Shun-Jen Cheng
Joint project of National Science Coucil of Taiwan with National Research Council of Canada and Department of Electrophysics, National Chiao Tung
University, Hsinchu 30050, Taiwan, R. O. C.
(Dated: November 15, 2006)
Abstract
In this report, we present the research results of the joint project between my research group at Department of Electrophysics, National Chiao Tung University, Hsinchu, and Quantum Theory group at Institute for Microstructural Sciences (IMS) of National Research Council, Ottawa. The project is partly carried out during the period of my visiting of IMS in this summer (July.2006-Sep.2006) under the support of NSC project No.NSC94-2112-M-009-011. In the studies, we reveal the characteristic emission spectra from a photoexcted semi-magnetic quantum dot containing one or two excitons interacting with two Mn2+ ions, calculated by using configuration interaction method. A journal paper based on the work is being in preparation.
PACS numbers:
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I. INTRODUCTION
In this work, we consider a 2D quantum dot conataining two Mn2+ impurities under photoexcitation and study the characteistic emission spectra from 1X-2Mn, and 2X-2Mn complexes in the semi-magnetic dots. Before performing the calculation of emission spectra, we first calculate the energy spectra and the eigen states of 0X-2Mn, 1X-2Mn, and 2X-2Mn complexes in dots.
In the calculations, we take the following parameters for II-VI SC QDs throughout this work: me = 0.106m0, mh = 0.424m0, = 10.6, ∆0 = 1meV, JeM(0) = 15meV · nm3, JhM(0) = 60meV · nm3, ωe = 4Ry∗, ωh = Ry∗, le = lh = l0 = p
Ry∗/~ωe· a∗B = 2.65nm , where Ry∗ = 12.8meV, a∗B = 5.29nm are the effective Rydberg and the effective Bohr radius for electron, respectively. The parameters of e-h exchange interaction: The energy splitting between the dark and bright X’s: δ0 = 32(∆0+94∆1z) = 1.0meV. The energy splitting between the bright X’s, arisen from deformation: δ1 = 34(∆1x− ∆1y) = 0 (symmetric dot considered).
The energy splitting between the dark X’s: δ2 = 34(∆1x+ ∆1y) = 0.1meV.
II. RELEVANT STUDIES IN LITERATURE
To date, we found the following papers relevant to the subject.
Experimental works:
1. the papers by the group of L. Besombes, Y. Leger, H.Mariette. Since 2000, the group demonstrate a series of experimental works of the emission spetra from II-VI self-assembled dots containing single Mn in magnetic fields. The characteristic spectrum of a single Mn interacting with single X, charged X, and bi-X are clearly identified, in comparison with the calculation based on simple effective spin Hamiltonian consisting of e-h exchnage, e-Mn, h-Mn spin-spin interactions. The features of the emission spectra differ from dot to dot, sensitively depending on the geometry of dot and the position of Mn. Only in the symmetric cases, six main lines with nearly equal energy spacing are observed in a spectrum.
In asymmetric cases, emssion lines fall into two groups, whose separation created by the ansiotropic part of e-h exchange interaction associated with the dot deformation. In the newly published PRL paper in 2006 [PRL 97,107401(2006)], they demonstarte the electrical control of the carrier number in dot and the measured characteristic emission spectrum of
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X, X+, X−, and XX, interacting with a single Mn, in a dot. Theory based on simple effective spin Hamiltonian well interpret the experimental results.
2. the pre-print by P.Wojnar, entitled “magneic QDs containing only few Mn ions”. In the work, micro-PL spectroscopy is carried out on individual II-V CdMnTe/ZnCdTe QDs in magnetic fields ranging from B = −6T to 6T. They observed the giant Zeeman effect in the PL spectra, i.e. the significant red shift of PL peaks with respect to increasing magnetic field. They proposed a model Hamiltonian to simulate the evolution of energyies, degree of circular polarization, and line width of the PL peaks with magnetic field. In the model, the ensemble of Mn ions is modeled as Mn cluster with a total effective spin ~S (it is however not very clear yet to me how they determine the spin value S). The interaction between carriers (electron and hole) and the model Mn cluster follows the Heisenberg model Hamiltonian.
The anti-ferromagnetc interaction between Mn’s is neglected here. The emission spectra are calculated by using the formulation of the Fermi’s golden rule. Comparing the simulation with the experimental results, it was estimated that a large (small) dot in the dot ensemble contains the number of Mn NM n ∼ 25 (NM n ∼ 5).
Theoretical works:
1. papers by A. O. Govorov et al (e.g. PRB, 70, 035321 (2004), “Optical probong of the spin state of a single magnetic impurity in a self-assembled quantum dot”.) In the first part of the paper, the authors calculate the energy spectrum of a 1X-1Mn asymmetric dot for different location of Mn by diagonalizing the spin Hamiltonian (both isotropic and anisotropic parts of e-h exchange interaction are considered). Then they solve the master equation to study the dynamic processes (pumping, emission, and relaxation) of 1Mn dot subject to a short pumping pulse. They showed that, with the resonant pumping technique, one can manipulate selectively individual spin of a Mn impurity, and a Mn impurity can act as qubit.
2. PRB, 73, 045301 (2006), J. Fernandez-Rossier, “Single-exciton spectroscopy of semi-magnetic quantum dots”. The paper presents the theoretical studies of the emission spectra from a single X in dot, interacting with few Mn’s(NM n = 1, 2, 3, 4). The authors took the hard-wall quantum box model for dot confining potental. For valence hole states, the (slight) HH-LH intermixing is considered.Although the X-Mn Hamiltonian is partly expressed in second quantization, the single X itself (to my understanding) is treated in single-particle picture (no higher shell scatterings due to e-e, h-h, e-h, e-Mn, and h-Mn interactions are
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considered). In the e-h exchange interaction, only the isotropic part of the e-h interaction is taken into account. The e-Mn and h-Mn interactions are modeled by spin-spin Heisen-berg Hamiltonian. The interaction between Mn’s is neglected because the sufficiently long distances between Mn’s are assumed for diluted magnetic dots (small number of Mn in each dot). Emission spectra are calculated using the Fermi’s golden rule. In the case of X-2Mn dots, the emission spectra versus the adjustable parameter r ≡ |ψ0(r2)|2/|ψ0(r1)| = 0.1 ∼ 1 are discussed, where r1(r2) is the position of the Mn#1 ( Mn#2). In symmetry case r = 1, 11 lines are observed in the calculated spectrum.
III. ENERGY SPECTRA
A. Two Mn’s QDs
We start with a carrier-free dot with two Mn2+ impurities located at postions ~R1 and R~2 in a dot, respectively. The interactions between Mn’s and between Mn and carriers in quantum dots are modeled by the spin-spin exchange Hamiltonian (For II-VI semi-magnetic semiconductors (SCs), Mn2+impurities substitute the divalent cations of SC and the induced electrostatic potential is negigible.) The Hamiltonian of a two-Mn dot system is thus written as
HM M = +|JM M(R12)| ~M1· ~M2, (1) where ~Mi denote the spin of the ith Mn, R12 ≡ | ~R2 − ~R1| is the distance between the two Mn2+ ions, and JM M(R12) = JM M(0) exp{−λ[R12/a0 − 1]} is the coupling constant of the anti-ferromagnetic Mn-Mn interaction, where a0 is the lattice constant and JM M(0) is a nearest-neighbor Mn-Mn interaction.[F.Qu PRL2006] The positive sign before the coupling constant in Eq.(1) indicates the anti-ferromangetic nature of the Mn-Mn interaction.
One can derive the energy spectrum of the 2Mn system with the Hamiltonian Eq.(1) analytically, which is given by
EM,Mz = |JM M(R12)|
2 [M (M + 1) − 35
2 ] , (2)
where M = 0, 1, ..5 is the total spin of the two-Mn system ( ~M ≡ ~M1+ ~M2) and Mz is its z-component. Accordingly, the ground state (GS) is that one with zero total spin of Mn’s M = 0, an anti-ferromagnetic (AF) state. Fig. 1 shows the calculated energy spectra of
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0 1 2 3 4 5
FIG. 1: The energy spectra of a quantum dot containing two Mn ions (a) vs the distance between Mn’s R12 (b) vs the total spin of the Mn’s for four different values ofR12 [marked with colored triangles in (a)]. See text for dot parameters.
the two-Mn systems with different R12. With increasing R12, the energy difference between states with spin M and (M + 1) decreases (E.g., the energy difference between the GS (M = 0) and the first excited state (M = 1) is given by |JM M(R12)|, which decreases with increasing with R12). For long R12&0.4l0, all of the 2Mn states becomes nearly degenerate.
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B. Single exciton interacting with two Mn’s
Under photoexcitation, excitons are created in the semi-magnetic QD and interact with Mn’s via the sp-d spin-spin exchange interaction. Experimetally, the number of exciton can be controlled by adjusting the power of excitation. Here, we consider a semi-magnetic 2D quantum dot with a single exciton and two Mn2+ ions. The Hmiltonian is expressed as
H1X2M n= HX0 + HeM + HhM + HM M , (3)
where HX0 = Te+Th−Vc(~re− ~rh)+Hehexis the total energy of a single exciton in a non-magnetic (Mn-free) dot, the e-h exchange interaction is given by
Hehex = −|∆0|~s · ~j − X
i=x,y,z
|∆1i|ji3si, (4)
and the e-Mn and h-Mn interaction are given by
HeM = −|JeM2D|~s · ~M1δ(~re− ~R1) − |JeM2D|~s · ~M2δ(~re− ~R2) , (5) and
HhM = +|JhM2D|~j · ~M1δ( ~rh− ~R1) + |JhM2D|~j · ~M2δ( ~rh− ~R2) , (6) respectively, where the coupling constant are given by JeM/hM2D ≡ JeM/hM(0) 2/d with d the thickness of quantum dot. To seek for the eigen solutions of the 1X-2Mn system, we take the four 1X-configuration |sz = ±1/2; jz = ±3/2i, combined with 36 possible 2Mn config-urations |M1z; M2zi, as basis and diagonalize the corresponding Hamiltonian matrix. Here we consider the pure heavy-hole states in quantum dots and hole spin has only two possible values jz = ±3/2. We also assume that the electrons and holes are frozen in their lowest orbital (s-orbit) state of quantum dot (The assuption will be released as we consider the RKKY effect in bi-exciton states of Mn-doped quantum dots). In the approximation, we can omit the constant kinetic energiesTe/h and direct Coulomb interaction Vc terms for brevity.
In the basis, the matrix elements of the Hamiltonian are given by h M1z0; M2z0|hsz0; jz0|H1X2M n|sz; jzi|M1z; M2zi
= hM1z0; M2z0|hsz0; jz0|Hehex− |JeM(1)|~s · ~M1− |JeM(2)|~s · ~M2
+|JhM(1)|jzM1z+ |JhM(2)|jzM2z+ |JM M(R12)| ~M1· ~M2|sz; jzi|M1z; M2zi ,
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where JeM/hM(I) = JeM/hM2D |ψ0( ~RI)|2 (I = 1, 2) is the coupling constant for electron/hole interacting with the Ith Mn. In the model of parabolic confinement potential, the wave function ψ0 is givn by ψm=0,n=0(x, y) = ψ0(x)ψ0(y), where ψ0(ex) = e−fx2/4/(2πl02)1/4 with ex = x/l0 and l0 =p
Ry∗/~ω0a∗B, where ω0 is the confinement frequency of parabolic potential.
1. Ground states (GS’s)
The 1X-2Mn Hamiltonian contains different types of carrier-carrier interactions: the fer-romagnetic (FM) e-Mn interaction, the anti-ferfer-romagnetic (AF) h-Mn interaction, and the AF Mn-Mn interaction. For bright X with opposite spins of hole and electron, a FM state of 2Mn is favored, but in the competition with the AF interactions between Mn’s. The spin properties of 1X-2Mn GS’s depend on the relative strengths of e-Mn, h-Mn, Mn-Mn, and e-h exchange interactions, determined by the positions of Mn’s.
Let us consider QDs containing the two Mn’s: one (Mn#1) fixed at the position ~R1 = l0(1, 0) and the other one’s (Mn#2) position movable on the x-y plane. Fig. 2(a) shows the calculated GS energy of the 1X-2Mn dot as a function of the second Mn’s position ~R2. Fig. 3(a) shows the mean value of total Mn spin hM2iGS ≡ h1X2M n; GS |M2|1X2M n; GSi of the 1X-2Mn GS’s of the same dot as a function of ~R2.
In Fig. 2(a), we see that the GS energy has a dip as the Mn#2 is placed in the region around ~R1 with radius R12 .0.3l0. In Fig. 3(a), we see the corresponding hM2iGS ∼ 0 i.e.
vanishes, indicating the AF GS of the 2Mn’s. The two Mn’s are in the AF GS, regardless of the existence of spin exciton, because of the strong Mn-Mn interaction in the short range.
As Mn#2 is placed far from Mn#1 ( R12 &0.3l0), the 2Mn GS will be in the FM phase ( hM2iGS ∼ 30) because of strong X-Mn FM interaction and weak Mn-Mn AF interaction.
The absolute value of the GS energy |EGS1X2M n| ∼ Eb1X2M n is approximately equal to the binding energy of 1X-2Mn complex, i.e. exciton magnetic polaron.
2. Excited states (ES’s)
Assuming JeM/hM(1) = JeM/hM(2) = ¯JeM/hM, the 1X-2Mn Hamitonian is written as H1X2M n = −|∆0|sz· jz− X
i=x,y,z
|∆1i|ji3si − | ¯JeM|~s · ~M + | ¯JhM|jzMz+ |JM M| ~M1· ~M2. (7)
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FIG. 2: Ground state energy spectra of photoexcied quantum dots with two Mn ions. Here we consider the case that one of the two Mn’s (Mn#1)is located at the fixed position ~R1 = (l0,0) while the other one (Mn#2) can be moved to any position ~R2 on the x-y plane. The GS energy vs R~2 of (a) a single exciton (b) a bi-exciton in the Mn-doped quantum dots.
The typical vaues of the parameters for II-VI semimagnetic semconductors: |∆0| ∼
| ¯JhM| > | ¯JeM| ∼ |∆1i|.
Let us consider two special cases:
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0 bi-exciton in the same Mn-doped quantum dots as in Fig.2. The charcteristic emission patterns of four cases for different ~R2, labled by capital letter A,B, C, and D, are studied in this work.
1. JM M → 0: The Hamiltonian is approximated to
H1X2M n ≈ −|∆0|sz· jz+ | ¯JhM|jzMz− | ¯JeM|~s · ~M + .... (8) Notable is that, due to the e-h exchange interaction, the 1X GS is supposed to be the dark X state. The carrier-Mn interaction affect the GS properties because the 2Mn in the GS’s is FM (Mz = 5). Due to the e-Mn interaction, the dark X GS’s are (slightly) intermixed with
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the bright X states, and becomes optically active. Moreover, the e-Mn interaction reduce the energy separation (∼ 0.15meV) between the dark GS’s and the lowest bright X excited states.
2. d ∼ a0 and JM M large:
H1X2M n≈ −|∆0|sz · jz+ |JM M| ~M1· ~M2+ | ¯JhM|jzMz− | ¯JeM|~s · ~M + ..., (9) Fig. ??(b) gives a schematic illustration of the energy spectrum of a 1X-2Mn dot under the condition. Because the 2Mn GS’s are the AF states (M = 0), the carrier-Mn interaction vanishes for the GS’s and the dark GS’s is not intermixed with the bright excited states.
The GS’s remain optically inactive as those of the Mn-free dots. The energy separation between the dark GS’s and the lowest bright X excited states remains the same (= 1meV) as that for Mn-free dots.
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