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Parabolic dot in tilted magnetic field
Physica E 8 (2000) 230–238
www.elsevier.nl/locate/physe
Parabolic dot in tilted magnetic eld T. Pyragien˙e ∗ , A. Matulis Institute of Semiconductor Physics, GoÄstauto 11, 2600 Vilnius, Lithuania Received 8 March 2000; received in revised form 3 April 2000; accepted 10 April 2000
Abstract The electron motion, the power absorption spectrum and the oscillator strengths are considered in the case of the parabolic dot in a tilted magnetic eld by means of solving the equations of motion. It is shown that for a linear system the classical dynamic matrix diagonalization leads to the second quantization representation in quantum mechanical description. It enables to generalize the Fock single-electron wave function set for the case with the tilted magnetic eld. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 03.65.−w Keywords: Electron density; Excitation spectrum; Oscillator strengths
1. Introduction Quantum dots, or artiÿcial atoms, have been a subject of intense theoretical and experimental research over the last few years [1]. The useful instrument in spectroscopy experiments is the magnetic eld applied in perpendicular to the quantum dot plane direction which enables to trace easily the quantum dot properties dependence on various parameters. Recently the experiments with the magnetic eld in quantum dot plane (Voight con guration) gained interest [2– 4], and even the magneto-photoluminescence dependence on the tilted magnetic eld direction was investigated [5]. The tilted magnetic eld couples the in-plane electron motion to the vertical one, and thus, enables to recover the quantum dot excitations which are forbidden ∗
Corresponding author.
in the case of more symmetric perpendicular magnetic eld. To our knowledge the tilted magnetic eld has been used for the theoretical polaron spectrum studies in the parabolic quantum wells [6]. The theoretical description of electron system in a quantum dot is mainly based on the wave function expansion into the series of non-interacting electron wave functions. In the most popular case of the parabolic dots the energy spectrum and the single-electron wave functions were calculated by Fock [7]. The analytical solution was obtained due to the circular symmetry of the quantum dot which leads to the separation of the variables. The tilted magnetic eld breaks the symmetry, and consequently, makes the problem more dicult. In the case of quantum well the remaining translation symmetry in the well plane along the perpendicular to the magnetic eld direction is sucient for the separa-
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 1 5 0 - 8
T. Pyragien˙e, A. Matulis / Physica E 8 (2000) 230–238
tion of the variables and for obtaining the analytical solution as it is shown in Ref. [8]. That is not the case of the quantum dot, where in the tilted magnetic eld there is no more coordinate space symmetries left. Nevertheless, due to the parabolicity there are symmetries in the total phase rp-space, which is enabled to get the analytical solution and construct the single-electron wave function set in this more complicated case. The purpose of the present paper is to generalize the Fock single-electron wave function set for the case of the parabolic dot with the applied tilted magnetic eld, and to illustrate the in uence of the tilted eld on the electron energy spectrum and oscillator strengths. The consideration is carried out using the electron equations of motion and the close relation between the quantum mechanical and the classical descriptions of the parabolic systems. The paper is organized as follows. In Section 2 the problem is formulated, and in the next Section 3 the solution of Heisenberg equations of motion which actually coincide with the classical ones is outlined. Section 4 is devoted for Eigenvalue problem, and the quantum dot excitation spectrum and the oscillator strengths are illustrated in Section 5. In Section 6 the construction of wave function set is discussed and in the last, Section 7 the conclusions are given.
The magnetic eld symbol B actually stands for the ratio !c =!0 where !c is the electron–cyclotron frequency. When calculating the optical absorption we have to add the perturbation Hamiltonian HF = −(r · e0 )E exp(i!t) where r = {x; y; z} is the dimensionless electron coordinate vector, and the symbol E stands for the electric eld strength measured in ˜!0 =ea0 units. The symbol e0 = {ex ; ey ; ez } is the unit vector indicating the electric eld polarization.
3. Equations of motion It is known that the quantum mechanical description of the harmonic oscillator coincides with the classical one. In order to employ the classical analogies in the case of parabolic quantum dot with the tilted magnetic eld applied, we shall consider the quantum mechanical problem of the absorption using the Heisenberg representation, i.e. we shall solve the equations of motion for the single-electron operator – coordinate r(t) and momentum p(t). For the sake of convenience we shall put both operators together and de ne the vector operator with six components X = {r; p} = {x; y; z; px ; py ; pz }:
(2)
It enables to rewrite Hamiltonian (1) in terms of the following quadratic form:
2. Model We consider 3D parabolic quantum dot compressed in z-direction. The homogeneous magnetic eld B = {0; By ; Bz } = B{0; sin ’; cos ’} is applied in yz-plane where ’ is the angle between the magnetic eld and the z-direction as it is shown in the upper left corner of Fig. 1. The single electron in the dot is described by the following dimensionless Hamiltonian: H = 12 {(p + A)2 + (x2 + y2 + 2 z 2 )};
231
(1)
where the symbol A = {zBy ; xBz ; 0} stands for the vector potential. The energy is measured in ˜!0 units, where !0 is the characteristic frequency of the lateral con nement potential, and the unit of length is p a0 = ˜=m!0 . The parameter ¿ 1 characterizes the strength of the perpendicular con nement potential, or the perpendicular compression of the quantum dot.
H = 12 X + HX ;
(3)
where the superscript + indicates the Hermitian conjugate vector, and the symbol H stands for the Hamilton matrix which can be written down as the following block matrix: A B ; (4) H= B+ I where
0 1 + Bz2 0 0 Bz 0 ; B = 0 0 0 ; 1 0 A= 0 By 0 0 0 0 2 + By2 (5)
and I is 3 × 3 identity matrix.
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Fig. 1. Power absorption spectrum for various magnetic eld directions. The thickness of curves indicate the corresponding oscillator strengths.
The vector operator obeys the following equation of motion: 9 (6) − i X = [H + HF ; X ] = MX − if0 Eei!t ; 9t where the dynamic matrix M = SH is de ned as the product of Hamiltonian matrix (4) and the commutation matrix S = −[X ; X + ] with the elements Sij = [Xj ; Xi+ ], which can be written down as the following block matrix: 0 −iI S= : (7) iI 0 The symbol f0 = {0; e0 } is the analog of the polarization vector in the rp-space. Note that the Hamiltonian
matrix is real and symmetric while the matrix S is Hermitian and unitary, H+ = HT = H;
S+ = S−1 = −ST = S:
(8)
Here the superscript T indicates the transposed matrix. It is remarkable that equations of motion (6) exactly coincide with the classical ones if the vector X components are treated as the classical electron coordinates and momenta. Due to the linearity of Eq. (6) for the calculation of the absorption power only the non-homogeneous part of the solution is of importance. Assuming X (t) = X exp(i!t) it can be formally presented as X = iE(M − !)−1 f0 ;
(9)
T. Pyragien˙e, A. Matulis / Physica E 8 (2000) 230–238
and the absorption power is E E Re(e0 · r) ˙ = Re(!f0+ SX ) 2 2 E2 Re{i!f0+ S(M − !)−1 f0 }: (10) = 2 As the non-homogeneous term of Eq. (6) (and the obtained solution (9) as well) does not include any quantum mechanical operator, averaging over the system ground state |i0 is trivial, and consequently, the obtained absorption power coincides exactly with the classical one. The most simple way to nd the solution of Eq. (6) is to make use of the eigenvectors of the dynamic matrix obtained by solving the eigenvalue problem P=
{M − !n }Un = 0;
(11)
Vn+ {M − !n } = 0:
(12)
Note that, as the dynamic matrix is not Hermitian (M+ = H+ S+ = HS 6= M) both eigenvectors (left and right) have to be de ned. The above eigenvectors enable to rewrite absorption power (10) as a sum over the dynamic matrix eigenstates P=
E 2 P i!(f0+ SUn )(Vn+ f0 ) Re : 2 (!n − !)(Vn+ Un ) n
(13)
The obtained formal expansion has sense only in the case when the dissipation is properly taken into account. In the case of weak dissipation the power absorption can be obtained by some special trick – by adding a small imaginary shift to the external eld frequency (! → ! − i, → 0). It enables to replace the denominators in expression (13) by the corresponding Lorentzian functions and rewrite the power absorption in the following nal form: fn 1 P ; P(!) = E 2 2 2 2 n (! − !n ) +
(14)
where the symbol fn =
!n (f0+ SUn )(Vn+ f0 ) (Vn+ Un )
(15)
stands for the quantum (and the classical as well, see the Appendix in Ref. [9]) oscillator strength. Left and right eigenvectors can always be chosen orthogonal and normalized, namely, (Vn+ Um ) = nm . Moreover, in the case of weak dissipation ( → 0) the
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dynamic matrix should have P the complete eigenvector set what means that n Un Vn+ becomes just the identity matrix. Having that in mind (and expressions (11) and (4) as well) we see that P P P fn = (f0+ SMUn )(Vn+ f0 ) = (f0+ Hf0 ) n
n 2
= e = 1;
n
(16)
and the oscillator strengths obey the standard sum rule.
4. The eigenvalue problem The eigenvalues can be found zeroing the determinant of matrix {M − !} in Eq. (11) which immediately leads to the equation !6 − (2 + 2 + B2 )!4 + (1 + 2 2 + 2 Bz2 + By2 )!2 − 2 = 0:
(17)
That cubic for !2 equation can be easily solved using the Cardano expressions [10]. It follows from Eq. (17) that there are three pairs of real eigenvalues ±!n (n = 1; 2; 3). Actually it is valid for more general case, and it is a consequence of the symmetry properties (8). Moreover, taking complex conjugate of expression (11) one can easily check that there are two complex conjugate eigenvectors Un and Un∗ corresponding to the pair of opposite sign eigenvalues (±!n ). The most simple way to obtain the eigenvectors is to convert the initial 6 × 6 eigenvalue problem (11) into 3 × 3 one. For that purpose we rewrite it taking into account the explicit expressions of matrices (4) and (7) and denoting Un = {rn ; pn }. So, instead of expression (11) we obtain the equation rn A B(!n ) = 0; (18) pn B+ (!n ) I where B(!) = B + i!I. The obtained equation enables to substitute pn = −B + (!n )rn ;
(19)
and to transform it into more simple 3 × 3 matrix equation {A − B(!n )B+ (!n )}rn = 0:
(20)
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The straightforward solution of that equation leads to the following eigenvectors:
Table 1 Asymptotic expressions for frequencies
x n = (1 − !n2 )( 2 − !n2 );
!n
B→0
yn = i!n ( 2 − !n2 )Bz ;
1
+
zn = −i!n (1 − !n2 )By :
(21)
The left eigenvectors can be easily expressed through the right ones. Indeed, calculating the Hermitian conjugate of Eq. (12) we obtain {HS − !n }Vn = {M − !n }SVn = 0:
!n |f0+ SUn |2 ; (Un+ SUn )
(23)
or with the short eigenvectors as fn =
!n |e0 B+ (!n )rn |2 : 2Im{rn B(!n )rn }
(24)
In our special quantum dot with tilted magnetic eld case taking into account expression (21) it reduces to fn =
!n2 |e0 rn |2 : 2(!n2 |x n |2 + |yn |2 + 2 |zn |2 )
3
By2
2( 2 − 1) Bz 1+ 2 Bz 1− 2
B 1p 2 2
Bz + By2 B
p
2 Bz2 + By2
(22)
Comparing the obtained equation with Eq. (11) one may conclude that Vn = SUn , and rewrite expression (15) for the oscillator strength as fn =
2
B→∞
(25)
5. Power absorption spectrum and oscillator strengths The roots of bi-cubic equation (17) and expression (25) for oscillator strengths give us all necessary information about our system. In that section we shall discuss the qualitative peculiarities of the system behavior. In Fig. 1 the power absorption spectrum as the function of the magnetic eld strength for various magnetic eld directions is shown. The thickness of the curves indicate the oscillator strengths, each row corresponding to the di erent electric eld polarization. The spectrum branches with zero oscillator strengths as shown by dotted curves. In the case of the magnetic eld perpendicular to main dot plain (the rst column, ’ = 0) there are two separate electron motions. Two spectrum branches correspond to in-plane motion, and the third one (shown by the horizontal line) corresponds to the electron motion along the magnetic eld
direction (vertical motion, in this case). In tilted magnetic eld (the second column) electron motions are not independent, and consequently, the anti-crossing between two upper branches appears. And at last when the magnetic eld direction is in the main dot plain, the longitudinal mode (the horizontal line in the right column) becomes separated again. Two other coupled modes remind the in-plane modes of the previous perpendicular magnetic eld case, although now they are split in the small magnetic eld region due to di erent con nement potential strength in x- and z-direction. The physical meaning of the spectrum branches follows the weak and strong magnetic eld asymptotic expressions which are shown in Table 1. In the case of weak magnetic eld two degenerate in-plane motion branches demonstrate the linear Zeeman splitting proportional to the vertical magnetic eld component Bz . The single vertical motion branch undergoes the quadratic-Zeeman shift which is caused by the coupling of all motion modes in the tilted magnetic eld, and consequently, depends on its lateral component By . In the limit case of the strong magnetic eld the classical electron trajectories corresponding to the above three modes are shown in Fig. 2. The numbers on the trajectories indicate the spectrum branches as they are marked in the left lower part of Fig. 1. The Larmor circle (1) at the origin corresponds to the upper branch approaching proportional to B cyclotron frequency (thin dashed line). In the case of the middle branch (2) the electron oscillates along the magnetic eld direction with the weighted frequency. And the last lower branch (3) corresponds to the Larmor circle drift along the dot edge caused by the gradient of the electron potential energy with the guiding center velocity v = [V (r) × B]=B2 . Its
T. Pyragien˙e, A. Matulis / Physica E 8 (2000) 230–238
235
netic eld (the middle column) we see the behavior typical to the anti-crossing phenomena when the main oscillator strength smoothly switches from one mode to another. It is remarkable that there are regions of the magnetic eld strength where the electrical eld of any polarization excites more or less all spectrum branches. 6. Quantum mechanical states
Fig. 2. Electron motion in the limit case of strong magnetic eld. The thin dashed oval indicates the plane perpendicular to the magnetic eld. The numbers on electron trajectories correspond to the spectrum branches in Fig. 1.
frequency is inversely proportional to the magnetic eld strength averaged over the trajectory de ned by V (r) = const. In the case of the perpendicular magnetic eld (the rst column in Fig. 1) the oscillator strengths behavior is rather trivial. The longitudinal electric eld (polarized along the magnetic eld) excites only the vertical mode (2) while the electric eld with in-plane polarization (either in x- or y-direction) excites the two in-plane motion modes. In the case of small magnetic eld both of them have the same oscillator strengths, but when the magnetic eld strength increases, mostly the upper mode (in the limit case B → ∞ corresponding to the electron–cyclotron motion) is excited while the lower one – so called edge mode – rapidly looses its intensity. The similar behavior is seen in the case with the in-plane magnetic eld direction (the third column). Again the longitudinal electric eld excites only the single-longitudinal mode, while in the case of the perpendicular (to the magnetic eld) electric eld both perpendicular modes can be excited. However, they demonstrate di erent (as compared with the previous vertical magnetic eld case) oscillator strengths behaviour in the region of the small B due to the different physical nature of the spectrum branches in that limit region. Here the electric eld excites either the upper or the lower mode corresponding to its polarization. That is why in the case of electric eld polarized in the z-direction some broad peak in the edge mode oscillator strength is seen. In the case of the tilted mag-
The classical solution obtained in the previous section is sucient for the absorption power calculation. But if one looks for electron density distribution in a dot, or is going to perform the calculations of perturbed quantum dot properties, say, one likes to take into account the non-linear term of electron interaction, it is necessary to obtain the electron wave functions and the energy spectrum as well. In this section it is shown how to do that remaining in the framework of the above used equations of motion. For that purpose we shall apply to the Hamiltonian matrix (4) the same representation which makes the dynamic matrix diagonal, introducing the following transformation matrix: U = (U1 U2 U3 U1∗ U2∗ U3∗ ):
(26)
The columns of that matrix are just the right eigenvectors (11). Besides, the rst three of them correspond to the positive eigenvalues, while the last three – to the same eigenvalues taken with the opposite sign. The inverse transformation matrix U−1 can be constructed in the same way but using the left eigenvectors. We shall choose it as U−1 = RU+ S;
(27)
where R is the following diagonal block matrix: I 0 R= : (28) 0 −I Constructing the inverse matrix we assumed that the eigenvectors are normalized according to the condition, Un+ SUn = 1;
n = 1; 2; 3:
(29)
In agreement with expression (11) the transformation matrix obeys the following matrix equation: MU = UL;
(30)
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T. Pyragien˙e, A. Matulis / Physica E 8 (2000) 230–238
where the diagonal block matrix W 0 L= 0 −W
(31)
is composed of the dynamic matrix eigenvalues Wnm = nm !n . Now let us introduce new vector of operators + + Z = {a+ 1 ; a2 ; a3 ; a1 ; a2 ; a3 }
(32)
by means of the following transformation: X = UZ :
(33)
The commutators of those operators can be obtained in a formal way taking into account the previously de ned commutation matrix [X ; X + ] and eigenvector normalization condition (29). Indeed, one can write down [Z ; Z + ] = U−1 [X ; X + ](U−1 )+ = −U−1 SSUR = −R;
(34)
which immediately leads to the standard commutation relations for creation and annihilation operators [an ; a+ m ] = nm ;
[an ; am ] = 0:
(35)
In order to complete the quantum mechanical description let us apply transformation (33) for Hamiltonian (3) ˜ : 2H = X + HX = Z + U+ HUZ = Z + HZ
(36)
Taking into account expressions (27), (30) and commutation matrix properties (8) we present the new Hamiltonian matrix as ˜ = U+ SMU = U+ SUL = RL H
(37)
which leads to the following Hamiltonian: 3 1P + !n (an a+ n + an an ) 2 n=1 3 P 1 : !n a+ a + = n n 2 n=1
H=
(38)
Taking a look at expressions (35) and (38) we see that diagonalizing the classical equations of motion we arrived at the second quantization representation in quantum mechanical description of the problem. It is quite general statement for the linear systems which are described by the symmetric type (3) Hamiltonian, and thus, the presented technique is not restricted only to the parabolic 3D dot under the consideration.
The ground state wave function 0 can be constructed by means of the standard technique, i.e. it can be de ned by means of the following vector equation: a0 = 0
(39)
with the three-component vector operator de ned as a = {a1 ; a2 ; a3 }. The above equation corresponds to the absence of excited particles in the ground state. Now taking into account the transformation Z = U−1 X and using the last three rows of inverse matrix U−1 one can de ne two 3 × 3 matrices G and J, and rewrite Eq. (39) as (Gr + Jp)0 = 0:
(40)
It is easy to check that the above rst-order di erential equation set can be satis ed by the Gaussian type function 0 (r) = exp(− 12 r+ Cr);
(41)
with the quadratic form matrix in exponent de ned as C = iJ−1 G:
(42)
The speci c expression of that matrix is given in the appendix. It is remarkable that x-coordinate is coupled with other ones via imaginary matrix components only. Consequently, in the ground-state electron density expression (r) = |0 (r)| = exp(−rT Re Cr);
(43)
we have decoupled Gaussian electron distribution along the x-direction and its width is de ned by (vcxx )−1=2 coecient. As it is seen from expression (A.2) and Table 1 in the case of weak magnetic eld the width is de ned by the lateral con nement potential while in the asymptotic strong magnetic eld region it follows the magnetic length (proportional to B−1=2 ). The electron density in yz-plane is also Gaussian. Its width and orientation can be easily obtained by means of the diagonalization of D matrix. The qualitative picture of that electron density in the ground state is shown in Fig. 3. The quantum dot is indicated by a thick solid oval. The electron density is shown by two other ovals. The dashed one corresponds to the density in the case of zero magnetic eld. It is seen that it follows the dot con guration. The density in the case of strong magnetic eld is indicated by the shadowed oval. Now the density is compressed and
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237
7. Conclusions
Fig. 3. The ground state electron density in yz-plane: dashed ellipse indicates the density con guration in the case of weak magnetic eld, shadowed oval – strong magnetic eld case.
Fig. 4. The de ection of the density oval from the magnetic eld direction for various tilted magnetic eld angles ’: =8 – dashed, =4 – solid, and 3=8 – dotted curves.
elongated in nearly magnetic eld direction. It is remarkable that the density does not exactly follow the eld direction but is de ected towards the main dot plane where the con nement is weaker. The de ection angle dependence on the magnetic eld strength is shown in Fig. 4 for various magnetic eld angles. In the case of weak magnetic eld the de ection angle corresponds to the density elongated in the dot plane, and rapidly decreases with the increment of the magnetic eld strength. The wave functions of the excited states can be obtained in a standard way acting with operators a+ n on the ground state function (41).
The motion of electron in 3D parabolic dot compressed in the z-direction was considered in the case with the tilted magnetic eld applied by means of the Heizenberg equations of motion which actually coincides with the classical ones. Diagonalizing the dynamic matrix, the power absorption spectrum and the oscillator strengths were obtained. In the case of the perpendicular magnetic eld (directed along the z-direction) the excitation spectrum consists of two modes of in-plane motion (cyclotron and edge modes), and the magnetic eld is independent perpendicular mode. The tilted magnetic eld couples the cyclotron mode with the perpendicular one leading to the anti-crossing phenomena. The tilted magnetic eld also reduces the selection rules and facilitates the edge mode excitation. The dynamic matrix diagonalization leads to the second quantization representation in quantum mechanical description, and consequently, it enables in a quite standard way to construct the single-electron wave function set. We considered the electron density in the ground state. In the case of weak magnetic eld its shape follows the con nement potential pro le. In the opposite case of strong magnetic eld it becomes cigar-shaped, de ecting from the magnetic eld direction towards the direction of the weakest con nement potential. Appendix A. The ground-state wave function matrix The matrix C for ground-state wave function (41) can be calculated straightforwardly taking into account expressions (21) for right eigenvectors and inverse transformation matrix expression (27). That simple but cumbersome calculation leads to the following matrix: cxx −2il C=v : (A.1) −2il T D Here cxx = 2(!1 + !2 )(!2 + !3 )(!1 + !3 )= ;
(A.2)
l = {ly ; lz } is two-component vector with the following components: ly = Bz W3 ; lz = By W2 ;
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and the symbol D stands for 2 × 2 matrix with components dyy = W32 − 1= − 2 − By2 + Bz2 + 2; dzz = W12 + By2 − Bz2 ; dyz = dzy = −2By Bz : In the above expressions the following de nitions are used: v = =(W12 − B2 ); W1 = !1 + !2 + !3 + ; 1 1 1 1 + + + ; W2 = !1 !2 !3
1 W3 = !1 + !2 + !3 + :
References [1] L. Jacak, P. Hawrylak, A. Wojs, Quantum Dots, Springer, Berlin, 1998. [2] R. Rinaldi et al., Phys. Rev. Lett. 77 (1996) 342. [3] P.D. Wang et al., Phys. Rev. B 53 (1996) 16 458. [4] H. Lipsanen et al., Phys. Scripta T 79 (1999) 20. [5] L.R. Wilson et al., Phys. Rev. B 57 (1998) R2073. [6] G.-Q. Hai, F.M. Peeters, Phys. Rev. B 60 (1999) 8984. [7] V. Fock, Z. Phys. 47 (1928) 446. [8] J.C. Maan, in: G. Bauer, F. Kuchar, H. Heinrich (Eds.), Two-Dimensional Systems, Heterostructures and Superlattices, Springer, Berlin, 1984, p. 183. [9] B. Partoens, A. Matulis, F.M. Peeters, Phys. Rev. B 57 (1998) 13 039. [10] G. Korn, T. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, 1961.
www.elsevier.nl/locate/physe
Parabolic dot in tilted magnetic eld T. Pyragien˙e ∗ , A. Matulis Institute of Semiconductor Physics, GoÄstauto 11, 2600 Vilnius, Lithuania Received 8 March 2000; received in revised form 3 April 2000; accepted 10 April 2000
Abstract The electron motion, the power absorption spectrum and the oscillator strengths are considered in the case of the parabolic dot in a tilted magnetic eld by means of solving the equations of motion. It is shown that for a linear system the classical dynamic matrix diagonalization leads to the second quantization representation in quantum mechanical description. It enables to generalize the Fock single-electron wave function set for the case with the tilted magnetic eld. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 73.20.Dx; 03.65.−w Keywords: Electron density; Excitation spectrum; Oscillator strengths
1. Introduction Quantum dots, or artiÿcial atoms, have been a subject of intense theoretical and experimental research over the last few years [1]. The useful instrument in spectroscopy experiments is the magnetic eld applied in perpendicular to the quantum dot plane direction which enables to trace easily the quantum dot properties dependence on various parameters. Recently the experiments with the magnetic eld in quantum dot plane (Voight con guration) gained interest [2– 4], and even the magneto-photoluminescence dependence on the tilted magnetic eld direction was investigated [5]. The tilted magnetic eld couples the in-plane electron motion to the vertical one, and thus, enables to recover the quantum dot excitations which are forbidden ∗
Corresponding author.
in the case of more symmetric perpendicular magnetic eld. To our knowledge the tilted magnetic eld has been used for the theoretical polaron spectrum studies in the parabolic quantum wells [6]. The theoretical description of electron system in a quantum dot is mainly based on the wave function expansion into the series of non-interacting electron wave functions. In the most popular case of the parabolic dots the energy spectrum and the single-electron wave functions were calculated by Fock [7]. The analytical solution was obtained due to the circular symmetry of the quantum dot which leads to the separation of the variables. The tilted magnetic eld breaks the symmetry, and consequently, makes the problem more dicult. In the case of quantum well the remaining translation symmetry in the well plane along the perpendicular to the magnetic eld direction is sucient for the separa-
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 0 ) 0 0 1 5 0 - 8
T. Pyragien˙e, A. Matulis / Physica E 8 (2000) 230–238
tion of the variables and for obtaining the analytical solution as it is shown in Ref. [8]. That is not the case of the quantum dot, where in the tilted magnetic eld there is no more coordinate space symmetries left. Nevertheless, due to the parabolicity there are symmetries in the total phase rp-space, which is enabled to get the analytical solution and construct the single-electron wave function set in this more complicated case. The purpose of the present paper is to generalize the Fock single-electron wave function set for the case of the parabolic dot with the applied tilted magnetic eld, and to illustrate the in uence of the tilted eld on the electron energy spectrum and oscillator strengths. The consideration is carried out using the electron equations of motion and the close relation between the quantum mechanical and the classical descriptions of the parabolic systems. The paper is organized as follows. In Section 2 the problem is formulated, and in the next Section 3 the solution of Heisenberg equations of motion which actually coincide with the classical ones is outlined. Section 4 is devoted for Eigenvalue problem, and the quantum dot excitation spectrum and the oscillator strengths are illustrated in Section 5. In Section 6 the construction of wave function set is discussed and in the last, Section 7 the conclusions are given.
The magnetic eld symbol B actually stands for the ratio !c =!0 where !c is the electron–cyclotron frequency. When calculating the optical absorption we have to add the perturbation Hamiltonian HF = −(r · e0 )E exp(i!t) where r = {x; y; z} is the dimensionless electron coordinate vector, and the symbol E stands for the electric eld strength measured in ˜!0 =ea0 units. The symbol e0 = {ex ; ey ; ez } is the unit vector indicating the electric eld polarization.
3. Equations of motion It is known that the quantum mechanical description of the harmonic oscillator coincides with the classical one. In order to employ the classical analogies in the case of parabolic quantum dot with the tilted magnetic eld applied, we shall consider the quantum mechanical problem of the absorption using the Heisenberg representation, i.e. we shall solve the equations of motion for the single-electron operator – coordinate r(t) and momentum p(t). For the sake of convenience we shall put both operators together and de ne the vector operator with six components X = {r; p} = {x; y; z; px ; py ; pz }:
(2)
It enables to rewrite Hamiltonian (1) in terms of the following quadratic form:
2. Model We consider 3D parabolic quantum dot compressed in z-direction. The homogeneous magnetic eld B = {0; By ; Bz } = B{0; sin ’; cos ’} is applied in yz-plane where ’ is the angle between the magnetic eld and the z-direction as it is shown in the upper left corner of Fig. 1. The single electron in the dot is described by the following dimensionless Hamiltonian: H = 12 {(p + A)2 + (x2 + y2 + 2 z 2 )};
231
(1)
where the symbol A = {zBy ; xBz ; 0} stands for the vector potential. The energy is measured in ˜!0 units, where !0 is the characteristic frequency of the lateral con nement potential, and the unit of length is p a0 = ˜=m!0 . The parameter ¿ 1 characterizes the strength of the perpendicular con nement potential, or the perpendicular compression of the quantum dot.
H = 12 X + HX ;
(3)
where the superscript + indicates the Hermitian conjugate vector, and the symbol H stands for the Hamilton matrix which can be written down as the following block matrix: A B ; (4) H= B+ I where
0 1 + Bz2 0 0 Bz 0 ; B = 0 0 0 ; 1 0 A= 0 By 0 0 0 0 2 + By2 (5)
and I is 3 × 3 identity matrix.
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Fig. 1. Power absorption spectrum for various magnetic eld directions. The thickness of curves indicate the corresponding oscillator strengths.
The vector operator obeys the following equation of motion: 9 (6) − i X = [H + HF ; X ] = MX − if0 Eei!t ; 9t where the dynamic matrix M = SH is de ned as the product of Hamiltonian matrix (4) and the commutation matrix S = −[X ; X + ] with the elements Sij = [Xj ; Xi+ ], which can be written down as the following block matrix: 0 −iI S= : (7) iI 0 The symbol f0 = {0; e0 } is the analog of the polarization vector in the rp-space. Note that the Hamiltonian
matrix is real and symmetric while the matrix S is Hermitian and unitary, H+ = HT = H;
S+ = S−1 = −ST = S:
(8)
Here the superscript T indicates the transposed matrix. It is remarkable that equations of motion (6) exactly coincide with the classical ones if the vector X components are treated as the classical electron coordinates and momenta. Due to the linearity of Eq. (6) for the calculation of the absorption power only the non-homogeneous part of the solution is of importance. Assuming X (t) = X exp(i!t) it can be formally presented as X = iE(M − !)−1 f0 ;
(9)
T. Pyragien˙e, A. Matulis / Physica E 8 (2000) 230–238
and the absorption power is E E Re(e0 · r) ˙ = Re(!f0+ SX ) 2 2 E2 Re{i!f0+ S(M − !)−1 f0 }: (10) = 2 As the non-homogeneous term of Eq. (6) (and the obtained solution (9) as well) does not include any quantum mechanical operator, averaging over the system ground state |i0 is trivial, and consequently, the obtained absorption power coincides exactly with the classical one. The most simple way to nd the solution of Eq. (6) is to make use of the eigenvectors of the dynamic matrix obtained by solving the eigenvalue problem P=
{M − !n }Un = 0;
(11)
Vn+ {M − !n } = 0:
(12)
Note that, as the dynamic matrix is not Hermitian (M+ = H+ S+ = HS 6= M) both eigenvectors (left and right) have to be de ned. The above eigenvectors enable to rewrite absorption power (10) as a sum over the dynamic matrix eigenstates P=
E 2 P i!(f0+ SUn )(Vn+ f0 ) Re : 2 (!n − !)(Vn+ Un ) n
(13)
The obtained formal expansion has sense only in the case when the dissipation is properly taken into account. In the case of weak dissipation the power absorption can be obtained by some special trick – by adding a small imaginary shift to the external eld frequency (! → ! − i, → 0). It enables to replace the denominators in expression (13) by the corresponding Lorentzian functions and rewrite the power absorption in the following nal form: fn 1 P ; P(!) = E 2 2 2 2 n (! − !n ) +
(14)
where the symbol fn =
!n (f0+ SUn )(Vn+ f0 ) (Vn+ Un )
(15)
stands for the quantum (and the classical as well, see the Appendix in Ref. [9]) oscillator strength. Left and right eigenvectors can always be chosen orthogonal and normalized, namely, (Vn+ Um ) = nm . Moreover, in the case of weak dissipation ( → 0) the
233
dynamic matrix should have P the complete eigenvector set what means that n Un Vn+ becomes just the identity matrix. Having that in mind (and expressions (11) and (4) as well) we see that P P P fn = (f0+ SMUn )(Vn+ f0 ) = (f0+ Hf0 ) n
n 2
= e = 1;
n
(16)
and the oscillator strengths obey the standard sum rule.
4. The eigenvalue problem The eigenvalues can be found zeroing the determinant of matrix {M − !} in Eq. (11) which immediately leads to the equation !6 − (2 + 2 + B2 )!4 + (1 + 2 2 + 2 Bz2 + By2 )!2 − 2 = 0:
(17)
That cubic for !2 equation can be easily solved using the Cardano expressions [10]. It follows from Eq. (17) that there are three pairs of real eigenvalues ±!n (n = 1; 2; 3). Actually it is valid for more general case, and it is a consequence of the symmetry properties (8). Moreover, taking complex conjugate of expression (11) one can easily check that there are two complex conjugate eigenvectors Un and Un∗ corresponding to the pair of opposite sign eigenvalues (±!n ). The most simple way to obtain the eigenvectors is to convert the initial 6 × 6 eigenvalue problem (11) into 3 × 3 one. For that purpose we rewrite it taking into account the explicit expressions of matrices (4) and (7) and denoting Un = {rn ; pn }. So, instead of expression (11) we obtain the equation rn A B(!n ) = 0; (18) pn B+ (!n ) I where B(!) = B + i!I. The obtained equation enables to substitute pn = −B + (!n )rn ;
(19)
and to transform it into more simple 3 × 3 matrix equation {A − B(!n )B+ (!n )}rn = 0:
(20)
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The straightforward solution of that equation leads to the following eigenvectors:
Table 1 Asymptotic expressions for frequencies
x n = (1 − !n2 )( 2 − !n2 );
!n
B→0
yn = i!n ( 2 − !n2 )Bz ;
1
+
zn = −i!n (1 − !n2 )By :
(21)
The left eigenvectors can be easily expressed through the right ones. Indeed, calculating the Hermitian conjugate of Eq. (12) we obtain {HS − !n }Vn = {M − !n }SVn = 0:
!n |f0+ SUn |2 ; (Un+ SUn )
(23)
or with the short eigenvectors as fn =
!n |e0 B+ (!n )rn |2 : 2Im{rn B(!n )rn }
(24)
In our special quantum dot with tilted magnetic eld case taking into account expression (21) it reduces to fn =
!n2 |e0 rn |2 : 2(!n2 |x n |2 + |yn |2 + 2 |zn |2 )
3
By2
2( 2 − 1) Bz 1+ 2 Bz 1− 2
B 1p 2 2
Bz + By2 B
p
2 Bz2 + By2
(22)
Comparing the obtained equation with Eq. (11) one may conclude that Vn = SUn , and rewrite expression (15) for the oscillator strength as fn =
2
B→∞
(25)
5. Power absorption spectrum and oscillator strengths The roots of bi-cubic equation (17) and expression (25) for oscillator strengths give us all necessary information about our system. In that section we shall discuss the qualitative peculiarities of the system behavior. In Fig. 1 the power absorption spectrum as the function of the magnetic eld strength for various magnetic eld directions is shown. The thickness of the curves indicate the oscillator strengths, each row corresponding to the di erent electric eld polarization. The spectrum branches with zero oscillator strengths as shown by dotted curves. In the case of the magnetic eld perpendicular to main dot plain (the rst column, ’ = 0) there are two separate electron motions. Two spectrum branches correspond to in-plane motion, and the third one (shown by the horizontal line) corresponds to the electron motion along the magnetic eld
direction (vertical motion, in this case). In tilted magnetic eld (the second column) electron motions are not independent, and consequently, the anti-crossing between two upper branches appears. And at last when the magnetic eld direction is in the main dot plain, the longitudinal mode (the horizontal line in the right column) becomes separated again. Two other coupled modes remind the in-plane modes of the previous perpendicular magnetic eld case, although now they are split in the small magnetic eld region due to di erent con nement potential strength in x- and z-direction. The physical meaning of the spectrum branches follows the weak and strong magnetic eld asymptotic expressions which are shown in Table 1. In the case of weak magnetic eld two degenerate in-plane motion branches demonstrate the linear Zeeman splitting proportional to the vertical magnetic eld component Bz . The single vertical motion branch undergoes the quadratic-Zeeman shift which is caused by the coupling of all motion modes in the tilted magnetic eld, and consequently, depends on its lateral component By . In the limit case of the strong magnetic eld the classical electron trajectories corresponding to the above three modes are shown in Fig. 2. The numbers on the trajectories indicate the spectrum branches as they are marked in the left lower part of Fig. 1. The Larmor circle (1) at the origin corresponds to the upper branch approaching proportional to B cyclotron frequency (thin dashed line). In the case of the middle branch (2) the electron oscillates along the magnetic eld direction with the weighted frequency. And the last lower branch (3) corresponds to the Larmor circle drift along the dot edge caused by the gradient of the electron potential energy with the guiding center velocity v = [V (r) × B]=B2 . Its
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netic eld (the middle column) we see the behavior typical to the anti-crossing phenomena when the main oscillator strength smoothly switches from one mode to another. It is remarkable that there are regions of the magnetic eld strength where the electrical eld of any polarization excites more or less all spectrum branches. 6. Quantum mechanical states
Fig. 2. Electron motion in the limit case of strong magnetic eld. The thin dashed oval indicates the plane perpendicular to the magnetic eld. The numbers on electron trajectories correspond to the spectrum branches in Fig. 1.
frequency is inversely proportional to the magnetic eld strength averaged over the trajectory de ned by V (r) = const. In the case of the perpendicular magnetic eld (the rst column in Fig. 1) the oscillator strengths behavior is rather trivial. The longitudinal electric eld (polarized along the magnetic eld) excites only the vertical mode (2) while the electric eld with in-plane polarization (either in x- or y-direction) excites the two in-plane motion modes. In the case of small magnetic eld both of them have the same oscillator strengths, but when the magnetic eld strength increases, mostly the upper mode (in the limit case B → ∞ corresponding to the electron–cyclotron motion) is excited while the lower one – so called edge mode – rapidly looses its intensity. The similar behavior is seen in the case with the in-plane magnetic eld direction (the third column). Again the longitudinal electric eld excites only the single-longitudinal mode, while in the case of the perpendicular (to the magnetic eld) electric eld both perpendicular modes can be excited. However, they demonstrate di erent (as compared with the previous vertical magnetic eld case) oscillator strengths behaviour in the region of the small B due to the different physical nature of the spectrum branches in that limit region. Here the electric eld excites either the upper or the lower mode corresponding to its polarization. That is why in the case of electric eld polarized in the z-direction some broad peak in the edge mode oscillator strength is seen. In the case of the tilted mag-
The classical solution obtained in the previous section is sucient for the absorption power calculation. But if one looks for electron density distribution in a dot, or is going to perform the calculations of perturbed quantum dot properties, say, one likes to take into account the non-linear term of electron interaction, it is necessary to obtain the electron wave functions and the energy spectrum as well. In this section it is shown how to do that remaining in the framework of the above used equations of motion. For that purpose we shall apply to the Hamiltonian matrix (4) the same representation which makes the dynamic matrix diagonal, introducing the following transformation matrix: U = (U1 U2 U3 U1∗ U2∗ U3∗ ):
(26)
The columns of that matrix are just the right eigenvectors (11). Besides, the rst three of them correspond to the positive eigenvalues, while the last three – to the same eigenvalues taken with the opposite sign. The inverse transformation matrix U−1 can be constructed in the same way but using the left eigenvectors. We shall choose it as U−1 = RU+ S;
(27)
where R is the following diagonal block matrix: I 0 R= : (28) 0 −I Constructing the inverse matrix we assumed that the eigenvectors are normalized according to the condition, Un+ SUn = 1;
n = 1; 2; 3:
(29)
In agreement with expression (11) the transformation matrix obeys the following matrix equation: MU = UL;
(30)
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where the diagonal block matrix W 0 L= 0 −W
(31)
is composed of the dynamic matrix eigenvalues Wnm = nm !n . Now let us introduce new vector of operators + + Z = {a+ 1 ; a2 ; a3 ; a1 ; a2 ; a3 }
(32)
by means of the following transformation: X = UZ :
(33)
The commutators of those operators can be obtained in a formal way taking into account the previously de ned commutation matrix [X ; X + ] and eigenvector normalization condition (29). Indeed, one can write down [Z ; Z + ] = U−1 [X ; X + ](U−1 )+ = −U−1 SSUR = −R;
(34)
which immediately leads to the standard commutation relations for creation and annihilation operators [an ; a+ m ] = nm ;
[an ; am ] = 0:
(35)
In order to complete the quantum mechanical description let us apply transformation (33) for Hamiltonian (3) ˜ : 2H = X + HX = Z + U+ HUZ = Z + HZ
(36)
Taking into account expressions (27), (30) and commutation matrix properties (8) we present the new Hamiltonian matrix as ˜ = U+ SMU = U+ SUL = RL H
(37)
which leads to the following Hamiltonian: 3 1P + !n (an a+ n + an an ) 2 n=1 3 P 1 : !n a+ a + = n n 2 n=1
H=
(38)
Taking a look at expressions (35) and (38) we see that diagonalizing the classical equations of motion we arrived at the second quantization representation in quantum mechanical description of the problem. It is quite general statement for the linear systems which are described by the symmetric type (3) Hamiltonian, and thus, the presented technique is not restricted only to the parabolic 3D dot under the consideration.
The ground state wave function 0 can be constructed by means of the standard technique, i.e. it can be de ned by means of the following vector equation: a0 = 0
(39)
with the three-component vector operator de ned as a = {a1 ; a2 ; a3 }. The above equation corresponds to the absence of excited particles in the ground state. Now taking into account the transformation Z = U−1 X and using the last three rows of inverse matrix U−1 one can de ne two 3 × 3 matrices G and J, and rewrite Eq. (39) as (Gr + Jp)0 = 0:
(40)
It is easy to check that the above rst-order di erential equation set can be satis ed by the Gaussian type function 0 (r) = exp(− 12 r+ Cr);
(41)
with the quadratic form matrix in exponent de ned as C = iJ−1 G:
(42)
The speci c expression of that matrix is given in the appendix. It is remarkable that x-coordinate is coupled with other ones via imaginary matrix components only. Consequently, in the ground-state electron density expression (r) = |0 (r)| = exp(−rT Re Cr);
(43)
we have decoupled Gaussian electron distribution along the x-direction and its width is de ned by (vcxx )−1=2 coecient. As it is seen from expression (A.2) and Table 1 in the case of weak magnetic eld the width is de ned by the lateral con nement potential while in the asymptotic strong magnetic eld region it follows the magnetic length (proportional to B−1=2 ). The electron density in yz-plane is also Gaussian. Its width and orientation can be easily obtained by means of the diagonalization of D matrix. The qualitative picture of that electron density in the ground state is shown in Fig. 3. The quantum dot is indicated by a thick solid oval. The electron density is shown by two other ovals. The dashed one corresponds to the density in the case of zero magnetic eld. It is seen that it follows the dot con guration. The density in the case of strong magnetic eld is indicated by the shadowed oval. Now the density is compressed and
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7. Conclusions
Fig. 3. The ground state electron density in yz-plane: dashed ellipse indicates the density con guration in the case of weak magnetic eld, shadowed oval – strong magnetic eld case.
Fig. 4. The de ection of the density oval from the magnetic eld direction for various tilted magnetic eld angles ’: =8 – dashed, =4 – solid, and 3=8 – dotted curves.
elongated in nearly magnetic eld direction. It is remarkable that the density does not exactly follow the eld direction but is de ected towards the main dot plane where the con nement is weaker. The de ection angle dependence on the magnetic eld strength is shown in Fig. 4 for various magnetic eld angles. In the case of weak magnetic eld the de ection angle corresponds to the density elongated in the dot plane, and rapidly decreases with the increment of the magnetic eld strength. The wave functions of the excited states can be obtained in a standard way acting with operators a+ n on the ground state function (41).
The motion of electron in 3D parabolic dot compressed in the z-direction was considered in the case with the tilted magnetic eld applied by means of the Heizenberg equations of motion which actually coincides with the classical ones. Diagonalizing the dynamic matrix, the power absorption spectrum and the oscillator strengths were obtained. In the case of the perpendicular magnetic eld (directed along the z-direction) the excitation spectrum consists of two modes of in-plane motion (cyclotron and edge modes), and the magnetic eld is independent perpendicular mode. The tilted magnetic eld couples the cyclotron mode with the perpendicular one leading to the anti-crossing phenomena. The tilted magnetic eld also reduces the selection rules and facilitates the edge mode excitation. The dynamic matrix diagonalization leads to the second quantization representation in quantum mechanical description, and consequently, it enables in a quite standard way to construct the single-electron wave function set. We considered the electron density in the ground state. In the case of weak magnetic eld its shape follows the con nement potential pro le. In the opposite case of strong magnetic eld it becomes cigar-shaped, de ecting from the magnetic eld direction towards the direction of the weakest con nement potential. Appendix A. The ground-state wave function matrix The matrix C for ground-state wave function (41) can be calculated straightforwardly taking into account expressions (21) for right eigenvectors and inverse transformation matrix expression (27). That simple but cumbersome calculation leads to the following matrix: cxx −2il C=v : (A.1) −2il T D Here cxx = 2(!1 + !2 )(!2 + !3 )(!1 + !3 )= ;
(A.2)
l = {ly ; lz } is two-component vector with the following components: ly = Bz W3 ; lz = By W2 ;
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and the symbol D stands for 2 × 2 matrix with components dyy = W32 − 1= − 2 − By2 + Bz2 + 2; dzz = W12 + By2 − Bz2 ; dyz = dzy = −2By Bz : In the above expressions the following de nitions are used: v = =(W12 − B2 ); W1 = !1 + !2 + !3 + ; 1 1 1 1 + + + ; W2 = !1 !2 !3
1 W3 = !1 + !2 + !3 + :
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