Quantum Dots. The Technical University of Denmark, DTU Course Assignment. M.Sc. Student Morten Stilling

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1 Quantum Dots The Technical Universit of Denmark, DTU 111 Course Assignment M.Sc. Student Morten Stilling November 5,

2 Contents 1 Introduction A Description of the Quantum Dot.1 The Harmonic Oscillator Hamiltonian The Ladder Operators The Hamiltonian Expressed in Terms of the Ladder Operators. 5.4 The Eigenenergies of the Harmonic Oscillator The Angular Momentum Operator A Second Description of the Quantum Dot The New Ladder Operators The Operators ˆn + and ˆn The Hamiltonian Expressed in Terms of ˆn + and ˆn The Operator Representing the z-component of the Angular Momentum Expressed in Terms of ˆn + and ˆn The Eigenstates of the Operators Ĥ, ˆL z, ˆn + and ˆn The Eigenenergies of the Harmonic Oscillator The Application of a Magnetic Field The Hamiltonian in the Presence of a Magnetic Field The Hamiltonian Expressed in Terms of ˆn + and ˆn Eigenenergies in the Presence of a Magnetic Field A Discussion of Degenerac A Phenomenological Discussion of Degenerac A Phenomenological Discussion of the Lifting of Degenerac A Discussion of Figure 3 in the Review Article b R. C. Ashoori 19 6 A Discussion of the Review Article b R. C. Ashoori 19 7 Conclusion 1

3 1 Introduction Quantum dots have since the end of the 198s found growing interest within the scientific communit. Suddenl realizable due to new and improved semiconductor crstal growth techniques, quantum dots present new and challenging was of doing basic research and provide the means for creating new interesting applications. Among other things, quantum dots can be used as artificial atoms, helping our understanding of nanometer phenomena, and for making single-electron transistors and quantum dot lasers. Quantum dots also appear as a serious candidate for making nanoscale computing devices and the use of quantum dots as memor cells have been demonstrated in laboratories. In this small report a short description of the quantum dot is presented. The focus is on the description a quantum dot as a two-dimensional harmonic oscillator and is based on the review article b R. C. Ashoori [1. Some of the calculations done in this report, especiall some of the commutation relations, ma seem superfluous since the results can be deduced without doing an calculations. The calculations are done anwa for the simple purpose of practicing the craftmanship of quantum mechanics. In Section of this report a description of the quantum dot as a twodimensional harmonic oscillator is presented and the eigenenergies are found b the use of ladder operators. In Section 3 the description is modified b the introduction of new ladder operators and the operators ˆn + and ˆn. These operators are used to find an alternate form of the eigenenergies in which the angular momentum in the z-direction is a parameter. In Section 4 the eigenenergies are found in the presence of a magnetic field and in Section 5 the degenerac of the eigenstates is discussed. Finall, Section 6 provides a discussion of the review article b R. C. Ashoori. A Description of the Quantum Dot Different approaches can be taken when implementing quantum dots 1. Two approaches commonl used are the artificial paterning of thin film structures and the self-organized growth of strained heterostructures. The quantum dot described in [1 is created in a thin laer of Gallium Arsenide (GaAs semiconductor sandwiched between laers of Aluminum Gallium Arsenide (AlGaAs semiconductor. This confines the electrons verticall while an electric field is used to confine the electrons laterall. The quantum dot can be described as a two-dimensional harmonic oscillator in which electrons can be trapped b application of appropriate electromagnetic fields. This description will be used throghout this report. The spin of the electrons and the Coulomb interaction between electrons in the quantum dot are not considered in this report and the electron states are therefore considered to be totall uncorrelated. 1 See for example [ page 5.

4 .1 The Harmonic Oscillator Hamiltonian The Hamiltonian describing the two-dimensional harmonic oscillator is (ˆp x + ˆp Ĥ = m + 1 mω (ˆx + ŷ (1 where m is the effective mass of the electron and ω is the angular frecuenc of oscillation 3. The operators ˆx and ŷ are position operators and ˆp x and ˆp are momentum operators.. The Ladder Operators The ladder operators are defined â x 1 ( ˆx + i r r h ˆp x â 1 ( ŷ + i r r h ˆp ( (3 where r h/mω is the characteristic length. Since ˆx, ŷ, ˆp x and ˆp correspond to phsical observables the are self-adjoint, i.e. ˆx = ˆx, ŷ = ŷ, ˆp x = ˆp x and ˆp = ˆp. The adjoint of the ladder operators are then â x = 1 ( ˆx i r r h ˆp x â = 1 (ŷ r i r h ˆp = 1 ( ˆx i r r h ˆp x = 1 ( ŷ r i r h ˆp (4. (5 Note that the ladder operators are not self-adjoint and therefore cannot represent phsical observables. The commutation relations between the position and momentum operators are ([4 page 1 [ˆr i, ˆp j = i hδ ij (6 [ˆr i, ˆr j = (7 [ˆp i, ˆp j =. (8 The commutation relation between â x and â x is then [âx, â x = 1 ( ˆx + i r ( 1 ˆx r h ˆp x i r r h ˆp x 1 ( ˆx i r ( 1 ˆx r h ˆp x + i r r h ˆp x = 1 ( ˆx + r h ˆp x ī h ˆxˆp x + ī h ˆp xˆx r For GaAs m.67m where m is the free electron mass (according to [3 page For a good introduction to the quantum mechanical harmonic oscillator problem, including the definition of ω, please refer to [4 page 31 3

5 1 ( ˆx r = ī h [ˆx, ˆp x which, according to Equation (6, gives + r h ˆp x + ī h ˆxˆp x ī h ˆp xˆx [âx, â x = 1. (9 B analog, the commutation relation between â and â is [â, â = 1. (1 The commutation relation between â x and â is [â x, â = 1 ( ˆx + i r ( 1 ŷ r h ˆp x + i r r h ˆp 1 ( ŷ + i r ( 1 ˆx r h ˆp + i r r h ˆp x = 1 ( ˆxŷ r r h ˆp x ˆp + ī h ˆxˆp + ī h ˆp xŷ 1 (ŷˆx r r h ˆp ˆp x + ī h ˆp ˆx + hŷˆp ī x = 1 ( 1 [ˆx, ŷ r h [ˆp x, ˆp + ī h [ˆx, ˆp ī h [ŷ, ˆp x r which, according to Equations (6, (7 and (8, gives The commutation relation between â x and â is [â x, â = 1 ( ˆx i r ( 1 ŷ r h ˆp x = 1 = 1 1 ( ŷ i r r ( ˆxŷ r (ŷˆx 1 ( 1 r r r Using Equations (6, (7 and (8, [â x, â =. (11 i r r h ˆp 1 ( ˆx r i r h ˆp h ˆp x h ˆp x ˆp ī h ˆxˆp ī h ˆp xŷ r h ˆp ˆp x ī h ˆp ˆx hŷˆp ī x [ˆx, ŷ r h [ˆp x, ˆp ī h [ˆx, ˆp + ī h [ŷ, ˆp x. [â x, â =. (1 4

6 The commutation relation between â x and â is [âx, â = 1 ( ˆx + i r ( 1 ŷ r h ˆp x = 1 = 1 1 ( ŷ i r r ( ˆxŷ r (ŷˆx 1 ( 1 r r + r Using Equations (6, (7 and (8, i r r h ˆp 1 ( ˆx r + i r h ˆp h ˆp x h ˆp x ˆp ī h ˆxˆp + ī h ˆp xŷ + r h ˆp ˆp x ī h ˆp ˆx + hŷˆp ī x [ˆx, ŷ + r h [ˆp x, ˆp ī h [ˆx, ˆp ī h [ŷ, ˆp x. [âx, â =. (13 B analog, the commutation relation between â and â x is [â, â x =. (14 Note that the commutation relations of Equations (11, (1, (13 and (14 can be deduced without calculations since ˆx and ˆp x commute with ŷ and ˆp..3 The Hamiltonian Expressed in Terms of the Ladder Operators Until now, no phsical interpretation has been ascribed to the ladder operators. Indeed, since the are not self-adjoint operators the cannot represent phsical observables. The combinations â xâ x and â â, however, are self-adjoint 4 and it turns out that these combinations can be used to represent phsical observables. Postulate: The Hamiltonian can be written using the ladder operators, Ĥ = hω (â x â x + â â + 1. (15 Proof: Inserting the definitions of the ladder operators, ( 1 ˆx Ĥ = hω i r ( 1 ˆx r h ˆp x + i r r h ˆp x ( 1 ŷ + hω i r r = hω + hω ( ˆx r (ŷ r + r 1 ( ŷ r + i r h ˆp h ˆp h ˆp x + ī h ˆxˆp x ī h ˆp xˆx + r h ˆp + hŷˆp ī ī h ˆp ŷ + hω + hω 4 Generall, (  ˆB = ˆB  where  and ˆB are operators (see [5 page 14. 5

7 = hω ( r (ˆp h x + ˆp ( r = hω = hω + 1 r (ˆx + ŷ + ī h [ˆx, ˆp x + ī h [ŷ, ˆp + (ˆp h x + ˆp 1 + (ˆx r + ŷ + ī h (i h + ī h (i h + ( r (ˆp h x + ˆp 1 + (ˆx r + ŷ, and using the definition of the characteristic length, r h/mω, the Hamiltonian is recovered, (ˆp x + ˆp Ĥ = m + 1 mω (ˆx + ŷ QED. (16.4 The Eigenenergies of the Harmonic Oscillator Rewritting the Hamiltonian 5, Ĥ = ( ˆp x m + 1 mω ˆx + ( ˆp m + 1 mω ŷ Ĥ = Ĥx + Ĥ, (17 it is apparent that the two-dimensional harmonic oscillator Hamiltonian can be described as a sum of two one-dimensional harmonic oscillator Hamiltonians. It is also apparent that the two-dimensional harmonic oscillator is isotropic since the angular frequenc of oscillation, ω, is the same for both the x- and the -direction. This rotational smmetr around the z-axis has important consequences when dealing with the angular momentum (treated in the next section. The eigenenergies of a one-dimensional harmonic oscillator is known from basic quantum mechanics and will be stated here without further discussion 6 : ( E x = n x + 1 hω (18 ( E = n + 1 hω (19 with n x, n =, 1,, 3,. The eigenenerg of the two-dimensional harmonic oscillator is then E = E x + E ( = n x + 1 ( hω + n + 1 hω ( which gives E = E nxn = (n x + n + 1 hω (1 5 The approach used in this section is the same as the one used to find the eigenenergies of a three dimensional harmonic oscillator in [6 page See for example [4 page 35. 6

8 where n x, n =, 1,, 3,. Note that Equations (15 and (1 impl that the eigenvalues of the operators â xâ x and â â are n x and n, respectivel. Note also that all eigenenergies but one are degenerate. The ground state has energ E g = hω with (n x, n = (, ; the first exited state has energ E g = hω with (n x, n = (1, or (, 1; the second exited state has energ E g = 3 hω with (n x, n = (, or (1, 1 or (, and so forth. This degenerac is discussed further in Section 5..5 The Angular Momentum Operator The operator corresponding to the z-component of the angular momentum, ˆL z = ˆxˆp ŷˆp x, is the generator of rotations around the z-axis ([5 page 69. Since the potential is rotationall smmetric around the z-axis, as discussed in the previous section, an rotation around this axis will leave the sstem unchanged. Then, [Ĥ, ˆL z =, ( i.e. ˆL z commutes with the Hamiltonian. In order to express the angular momentum operator, ˆLz, in terms of the ladder operators defined in the previous section, the combinations â x â and â â x are examined: â x â = 1 ( ˆx + i r ( 1 ŷ r h ˆp x i r r h ˆp = 1 ( ˆxŷ r + r h ˆp x ˆp ī h ˆxˆp + ī h ˆp xŷ = 1 ( ˆxŷ + r h ˆp x ˆp ī h ˆxˆp + hŷˆp ī x (3 r where [ŷ, ˆp x = (Equation (6 has been used in the last step. Similarl, â â x = 1 ( ŷ + i r ( 1 ˆx r h ˆp i r r h ˆp x = 1 (ŷˆx r + r h ˆp ˆp x + ī h ˆp ˆx hŷˆp ī x = 1 ( ˆxŷ + r h ˆp x ˆp + ī h ˆxˆp hŷˆp ī x r (4 where [ˆx, ŷ = [ˆp x, ˆp = [ˆx, ˆp = (Equations (6, (7 and (8 has been used. Subtracting Equation (3 from Equation (4, it is apparent that â â x â x â = ī h ˆxˆp ī hŷˆp x, (5 ˆL z = ˆxˆp ŷˆp x = i h ( â â x â x â. (6 7

9 Smmetr arguments show that this operator must commute with the Hamiltonian. To verif this the commutation relation is derived presentl: [Ĥ, ˆL (â z = hω x â x + â â + 1 ( i h ( â â x â x â ( i h ( â â x â x â (â hω x â x + â â + 1 = (i h ω ( â xâ x â â x + â â â â x + â â x +(i h ω ( â xâ x â x â + â â â x â + â x â +(i h ω ( â â xâ xâ x + â â xâ â + â â x (i h ω ( â x â â xâ x + â x â â â + â x â = (i h ω ( â xâ x â xâ + â xâ â â + â xâ +(i h ω ( â xâ x â x â + â x â â â + â x â +(i h ω ( â xâ xâ x â + â xâ â â + â xâ (i h ω ( â x â xâ x â + â x â â â + â x â where [â x, â = [ â x, â = [âx, â = [â, â x = (Equations (11, (1, (13 and (14 have been used to move all â x and â x to the left. Writting the commutation relation in this wa it is possible to see that some of the terms cancel, [Ĥ, ˆL z = (i h ω ( â xâ x â xâ + â xâ â â +(i h ω ( â xâ x â x â + â x â â â +(i h ω ( â xâ xâ x â + â xâ â â (i h ω ( â x â xâ x â + â x â â â According to Equations (9 and (1 [âx, â x = 1 âx â x = 1 + â xâ x (7 [â, â = 1 â â = 1 + â â (8 and the commutation relation can be restated [Ĥ, ˆL z = (i h ω ( â (â x x â x â x â x â + â x (â â â â â +(i h ω (( â xâ x (âx â ( 1 + â x â x (âx â +(i h ω (( â x â ( 1 + â â (âx â (â â = (i h ω ( â [â x x, â x â + â x [â, â â (i h ω ( â x â + (i h ω ( â x â = (i h ω ( â x( 1â + â x(1â. Finall, the commutation relation between the Hamiltonian and the operator representing the z-component of the angular momentum has been found to be [Ĥ, ˆL z =. (9 8

10 3 A Second Description of the Quantum Dot Equation (1, showing the eigenenergies of the harmonic oscillator, cannot easill be used to predict the effects of appling a magnetic field over the quantum dot. In order to explain these effects some new operators are defined and the eigenenergies are described in terms of these. 3.1 The New Ladder Operators Two new ladder operators are introduced, â + = 1 (â x + iâ (3 â = 1 (â x iâ, (31 along with their respective adjoints, â + = 1 (â x iâ (3 â = 1 (â x + iâ. (33 The commutation relations of the new ladder operators are derived presentl. The commutation relation between â + and â + is [ â +, â + = 1 (â x + iâ 1 (â x iâ 1 (â x iâ 1 (â x + iâ = 1 (âx â x + â â iâ x â + iâ â x 1 (â x â x + â â iâ â x + iâ xâ = 1 ([âx, â x + [â, â i [âx, â + i [â, â x which, when using [ â x, â x = [â, â = 1 and [âx, â = [â, â x = (Equations (9, (1, (13 and (14, gives [ â +, â + = 1 (34 The commutation relation between â and â is [ â, â = 1 (â x iâ 1 (â x + iâ 1 (â x + iâ 1 (â x iâ = 1 (âx â x + â â + iâ x â iâ â x 1 (â x â x + â â + iâ â x iâ xâ = 1 ([âx, â x + [â, â + i [âx, â i [â, â x 9

11 which becomes [ â, â = 1 (35 again using Equations (9, (1, (13 and (14. The commutation relation between â + and â is [â +, â = 1 (â x + iâ 1 (â x iâ 1 (â x iâ 1 (â x + iâ = 1 (â xâ x + â â iâ x â + iâ â x 1 (â xâ x + â â + iâ x â iâ â x = i [â x, â which gives [â +, â = (36 since [â x, â = (Equation (11. The commutation relation between â + and â is [ â +, â = 1 (â x iâ 1 (â x + iâ 1 (â x + iâ 1 (â x iâ = 1 (â x â x + â â + iâ xâ iâ â x 1 (â x â x + â â iâ xâ + iâ â x = i [ â x, â which gives [ â +, â = (37 since [ â x, â = (Equation (1. The commutation relation between â + and â is [ â +, â = 1 (â x + iâ 1 (â x + iâ 1 (â x + iâ 1 (â x + iâ = 1 (âx â x â â + iâ x â + iâ â x 1 (â x â x â â + iâ â x + iâ xâ = 1 ([âx, â x [â, â + i [âx, â + i [â, â x 1

12 which gives [ â +, â =. (38 since [ â x, â x = [â, â = 1 and [âx, â = [â, â x = (Equations (9, (1, (13 and (14. The commutation relation between â and â + is [ â, â + = 1 (â x iâ 1 (â x iâ 1 (â x iâ 1 (â x iâ = 1 (âx â x â â iâ x â iâ â x 1 (â x â x â â iâ xâ iâ â x = 1 ([âx, â x [â, â i [âx, â i [â, â x which gives [ â, â + = (39 again using Equations (9, (1, (13 and (14. The result of this commutation relation analsis, Equations (34, (35, (36, (37, (38 and (39, shows that the new ladder operators, â + and â, have commutation relations characteristic of ladder operators. 3. The Operators ˆn + and ˆn It is usefull to define two new operators ˆn + â +â + (4 ˆn â â. (41 These can be used to express the Hamiltonian and the operator representing the z-component of the angular momentum. This will be done in the following sections. 3.3 The Hamiltonian Expressed in Terms of ˆn + and ˆn In order to express the Hamiltonian in terms of the operators ˆn + and ˆn the sum of the two operators is examined, ˆn + + ˆn = 1 (â x iâ 1 (â x + iâ + 1 (â x + iâ 1 (â x iâ = 1 (â x â x + â â + iâ xâ iâ â x + 1 (â x â x + â â iâ xâ + iâ â x = â xâ x + â â. (4 11

13 Using Equation (15 it is apparent that the Hamiltonian can be written Ĥ = hω (ˆn + + ˆn + 1. ( The Operator Representing the z-component of the Angular Momentum Expressed in Terms of ˆn + and ˆn In order to express the operator representing the z-component of the angular momentum in terms of the operators ˆn + and ˆn the difference between the two operators is examined, ˆn ˆn + = 1 (â x + iâ 1 (â x iâ 1 (â x iâ 1 (â x + iâ = 1 (â x â x + â â iâ xâ + iâ â x 1 (â x â x + â â + iâ xâ iâ â x = i ( â xâ + â â x = i ( â â x â x â. (44 where [ â x, â = [â, â x = (Equations (13 and (14 has been used in the last step. Using Equation (6 it is apparent that the operator representing the z- component of the angular momentum can be written ˆL z = h (ˆn ˆn +. ( The Eigenstates of the Operators Ĥ, ˆL z, ˆn + and ˆn It is possible to find a common set of eigenstates for operators that commute. In order to find out if this is possible for the operators Ĥ, ˆL z, ˆn + and ˆn the commutation relations between these operators are found presentl. The commutation relation between ˆn + and ˆn is [ˆn +, ˆn = â +â + â â â â â +â + = â +â + â â â +â + â â (46 where â + and â + in the second term [ have been moved [ to the [ left b using the commutation relations [â +, â = â +, â = â +, â = â, â + = (Equations (36, (37, (38 and (39. The commutation relation between ˆn + and ˆn is then [ˆn +, ˆn =. (47 This commutation relation will be used to derive the five other commutation relations. 1

14 The commutation relation between Ĥ and ˆL z is [Ĥ, ˆL z = hω (ˆn + + ˆn + 1 h (ˆn ˆn + h (ˆn ˆn + hω (ˆn + + ˆn + 1 = h ω (ˆn +ˆn + ˆn ˆn + ˆn ˆn +ˆn + ˆn ˆn + ˆn + h ω (ˆn ˆn + + ˆn ˆn + ˆn ˆn +ˆn + ˆn +ˆn ˆn + = h ω [ˆn +, ˆn (48 which, when using Equation (47, gives [Ĥ, ˆL z =. (49 The commutation relation between Ĥ and ˆn + is [Ĥ, ˆn + = hω (ˆn + + ˆn + 1 ˆn + ˆn + hω (ˆn + + ˆn + 1 = hω (ˆn +ˆn + + ˆn ˆn + + ˆn + hω (ˆn +ˆn + + ˆn +ˆn + ˆn + = hω [ˆn, ˆn + (5 which gives [Ĥ, ˆn + =. (51 The commutation relation between Ĥ and ˆn is [Ĥ, ˆn = hω (ˆn + + ˆn + 1 ˆn ˆn hω (ˆn + + ˆn + 1 = hω (ˆn +ˆn + ˆn ˆn + ˆn hω (ˆn ˆn + + ˆn ˆn + ˆn = hω [ˆn +, ˆn (5 which gives [Ĥ, ˆn =. (53 The commutation relation between ˆL z and ˆn + is [ˆLz, ˆn + = h (ˆn ˆn + ˆn + ˆn + h (ˆn ˆn + = h (ˆn ˆn + ˆn +ˆn + h (ˆn +ˆn ˆn +ˆn + = h [ˆn, ˆn + (54 which gives [ˆLz, ˆn + =. (55 The commutation relation between ˆL z and ˆn is [ˆLz, ˆn = h (ˆn ˆn + ˆn ˆn h (ˆn ˆn + = h (ˆn ˆn ˆn +ˆn h (ˆn ˆn ˆn ˆn + = h [ˆn, ˆn + (56 13

15 which gives [ˆLz, ˆn =. (57 The operators Ĥ, ˆL z, ˆn + and ˆn all commute and therefore it is possible to find a complete set of common eigenstates. 3.6 The Eigenenergies of the Harmonic Oscillator The eigenenergies of the harmonic oscillator can be derived from the timeindependent Schrödinger equation Ĥ Ψ = E Ψ hω (ˆn + + ˆn + 1 Ψ = E Ψ (58 The eigenvalues of the operators ˆn +, ˆn and ˆL z are n +, n and l h, respectivel, where n +, n =, 1,, 3, and l = n n +. The values of l follow directl from Equation (45. The quantum number l can be either positive or negative. Here, the two different situations are considered separatel and a joint description is provided. Positive l Rearranging Equation (45, ˆL z = h (ˆn ˆn + ˆn = 1 h ˆL z + ˆn + (59 the time-independent Schrödinger equation can be written ( 1 hω (ˆn + + h ˆL z + ˆn Ψ = E Ψ hω (ˆn h ˆL z + 1 Ψ = E Ψ hω ( n h hl + 1 Ψ = E Ψ (6 and the eigenenergies of the harmonic oscillator can be written where l is positive and n + < n. E = E n+,l = hω (n + + l + 1 (61 Negative l Again, rearranging Equation (45, ˆL z = h (ˆn ˆn + ˆn + = n 1 h ˆL z (6 the time-independent Schrödinger equation can be written (( hω n 1 h ˆL z + ˆn + 1 Ψ = E Ψ 14

16 hω (ˆn 1 h ˆL z + 1 Ψ = E Ψ ( hω n 1 h hl + 1 Ψ = E Ψ (63 and the eigenenergies of the harmonic oscillator can be written where l is negative and n < n +. E = E n,l = hω (n l + 1 (64 Joint Description Defining the quantum number n min (n +, n, i.e. the smallest of the quantum numbers n + and n, the eigenenergies can be written E = E n,l = hω (n + l + 1. (65 where n, l =, 1,, 3,. Note that the eigenenergies are still degenerate as, of course, nothing has phsicall changed the sstem onl the description of the sstem has changed. The degenerac is discussed further in Section 5. 4 The Application of a Magnetic Field If a magnetic field of size B is applied in the z-direction, i.e. B z = B and B x = B =, the eigenenergies of the sstem changes. A gauge is chosen so that the vector potential is of the form A = A x. (66 The magnetic field can be found from the vector potential ([5 page 37, B = A = A = A x z ( z x x x (. (67 Then, the magnetic field is directed along the z-axis (as required and the constant A can be expressed A = B. (68 15

17 4.1 The Hamiltonian in the Presence of a Magnetic Field The generic form of the Hamiltonian for an electron in an electromagnetic field is ([5 page 39 Ĥ = 1 e (ˆ p + ˆ A eφ (69 m c which in the present situation becomes (φ is the scalar potential and eφ is therefore the potential energ function Ĥ = 1 ( (ˆp x + e ( m c Âx + ˆp + e c  + 1 mω (ˆx + ŷ. (7 Using Âx = A ŷ = Bŷ and  = A ˆx = Bˆx the Hamiltonian can be written ( 1 Ĥ = (ˆp x eb ( m c ŷ + ˆp + eb c ˆx + 1 mω (ˆx + ŷ ( ( ( 1 eb eb = ˆp x + ŷ ŷˆp x m c c ( + 1 ( ( eb eb ˆp + ˆx + ˆxˆp m c c + 1 mω (ˆx + ŷ (71 where [ˆx, ˆp = [ŷ, ˆp x = (Equation (6 has been used. Rewritting, 1 Ĥ = (ˆp m x + ˆp 1 + mω (ˆx + ŷ + 1 ( eb m1 (ˆx + ŷ + 1 ( eb (ˆxˆp ŷˆp x (7 4 mc mc which, when using ˆL z = ˆxˆp ŷˆp x and defining the cclotron frequenc ω c eb/mc, becomes Ĥ = (ˆp x + ˆp m + 1 ( m ω ω c (ˆx + ŷ + 1 ω ˆL c z (73 or, using the new angular frequenc ω = ω ω c, Ĥ = (ˆp x + ˆp m + 1 mω (ˆx + ŷ + 1 ω c ˆL z. (74 4. The Hamiltonian Expressed in Terms of ˆn + and ˆn Using the new characteristic length, r = h/mω, the ladder operators can be redefined, â x = 1 ( ˆx r + i r h ˆp x (75 â = 1 (ŷ r + i r h ˆp, (76 16

18 and the operators â +, â, ˆn + and ˆn, â + = 1 (â x + iâ (77 â = 1 (â x iâ (78 ˆn + = â +â + (79 ˆn = â â, (8 are now expressed in terms of the new ladder operators. It is apparent that the two first terms in the Hamiltonian, Equation (74, still represent a harmonic oscillator, albeit with a different angular frequenc ω = ω ω c. Using the previous result, Equation (43, the Hamiltonian can imidiatel be written Ĥ = hω (ˆn + + ˆn ω c ˆL z (81 or Ĥ = hω (ˆn + + ˆn ω c h (ˆn ˆn +. (8 4.3 Eigenenergies in the Presence of a Magnetic Field Using Equation (81 and the analsis of Section 3.6 the eigenenergies of the harmonic oscillator in a magnetic field can be deduced, E = E n,l = hω (n + l l hω c. (83 where n =, 1,, 3, and l =,, 1,, 1,,. 5 A Discussion of Degenerac Equation (65 gives the eigenenergies of the quantum dot with no applied magnetic field, E = E n,l = hω (n + l + 1 (84 where n, l =, 1,, 3,. Equation (83 gives the eigenenergies of the quantum dot in the presence of a magnetic field of size B in the z-direction, E = E n,l = hω (n + l l hω c (85 where n =, 1,, 3, and l =,, 1,, 1,,. The eigenenergies of the first 15 states have been calculated and tabulated in Table 1. It is apparent that the exited states are all degenerate when there is no magnetic field applied (second column and that this degenerac is lifted when a magnetic field is applied in the z-direction (third column. A phenomenological explanation for this lifting of degenerac is presented presentl. 17

19 (n, l E n,l (B = E n,l (B (, hω hω (,-1 hω hω hωc (,1 hω hω + hωc (,- 3 hω 3 hω hωc (1, 3 hω 3 hω (, 3 hω 3 hω + hωc (,-3 4 hω 4 hω 3 hωc (1,-1 4 hω 4 hω hωc (1,1 4 hω 4 hω + hωc (,3 4 hω 4 hω + 3 hωc (,-4 5 hω 5 hω 4 hωc (1,- 5 hω 5 hω hωc (, 5 hω 5 hω (1, 5 hω 5 hω + hωc (,4 5 hω 5 hω + 4 hωc Table 1: Energ of the ground state and the first 14 exited states with or without an applied magnetic field. The second column shows that the exited states are degenerate when B =. The third column shows that the degenerac is lifted when B. 5.1 A Phenomenological Discussion of Degenerac Imagine the quantum dot potential as a bowl, rotationall smmetric around the z-axis. The electron is brought to oscillate back and forth in the bowl while simultaneousl rotating around in the bowl. This rotation of electric charge sets up a magnetic field in the positive or the negative z-direction depending on the direction of rotation represented b positive or negative values of l. When there is no applied magnetic field the energ of the sstem does not depend on the direction of the induced field and so does not depend on the direction of rotation. Then, states with +l and l are of the same energ impling a two-fold degenerac of all states except the ones with l = (which can be thought of as not rotating. Furthermore, if the oscillator quantum number, n, is decreased b one while the rotational quantum number, l, is increased b two the energ remains the same. This is apparent in Equation (65 and can be pictured as oscillating less while rotating more. This implies a degenerac for all states with n 1. The total degenerac is g = α + 1 where α denotes the energ level and E α = E is the ground state energ. This can be deduced from the arguments above but is more easill observed in Table A Phenomenological Discussion of the Lifting of Degenerac When a magnetic field is applied in the z-direction the energ of the sstem depends on the direction of the induced magnetic field and thereb on the 18

20 direction of rotation. Then, states with +l and l are nolonger of the same energ and this source of degenerac has been removed. This is apparent in Equation (83 where the last term shows the splitting of the energ levels that where previousl degenerate. The second source of degenerac, the decrementation of n b one and the incrementation of l b two, nolonger balance since there is now an extra term involving l (again the last term of Equation (83. The degenerac caused b this balancing is therefore lifted as well. The effect of appling a magnetic field is then a total lifting of degenerac 7 as can be observed in the third row of Table 1. Also, the angular frequenc of oscillation, ω, is replaced b the larger frequenc ω and there is a general raising of the energ levels. The energ levels, the degenerac and the lifting of the degenerac is shown in Figure 3 in the review article b R. C. Ashoori. This figure will be discussed in the next section. 5.3 A Discussion of Figure 3 in the Review Article b R. C. Ashoori Figure 3a in [1 shows the orbits of states with a particular value of n. It shows that the mean radius of the orbits increase with l. Figure 3b in [1 shows the energ levels when there is no applied magnetic field. The dots corresponding to states with the same value of n are connected, the left branch corresponds to negative values of l, the right branch corresponds to positive values of l and the lowest dot is the state with n, l = (,. It is apparent that all exited states are degenerate in accordance with the discussion of the previous section. Figure 3c in [1 shows the energ levels when a magnetic field is applied. It is apparent that the V s of Figure 3b are now distorted towards the left showing a lifting of degenerac. The states in the left branches (with negative values of l are lowered in energ relative to the states in the right branches (with positive values of l. This is in accordance with the discussion of the previous section and with the result of Equation (83. Figure 3d in [1 shows a colour coded image from a Single-electron Capacitance Spectroscop (SECS experiment. It shows the energ levels of a quantum dot containing 7 3 electrons as a function of applied magnetic field. This experiment will be discussed along with the Gated Transport Spectroscop (GTS experiment in the next section. 6 A Discussion of the Review Article b R. C. Ashoori The review article written b R. C. Ashoori and published in Nature on Februar 1st 1996 [1 describes two experiments performed on quantum dots, the results and the interpretations of the results. The experiments are called Single-electron 7 Naturall, when lifting the degenerac some of the states ma be raised/lowered into new degeneracies at certain values of B. 19

21 Capacitance Spectroscop (SECS and Gated Transport Spectroscop (GTS and are described briefl here. In SECS experiments a DC bias voltage is applied across the quantum dot to be examined. As the bias voltage is increased the number of electrons in the quantum dot is increased as electrons tunnel into the dot. B superimposing a small AC voltage on the bias voltage, and measuring the capacitance of the quantum dot, it is possible to find the bias voltages at which electrons are added to the dot. This is possible since the AC voltage will cause electrons to tunnel in and out of the dot at these critical bias voltages. The capacitance spectrum can be interpreted as showing the energies at which electrons fill up higher energ levels in the quantum dot and therefore represents the energ spectrum of the quantum dot. The SECS experiment is capable of detecting even the entr of the first electron making it possible to know the exact number of electrons in the dot. In GTS experiments a DC bias voltage is again applied across the quantum dot. In this scheme two probes are placed in close proximit to the quantum dot and a small DC probing voltage is applied between the probes. At the critical bias voltages electrons tunnel into and out of the quantum dot from the probes causing a current to flow in the probing circuit. Then, the bias voltages at which electrons can be added to the quantum dot can be interpreted as showing the energ spectrum of the dot. Unlike the SECS experiment it has et proven practicall impossible to detect the entr of the first electron and most GTS experiments start out with 5 or more electrons in the dot. The simple harmonic oscillator model discussed in this report and in [1 does not consider the effects of electron spin or electron-electron coulomb forces. These effects are not negligible and especiall the coulomb interaction between the electrons plas an important role in these large artificial atoms. State transitions that are not expected b the simple model are observed, spin-flips occur and it ma even seem that electrons attract each other. All in all, the analsis of quantum dots require more accurate models in order to better explain the observations of measurements. Please note that h has been accidentall replaced b η in Equation ( of the article. 7 Conclusion A description of quantum dots as two-dimensional harmonic oscillators has provided theoretical energ spectra with and without applied magnetic fields. The degenerac has been seen to be lifted b the magnetic field and a phenomenological explanation for this has been given. The review article b R. C. Ashoori has been discussed and it follows that more accurate theoretical descriptions are necessar in order to explain some of the data observed in experiments. The simple harmonic oscillator model does, however, provide a first glance into the magical world of quantum dots.

22 References [1 R. C. Ashoori, Electrons in Artificial Atoms, Nature, Februar 1996 [ D. Bimberg, M. Grundmann and N. N. Ledentsov, Quantum Dot Heterostructures, John Wile and Sons, 1999, ISBN: [3 Jasprit Singh, Semiconductor Optoelectronics Phsics and Technolog, McGraw-Hill, 1995, ISBN: [4 David J. Graffiths, Introduction to Quantum Mechanics, Prentice Hall, 1995, ISBN: [5 Leslie E. Ballentine, Quantum Mechanics A Modern Development, World Scientific Publishing, 1998, ISBN: [6 B. H. Bransden and C. J. Joachain, Quantum Mechanics, second edition, Prentice Hall,, ISBN: