coulomb coupling. It is also important to emphasize the fact that the Coulomb interaction seems to play the most important role for the propagation of plasmons in chains of quantum dots, which are complex quasi-one dimensional systems set up of zero dimensional ones. It is also found that the lattice constant of the one dimensional arrays defines the screening and therefore, the limit to allow the existence of plasmons, having a range.
(24) 18. 2. APPLICATIONS. Figure 13. Dispersion relation vs qx for a distance of 10 nm between dots. around the 50 nm. On the other hand, the geometry of the quantum dots, is also a defining factor, because it determines how the electrons are going to be confined within the dots, and is this confinement the one that modulates the excitations. In other words, on the mathematical expression, the confinement is conceal in the Aii0 , and as we mention before this factor modulates de Coulomb interaction. Even though, in the tunneling plasmon, the behavior is ruled out by the dependency on qx of the energies, the Aii0 modulates how fast or slow the excitation grows with qx and decrease with d. Finally, optical experiments can be perform on these systems in order to confirm the predicted behavior in our results. Perspectives of this work are to remake the calculations for several geometric shapes. To extend the model by introducing the exact expression of the coulomb interaction in the quasi one dimensional system to examine the limits of the calculations to purely one and two dimensions systems..
(25) APPENDIX A. Mathematical Steps from Chapter 1 1. Fourier Transformation of The Potential To do the Fourier transformation of the potential,in the x direction, it is important to observe that it is a Convolution product. e2 V (~r) = . Z. g(~ r0 ). z }| { 1 n(~r0 ) d~r0 |~r − ~r0 | | {z }. (A.1). f (~ r −~ r0 ). This means that the Fourier transformation of it, is the ordinary product of the transformations of g(~r0 ) and f (~r − r̄0 ), separately. However, the Fourier transform of f (~r − r̄0 ) is not trivial and is necessary to use Green’s functions 1 = G(~r − ~r0 ) |~r − ~r0 |. (A.2). by definition ∇2 G(~r − ~r0 ) = −4πδ(~r − ~r0 ) where. δ(~r − ~r0 ) =. Z. d3 q iq(~r−~r0 ) e (2π)3. so Z 3 4π d q iq(~r−~r0 ) e G(~r − ~r ) = (2π)3 q2 Z Z Z 0 0 dqx iq(x−x0 ) dqy dqz ei(qy (y−y )+qz (z−z )) = 4π e (2π)2 2π qx2 + qy2 + qz2 | {z } 0. F. using cylindrical coordinates. 19. (A.3).
(26) 20. A. MATHEMATICAL STEPS FROM CHAPTER 1. F −→. Z Z. Z. 0. 2π 0. Z. ∞. 0. dϕdqρ qρ ei~qρ ·(~ρ−~ρ ) 2π qx2 + qρ2. Z. 0. ∞. 0. dϕdqρ qρ eiqρ (ρ−ρ )cosϕ 2π qx2 + qρ2. dqρ qρ J0 [qρ (ρ − ρ0 )] 2π 2π (qρ + iqx )(qρ − iqx ). solving by poles F −→ 2K0 [qx (ρ − ρ0 )] inserting equation A.4 into equation A.3 gives Z 1 dqx iq(x−x0 ) e (2K0 [qx (ρ − ρ0 )]) = 4π |~r − ~r0 | (2π)2. (A.4). (A.5). so now the Fourier transformation is straight, as the reader can see equation A.4 is the cotrasformation of (2K0 [qx (ρ − ρ0 )]); therefore, the transformation of 1 f (~r − r̄0 ) = |~r−~ r 0 | is going to be f (qx ) = 4K0 [qx (ρ − ρ0 )]. (A.6). On the other hand, the Fourier transformation of g(~r0 ) is just g(qx ) = n(qx , y 0 , z 0 ). (A.7). Subsequently, the Fourier transformation of the potential, in the x direction, is expressed as V (qx , y, z) =. 4e2 . Z Z. h p i p y 2 + z 2 − y 02 + z 02 dy 0 dz 0 n(qx , y 0 , z 0 )K0 qx. (A.8). 2. The Self-Consistent Approximation by Ehrenreich and Cohen [6]1The self consistent approximation begins considering the Liouville’s equation for one particle i~. ∂ρ = [H, ρ] ∂t. (A.9). which describes the motion of the density matrix subject to a self consistent potential V (x, t), where ρ is the density matrix operator. The Hamiltonian of a system of particles is H = H0 + V (x, t) (A.10) 1This section is the author interpretation of Ehrenreich and Cohen original article.
(27) 2. THE SELF-CONSISTENT APPROXIMATION BY EHRENREICH AND COHEN. 21. where H0 =. E E E −1 p2 ~ ⇒ H0 ~k = Ek ~k ⇒ ~k = Ω 2 eik·~x 2m. is the Hamiltonian for a free electron. The density matrix operator ρ can be expanded in terms of an unperturbed density matrix ρ(0) and a density matrix ρ(1) that includes the perturbation ρ = ρ(0) + ρ(1). (A.11). ρ(0) has the following property E E ρ(0) ~k = f0 (Ek ) ~k where f0 (Ek ) is the distribution function. Using Fourier analysis over the potential X 0 V (q 0 , t)e−iq x (A.12) V (x, t) = q0. and expanding equation A.9, but neglecting the products V ρ(1) h i h i h i ∂ (0) i~ ρ + ρ(1) = H0 , ρ(0) + V, ρ(0) + H0 , ρ(1) ∂t E E and then applying it to the matrix elements ~k and ~k + ~q gives. (A.13). D E D h i h i h i E ~k i~ ∂ ρ(0) + ρ(1) ~k + ~q = ~k ~k + ~q H0 , ρ(0) + V, ρ(0) + H0 , ρ(1) ∂t (A.14). now it is necessary to take each term in A.14 separately. First, the one on the left-hand side E E D E ∂ D~ (0) ~ ∂ D~ (0) k + ~q + ~k ρ(1) ~k + ~q k ρ + ρ(1) ~k + ~q = i~ k ρ ∂t ∂t E E D D ∂ = i~ f (Ek+q ) ~k|~k + ~q + ~k ρ(1) ~k + ~q ∂t E E ∂ D~ (0) ∂ D~ (1) ~ i~ k ρ + ρ(1) ~k + ~q = i~ k ρ k + ~q (A.15) ∂t ∂t i~. second, the first term in the right-hand side E D i E D E D h ~k H0 , ρ(0) ~k + ~q = ~k H0 ρ(0) ~k + ~q − ~k ρ(0) H0 ~k + ~q E D E D = f (Ek+q )Ek+q ~k|~k + Ek+q f (Ek+q )~q − ~k|~k + ~q D h i E ~k H0 , ρ(0) ~k + ~q = 0 (A.16) next, the second term in the right-hand side.
(28) 22. A. MATHEMATICAL STEPS FROM CHAPTER 1. D h i E D E D E ~k V, ρ(0) ~k + ~q = ~k V ρ(0) ~k + ~q − ~k ρ(0) V ~k + ~q E D E D = ~k V f (Ek+q ) ~k + ~q − ~k f (Ek )V ~k + ~q D E = [f (Ek+q ) − f (Ek )] ~k V ~k + ~q E i D h ~k V, ρ(0) ~k + ~q = [f (Ek+q ) − f (Ek )] V (q, t) (A.17) finally, the third term on the right-hand side i E D E D E D h ~k H0 , ρ(1) ~k + ~q = ~k H0 ρ(1) ~k + ~q − ~k ρ(1) H0 ~k + ~q D E D E = ~k Ek ρ(1) ~k + ~q − ~k ρ(1) Ek+q ~k + ~q D E i D h E ~k H0 , ρ(1) ~k + ~q = (Ek − Ek+q ) ~k ρ(1) ~k + ~q (A.18) then rearranging equations A.15, A.16, A.17 and A.18 into A.14 gives E E D ∂ D~ (1) ~ k ρ k + ~q = [f (Ek+q ) − f (Ek )] V (q, t) + (Ek − Ek+q ) ~k ρ(1) ~k + ~q ∂t (A.19) D E This equation can be used to obtain a solution for ~k ρ(1) ~k + ~q . However, the interest is to find a self consistent solution for the electron density and the potential. In order to do so it is important to notice that the potential V has two parts; an external one V0 , and an screening potential Vs related to the induced charge in the electron density i~. E n o X XD e−iqx n = T r δ(x~e − ~x)ρ(1) = Ω−1 k~0 ρ(1) k~0 + ~q q. (A.20). k0. D E Which is determined by ~k ρ(1) ~k + ~q . In the case of Coulomb interaction, it can E D be assumed that V0 (q, t) → eαt eiωt . Moreover, if ~k ρ(1) ~k + ~q and Vs have the same time dependence of V0 , equation A.19 becomes i~. E ∂ D~ (1) ~ k + ~q e(iω+α)t = [f (Ek+q ) − f (Ek )]V (q, t)e(iω+α)t k ρ ∂t E D + (Ek − Ek+q ) ~k ρ(1) ~k + ~q e(iω+α)t. (A.21). Taking the time derivative E D i~(iω + α) ~k ρ(1) ~k + ~q e(iω+α)t = [f (Ek+q ) − f (Ek )]V (q, t)e(iω+α)t E D + (Ek − Ek+q ) ~k ρ(1) ~k + ~q e(iω+α)t.
(29) 2. THE SELF-CONSISTENT APPROXIMATION BY EHRENREICH AND COHEN. 23. D E Taking all the terms with ~k ρ(1) ~k + ~q in the right hand side. E D ~k ρ(1) ~k + ~q (Ek+q − Ek − ~ω + i~α) = [f (Ek+q ) − f (Ek )]V (q, t). To finally obtain E D ~k ρ(1) ~k + ~q =. [f (Ek+q ) − f (Ek )] V (q, t) (Ek+q − Ek − ~ω + i~α). (A.22). So, the electron density is going to be n(q, t) = Ω−1. X k. [f (Ek+q ) − f (Ek )] V (q, t) (Ek+q − Ek − ~ω + i~α). (A.23). where α = 0 stands for an adiabatic perturbation, in other words stands for a system were no external perturbations are consider. Although the original article does far more than this, equation A.23 is the part of interest to this text, as a matter of fact it is the pillar of the collective theory presented in this chapter, because, as mentioned before, it gives a self consistent solution for the potential and the electron density. It is also important to notice that the application of A.23 in this work is in the adiabatic approximation, so the time dependence is neglected. However, all the above can be used to calculate the dependence of the dielectric constant on the wave number q and frequency ω. The electric field, which is assumed to be polarized causes a displacement of the electric charge from its equilibrium position, the resulting polarization per unit volume is given by p(q, t) =. 1 [(ω, q) − 1]E(q, t) 4π. (A.24). where (w, q) is the dielectric function or response function of the system, p(q, t) is polarization relate to the induce charge by ien(q, t) q. (A.25). −iqV (q, t) e. (A.26). p(q, t) = and the electric field is E(q, t) =. replacing A.25 and A.26 into A.24 and simplifying for (ω, q), gives (ω, q) = 1 −. 4πe2 n(q, t) q 2 V (q, t). to finally, replace equation A.23 , so the dielectric function is. (A.27).
(30) 24. A. MATHEMATICAL STEPS FROM CHAPTER 1. (ω, t) = 1 − vg. X k. [f (Ek+q ) − f (Ek )] (Ek+q − Ek − ~ω + i~α). (A.28). 2. where vg = 4πe q 2 is the 3D potential. Ecuation A.28 has to be equal to cero in order to have Plasmon oscilations.[7] Going back to de development of the theory for collective excitation in one-dimensional arrays, from equation A.23 the density response is X X 2 2 n(qx , y 0 , z 0 ) = 2 |ψy0 | |ψz0 | hkx , i| V (x) |kx0 , i0 i hkx0 , i0 | eiqx x |kx , ii i,i0 kx ,kx0. ×. f (i0 ) − f (i) E(i0 ) − E(i) + ~ω. (A.29). 3. Calculation of the Matrix Elements. hkx0 , i0 | eiqx x |kx , ii = =. Z. ψi∗0 ,kx0 (x)eiqx x ψi,kx (x)dx. Z X. e−ikx Ld ϕ∗i0 (x − Ld)eiqx x. L. =. X. 0 id(kx l−kx L). e. l. Z. X. eikx ld ϕi (x − ld)dx. l. ϕ∗i0 (x)eiqx x ϕi (x. − ld)dx. It is important to notice that the systems under study in the first section of chapter 1, are without tunneling. In other words, the wave functions of different dots do not overlap. Consequently, the terms in the above summation where l 6= 0 are neglected. When everything is put together in the potential expression(equation 1.10,Chapter 1), the exponential containing L eliminates itself with an exponential coming from the matrix element hkx , i| V (x) |kx0 , i0 i. Thus hkx0 , i0 | eiqx x. |kx , ii =. Z. ϕ∗i0 (x)eiqx x ϕi (x)dx. Finally, since the array is periodic, a condition over the kx0 , kx and qx must be set. So the matrix element is hkx0 , i0 | eiqx x |kx , ii = δkx0 −kx −qx ,nG. Z. ϕ∗i0 (x)eiqx x ϕi (x)dx. hkx0 , i0 | eiqx x |kx , ii = δkx0 −kx −qx ,nG Aii0 (qx ). (A.30). However, in the second section, tunneling is allowed in the systems, and the summation is restricted to the first neighbors.Thus, the matrix element is going to be.
(31) 3. CALCULATION OF THE MATRIX ELEMENTS. hkx0 , i0 | eiqx x. |kx , ii = δkx0 −kx −qx ,nG. X. 0 id(kx l−kx L). e. l. hkx0 , i0 | eiqx x |kx , ii = δkx0 −kx −qx ,nG. X. Z. 25. ϕ∗i0 (x)eiqx x ϕi (x − ld)dx 0. eid(kx l−kx L) Aii0 (qx ). (A.31). l. With this done the calculation of matrix element of the potential is straight forward hkx , i| V (x) |kx0 , i0 i =. X. V (qx0 ) hkx0 , i0 | e−iqx x |kx , ii. 0 qx. replacing the expression of hkx0 , i0 | eiqx x |kx , ii for systems with and without tunneling gives hkx , i| V. (x) |kx0 , i0 i. =. X. V. (qx0 )δkx0 −kx −qx ,nG. 0 qx. hkx , i| V (x) |kx0 , i0 i =. X n0. Z. ϕ∗i0 (x)e−iqx x ϕi (x)dx. V (qx + n0 G)A∗ii0 (qx + n0 G). (A.32).
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(33) APPENDIX B. Mathematical Steps from Chapter 2 1. Wave Functions. and. The confinement in the y and z directions is parabolic of the form 2 m E10 y2 V = 2 ~ V =. m 2. . E10 ~. 2. 14. e. z2. therefore, the wave functions are ϕ0 (y) = ϕ0 (z) =. . . mE10 π~2 mE10 π~2. 14. e. −1 mE10 2 ~2. y2. −1 mE10 2 ~2. z2. and for a parabolic confinement of the form 2 m E10 V = x2 2 ~ in the x direction, the wave functions are; for the ground state 1 mE10 2 mE10 4 −1 e 2 ~2 x ϕ0 (x) = π~2 and for the first exited level ϕ1 (x) =. 4 π. . mE10 π~2. 3 ! 14. e. −1 mE10 2 ~2. y2. 2. Solution of Equation 2.15 C11 Π00 (ω) − 1 C12 Π(ω, qx ) C21 Π00 (ω) C22 Π(ω, qx ) − 1. =0. (B.1). 0 = C11 C22 Π00 (ω)Π(ω, qx ) − C11 Π00 (ω) − C22 Π(ω, qx ) − C12 C21 Π00 (ω)Π(ω, qx ) + 1 27.
(34) 28. B. MATHEMATICAL STEPS FROM CHAPTER 2. C11 Π10 + C22 Π01 = (C11 C22 − C12 C21 )Π10 Π01 + 1 ns0 (0) − ns0 (qx ) C11 (ns0 (0) − ns0 (qx )) C22 (ns0 − ns1 /2)E10 = (C11 C22 − C12 C21 ) + 2 E00 + ~ω (~ω)2 − E10 E00 + ~ω (ns0 − ns1 /2)E10 +1 2 (~ω)2 − E10 2 ((~ω)2 − E10 )C11 (ns0 (0) − ns0 (qx )) + (E00 + ~ω)C22 (ns0 − ns1 /2)E10 = 2 − E2 E 2 (~ω)3 − ~ωE10 10 00 + (~ω) E00 2 2 − E10 E00 + (~ω)2 E00 (C11 C22 − C12 C21 )(ns0 (0) − ns0 (qx ))(ns0 − ns1 /2)E10 + (~ω)3 − ~ωE10 2 2 3 2 (~ω) − ~ωE10 − E10 E00 + (~ω) E00. but C11 C22 − C12 C21 = 0, therefore 2 0 = (~ω)3 + (~ω)2 (E00 − C11 ns0 ) − ~ω(C22 (ns0 − ns1 /2)E10 + E10 ) 2 2 + E10 C11 (ns0 (0) − ns0 (qx )) − E10 E00 − E00 C22 (ns0 − ns1 /2)E10 (B.2).
(35) Bibliography [1] E. Ozbay, Plasmonics: merging photonics and electronics at nanoscale dimensions, Science 311, 189 (2006) [2] W. M. Que, G. Kirczenow, Theory of collective excitations in a two-dimensional array of quantum dots, Phys. Rev. B 38, 3614 (1988) [3] N. Kim, S. E. Ulloa, Collective modes in tunneling quantum-dots arrays, Phys. Rev. B 48, 11987 (1993) [4] S. Das Sarma, Wu-yan Lai Screening and elementary excitations in narrow-channel semiconductor microstructures, Phys. Rev. B 32, 1401 (1985) [5] Ting-Ting Kang Plasmons vertically coupled InAs/GaAs quantum dots, Phys. Rev. B 76, 075345 (2007) [6] M.H. Cohen, H. Ehrenreich, Self-Consistent Field Approach to the Many-Electron Problem,Phys. Rev. B 115, 738 (1959) [7] W. Luis Mochn, Plasmons, Centro de Ciencias Fisicas, Universidad Nacional Autnoma the mexico. [8] M. S. Kushwaha, H. Sakaki, Theory of minigap plasmons in a two-dimensional electron gas subject to a one-dimensional periodic potential, Phys. Rev. B 69, 155331 (2004) [9] W. Hansen, M. Horst, J.P. Kotthaus, U. Merkt, C. Sikorski and K. Ploog , Phys. Rev. Lett 58, 2586 (1987) [10] J. F. Nossa , Meta-Materiales Basados en Puntos Cuanticos: Geometria y Acoplamiento, Universidad de los Andes, Departamento de Fisica, Tesis de Maestria en Ciencias-Fisica(2007) [11] W. M. Que, G. Kirczenow, Theory of plasmons in lateral multiwire superlattices, Phys. Rev. B 37, 7153 (1988). 29.
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