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Intersubband optical absorption in Gaussian GaAs quantum dot in the presence of magnetic, electrical and AB flux fields
Physica B 575 (2019) 411699
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Physica B: Physics of Condensed Matter journal homepage: http://www.elsevier.com/locate/physb
Intersubband optical absorption in Gaussian GaAs quantum dot in the presence of magnetic, electrical and AB flux fields Boda Aalu Department of Physics, Indian Institute of Technology Hyderabad, Kandi, 502285, Sangareddy, Telangana, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Energetic spectrum Absorption threshold frequency Magnetic Electrical and AB flux fields Gaussian quantum dot
We theoretically study the spectral properties of single electron two-dimensional (2D) Gaussian quantum dot (GQD) in the presence of applied magnetic, electrical field and along with an Aharonov–Bohm (AB) flux field. We have calculated the exact solutions for the normalized wave functions and energy levels by using the Nikifor ov–Uvarov (NU) method within the effective-mass approximation and compared the results with parabolic po tential (PP) model. Based on the calculated energy spectrum and the wave function, we have obtained the intersubband light absorption coefficient ðK ðϖÞÞ and the value of absorption threshold frequency ðϖÞ. The main and important object of the present work is to study the effect of the GQD size ðRÞ and the strength of the po tential ðV0 Þ on the energetic spectrum and the absorption threshold frequency ðϖÞ. According to the present work results, the ground state (GS) energy and the ðϖÞ shows that the size of the QD, depth of the potential and electric field plays an important role.
1. Introduction The subject of low-dimensional structures (LDS) such as quantum dots (QDs) has recently been attracted more and theoretical efforts have been done to interpret their magnetic, electronic and optical properties. The unabated interest continued particularly in the last three decades, in the presence of electric field and magnetic field or so essentially for two reasons. Firstly, the applied magnetic field gives an extra potential to the system which can change the optical and transport properties of charge carries in QDs. Secondly, applying an electric field gives rise to charge carrier redistribution that makes the shift to the energy of quantum levels which experimentally control and change the intensity of opto electronic devices [1,2]. It is very necessary to investigate the influence of magnetic and electric fields on the charge carries in Gaussian quan tum dots (GQDs). Currently, the experimental research is made to calculate the linear, nonlinear optical and other properties of LDS for the purpose of fabrication and the proper working state of optical, electronic devices [3–14]. One of the most interesting electrical and optical properties is the ground state energy and the light intersubband absorption coefficient (LIAC). The previous works mainly focused on the LIAC with restricted geometries of spherical [15–18], parabolic, rectangular and cylindrical
QDs [19] and LDS such as quantum wires (QW), wells, antiwells, and antidots [20–22] in the absence and presence of magnetic field (B) [1,2]. To obtain more information about the optical properties of nano structures, the reader can also refer to Refs. [23–26]. Some recent experiments have indicated that the confining potential in a QD is not really harmonic but rather anharmonic and has a finite depth [27,28]. Recently Adamowski et al. [29] have proposed a Gaussian confining potential for the investigation of the properties of excess electrons in QD. This potential has a central minimum and a finite depth and in the neighbourhood of the dot, the centre would behave like a parabolic potential and would thus approximately satisfy the gener alized Kohn theorem. Furthermore, in contrast to the rectangular po tential well, it is continuous at the dot boundary and this makes it easier to handle mathematically. Also, the force experienced by the particles within this potential well is nonzero, which is again a desirable feature. The other advantages with the Gaussian confining potential vis-a-vis a parabolic potential are that the former can describe, in addition to the excitations, the ionization and tunnelling processes. Masumoto and Takagahara [30] have shown that for small QDs, the Gaussian potential is indeed a good approximation for the confining potential. The Gaussian potential has already been used by several authors as the model for confinement to study the electronic properties of a QD
E-mail address: [email protected]. https://doi.org/10.1016/j.physb.2019.411699 Received 18 July 2019; Received in revised form 5 September 2019; Accepted 14 September 2019 Available online 19 September 2019 0921-4526/© 2019 Elsevier B.V. All rights reserved.
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Physica B: Physics of Condensed Matter 575 (2019) 411699
[31–47]. However, the effect of magnetic field on the GS energy and the light intersubband absorption of QDs are given considerable attention. But, the light intersubband absorption of Gaussian quantum dot (GQD) system under the influence of magnetic, electric and AB flux fields has not been reported theoretically so far in this direction. In this work, we study the GS energy and intersubband light ab sorption in a GQD system in the presence of magnetic, electric and AB flux fields. We investigated the wave functions and the light intersub band absorption coefficient to show the dependence of the GS energy and the threshold frequency of absorption on the strengths of applied magnetic, electric fields and also on size, the strength of the QD potential for different AB flux values. We also compared Gaussian potential model results with parabolic potential model results. The theoretical calcula tions are presented in Section. 2. The numerical discussion of results is presented in Section. 3. Finally, the conclusions are briefly given in Section. 4.
function of the harmonic oscillator (HO). The GPM problem now re duces to an effective PPM problem. The final confinement potential is taken the form � 1 1 Vðρ; zÞ ¼ m� ω2 r2 ¼ m� ω2 ρ2 þ z2 ; 2 2
(5)
The SE (1) with confinement potential (5) in cylindrical coordinates is given by � � � ℏ2 1 ∂ ∂2 1 ∂2 ∂2 iℏωc þ þ þ ψ ð ρ ; φ; zÞ 2m* ρ ∂ρ ∂ρ2 ρ2 ∂φ2 ∂z2 2 � � * iℏeΦAB 1 ∂ψ ðρ; φ; zÞ m 2 2 m* 2 2 e2 Φ2 1 e2 BΦAB þ þ þ Ω ρ þ ω z þ 2 AB 2πm* c ρ2 ∂φ 2 2 8π m* c2 ρ2 4πm* c2 � eF Z V0 � ψ ðρ; φ; zÞ ¼ Eψ ðρ; φ; zÞ; (6)
2. Theoretical model
where
The Schrodinger equation (SE) for a single electron GQD in the presence of magnetic, electrical and AB flux field in cylindrical co ordinates can be written as � � 1 � e �2 p þ A eF :Z þ VðrÞ ψ ðρ; φ; zÞ ¼ Eψ ðρ; φ; zÞ; (1) 2m* c
Ω2 ¼ ω2 þ
ω2 ¼ ð1
ω2c 4
;
κÞω20 þ 2κV0 Ω ℏ þ 2m* ΩR2
�
(7) 1
;
and, ωc ¼ meB* c is the cyclotron frequency, ψ ðρ; φ; zÞ is an electron wave function can be presented as
where p is the momentum of the charged electron in the applied mag netic field B ¼ r � A, A is the vector potential chosen in symmetric � � gauge as A ¼ Aρ ¼ Az ¼ 0; Aφ ¼ B2ρ The vector potential A can be
(8)
ψ ðρ; φ; zÞ ¼ R ðρ; φÞX ðZÞ;
and by separating variables in Eqn. (8), we obtain the equations that determine R ðρ; φÞ and X ðZÞ. � � � � � * ℏ2 1 ∂R ∂2 R 1 ∂2 R iℏωc iℏeΦAB 1 ∂R m 2 2 þ 2 þ 2 þ þ Ωρ * 2 * 2 2m ρ ∂ρ ∂ρ ρ ∂φ 2 2πm c ρ ∂φ 2 � e2 Φ2 1 e2 BΦAB R þ 2 AB þ * 2 8π m c ρ2 4πm* c2
represented as a sum of two terms, A ¼ A1 þ A2 so that r �A1 ¼ B; and r � A2 ¼ 0. Where B is the applied magnetic field in the z-direction and it is parallel to the two plane-parallel electrodes of infinite extent, and A2 gives the additional magnetic flux ΦAB generated by a solenoid inserted inside the QD. Therefore, the vector potentials have azimuthal compo nents, given by Refs. [48–50]. � � Bρ A1 ¼ A1ρ ¼ 0; A1φ ¼ ; A1z ¼ 0 ; 2 � � ΦAB ; A2z ¼ 0 ; A2 ¼ A2ρ ¼ 0; A2φ ¼ (2) 2πρ � � Bρ ΦAB A ¼ A1 þ A2 ¼ Aρ ¼ 0; Aφ ¼ ; A2 ¼ 0 ; þ 2 2πρ
¼ Eρ ðR Þ; (9) ℏ2 ∂ 2 X m* ω2 z2 þ X * 2 2m ∂z 2
eF ZX
V0 X ¼ Ez X ;
(10)
The solutions of Eqns. (9) and (10) are known and look like [51]. sffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffi � a ðjmjþαþ1Þ jmj þ α þ nρ imφ ρ22 jmjþα 4a � ρ R ðρ; φÞX ðZÞ ¼ pffiffiffiffiffi F nρ ; jmj e α jmjþ 2 jmj!nρ ! 2π � �2 � � �� �� � 1 z eF 2b2 2 2 * 2 ρ b eF m ω þ α þ 1; 2 � 1 pffiffiffiffiffiffiffiffiffiffiffie H nz z b ; m* ω2 2a π4 2nz nz !
b and now we as The electric field is applied in z-direction F ¼ F Z sume, the deviation of Gaussian potential (GP) from the parabolic po tential (PP) is small so that it can be treated as a PP plus a perturbation term. This is a reasonably good assumption for small r. For a QD, r will be generally small, then the Hamiltonian is now becoming a sum of a solvable harmonic 2D oscillator type (r-direction) and a shifted 1D oscillator (z-direction) type. It can be considered as a better approxi mation. The confinement potential is taken as 2 13 0
(11) qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi where a ¼ mℏ* Ω ; b ¼ mℏ* ω ; α ¼ ΦΦAB0 ; Φ0 ¼ hce; and Φ0 is quantum flux.
where ω20 ¼ V0 =m* R2 ; V0 and R are respectively the strength and range of the confining potential of the QD and κ ¼ 0 for a parabolic potential model (PPM) and κ ¼ 1 for a Gaussian potential model (GPM). More effectively we write it as " # 2 2 V0 1 * 2 〈e r =2R 〉 2 κ 2 m ω0 V0 (4) r ; 〈r2 〉 〈r 〉 2
Fða; b; xÞ→ confluent hypergeometric function. nρ ; m are the radial and magnetic quantum numbers respectively. Hnz ½x�→ Hermit polynomial and nz is the quantum number. In order to solve Eqns. (9) and (10), we have used NU method [49,52] which is also recently used by Ref. [51]. We exactly follow the same treatment to get the final energy levels. "� � rffiffiffiffi2ffiffiffiffiffiffiffiffiffi !# ωc jmj þ α þ 1 Eðn; m; nz ; α; F Þ ¼ ℏω n þ þ4 2 ω2 �� � m þ α��ωc � 1 e2 F 2 þ ℏω þ nz þ V0 : (12) 2 2 ω 2m* ω2
� � where r2 is the expectation value of r2 with respect to GS wave
Eqns. (11) and (12), obtained for charge carriers wave functions and energy spectrum in a 2D-QD in the external magnetic and electrical
1 VðrÞ ¼ m* ω20 r2 2
V0
B 61 κ4 m* ω20 r2 þ V0 @e 2
r2 2R2
C7 1A5;
(3)
2
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Physica B: Physics of Condensed Matter 575 (2019) 411699
Fig. 1. In (a) Energy (b) Magnetic moment ðM ¼ ∂E=∂BÞ (c) Magnetic sus ceptibility ðS ¼ ∂M=∂BÞ as a function of the magnetic field for different AB flux fields α; n ¼ nz ¼ m ¼ 0: Dashed curves for the PP model and solid curves for the GP model.
Fig. 2. Energies as a function of the magnetic field for ðaÞ various quantum numbers nz ; n ¼ m ¼ 0: and ðbÞ different magnetic quantum numbers m in the absence of electric field (F ¼ 0Þ; n ¼ nz ¼ 0: Dashed curves for the PP model and solid curves for the GP model.
fields together with AB flux field allow us to calculate the direct inter subband light absorption coefficient K ðϖÞ and the threshold frequency ðϖÞ of absorption in the present system. We have an expression for the light absorption coefficient [13,14]. �2 � X X �� � K ðϖÞ ¼ N �ψ en;m;nz ðρ; φ; zÞψ hn’ ;m’ ;n’z ðρ; φ; zÞρdρ dφ dz� � δ Δ
0 1 � � sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � �m þ α� eB Eg jmj þ α þ 1 @ e2 B2 2 þ 4ωe A þ ω nz þ nþ þ ϖ¼ e 2 2 m* c ℏ 2 m* c2 1 0 � � � rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 e2 F 2 jm’ j þ α þ 1 @ e2 B2 ’ 2A þ þ n þ þ 4ωh 2 2 2 2ℏm* ω2e m*’ c2 � ’ � � � m þα eB 1 e2 F 2 V0 ’ þ ω þ 2 ; n þ h z *’ m c 2 2 2ℏm*’ ω2h ℏ (14)
n;m;nz n’ m’ n’z
e En;m;n z
� Enh’ ;m’ ;n’z ;
(13)
where N → is a quantity proportional to the square of dipole moment matrix element modulus. Δ ¼ ℏϖ E g; E g → is the width of the � forbidden energy gap, ϖ→ is incident light frequency, ψ en;m;nz ; � �� � ψ hn’ ;m’ ;n’ → is wave function of electron and (hole), Een;m;nz ; z � �� Ehn’ ;m’ ;n’ → is the energy of the electron and (hole). We finally
To obtain the ground state threshold frequency value of absorption, we have taken n ¼ nz ¼ m ¼ 0 in the above equation then � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � � � � Eg αþ1 � e2 F 2 1 1 Ω2e þ 4ω2e ϖ 000 ¼ þ *’ 2 þ * 2 2 m ωe m ωh ℏ 2ℏ � ��qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � αþ1 α ω 2V0 e þ ωh 2 þ Ωh þ 4ω2h þ ðΩe þ Ωh Þ þ : (15) 2 2 2 ℏ
z
calculate the threshold frequency value of absorption is given by
3
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Physica B: Physics of Condensed Matter 575 (2019) 411699
Fig. 4. Energies vs. dot size R; for different AB flux fields α; n ¼ nz ¼ m ¼ F ¼ 0: Dashed curves for PP model and solid curves for the GP model.
function of the applied magnetic field for different values of the quan tum numbers nz in 2ðaÞ, and various values of the m in 2ðbÞ for the same set of parameter values of the QD ðR ¼ 10nm; V0 ¼ 5meV ; α ¼ 2Þ. In all the cases, the GS Eigen energies (in units meV) increase non-linearly as the applied magnetic field increases. The parabolic potential model (PPM) with κ ¼ 0 is overestimated the GS energies for higher values of nz and m particularly at a low magnetic field. But, at the high magnetic field, the GS energies are the same for GPM and PPM irrespective of the value of nz and m. As shown in Fig. 2(a) the GS energy ðn ¼ m ¼ 0Þ leads to a phase transition to high-lying states n > 0 as the nz quantum number varies in the ascending order. As shown in Fig. 2(b), The GS energies are enhanced as the applied magnetic field and the magnetic quantum number m increases when ωc > ω and we also found the crossing between m ¼ 1; m ¼ 1 when ωc < ω states. In Fig. 3(a), the GS energy ðnz ¼ n ¼ m ¼ 0Þ (in units meV) of an electron in a Gaussian potential model (GPM) κ ¼ 1 is plotted vs external magnetic field, B; for two different values of the AB flux field and in the
Fig. 3. ðaÞ Energies as a function of the magnetic field for α ¼ 0; 2; n ¼ nz ¼ m ¼ 0: ðbÞ Energies as a function of the electric field for various dashed curves for B ¼ 0T and solid curves for B ¼ 1T.
3. Numerical discussion of results The theoretical method discussed in section 2 is quite general and it can be applied to any quantum dot (QD). But, for the realness, we apply it to a GaAs QD for which the material parameters: m*h ¼ 0:09me (hole
mass), m*e ¼ 0:067me (electron mass) and E g ¼ 1:52eV (the width of the forbidden energy gap). In Fig. 1(a–c) We have plotted the ground state energy, magnetic moment and magnetic susceptibility, respectively as a function of the magnetic field for different values of the AB flux α: As shown in Fig. 1(a) the GS energy ðnz ¼ n ¼ m ¼ 0Þ leads to a phase transition to high-lying states ði:e: n > 0Þ in the presence of AB flux field. In Fig. 1(b) we plot the magnetic moment of an electron, in a GQD as a function of the B. It is clear from the curves of an electron system that they show diamagnetic behaviour. This figure also shows that the magnetic moment for an electron system decreases as the magnetic field increases. In Fig. 1(c), we also plot the magnetic susceptibility of an electron in a GQD as a function of the B: Firstly, we observe that in an electron system the magnetic susceptibility is diamagnetic. Secondly, we observe that it increase even for small values of the magnetic field. After the certain critical value of the magnetic field, the increase in the magnetic susceptibility cut down and finally, it saturates to the constant value. In Fig. 2(a–b) We have plotted the ground state (GS) energies as a
Fig. 5. Energies vs. potential strength V0 ; for different AB flux fields α; n ¼ nz ¼ m ¼ F ¼ 0: Dashed curves for PP model and solid curves for the GP model. 4
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Physica B: Physics of Condensed Matter 575 (2019) 411699
Fig. 6. Threshold frequency of absorption ϖ 000 vs. applied magnetic field for ðaÞ GP model κ ¼ 1; ðbÞ PP model κ ¼ 0 and R ¼ 10nm; V0 ¼ 5meV; F ¼ 0 in both the cases.
Fig. 9. Threshold frequency of absorption ϖ 000 as a function of potential strength V0 for the GP model κ ¼ 1. Dashed curves for F ¼ 0 and Solid curves for F ¼ 600kV=cm:
the electric field increases, after a certain value of the electric field the energy decreases very rapidly. With B ¼ 1T the GS energy increases because the applied magnetic field provides an extra potential to the system. In Fig. 4, the GS energy of a single electron QD is plotted vs the QD size, R. In Fig. 5, the GS energy of the same system is plotted vs the strength of the QD potential, V0: In these figures, the parameters are α ¼ 0; 1; 2; 3; 4; F ¼ 0; B ¼ 2T. According to the uncertainty principle, as the position coordinate of the electron decreases the momentum of the electron increases. As we see, for decreasing dot size the GS energy in creases. The well-known fact is that the GS energy decreases as the depth of the QD potential increases and it is true for α ¼ 0: But, for other values of the AB flux α ¼ 1; 2; 3; 4 as the QD potential, V0 increases the GS energy increases. In both cases, the PPM overestimates the GS energy. We plot the GS threshold frequency of absorption (TFA) ϖ 000 vs the magnetic field B; in Fig. 6. One can see from Fig. 6 (a) (Fig. 6 (b)) that the dependence of TFA ϖ 000 on the field, B is nonlinear for GPM and (linear) for PPM. The important behaviour in the application of AB flux field ΦAB leads to a group of the phase transition (PT) for the GS n ¼ 0, pre dominantly, α ¼ ΦΦAB0 ¼ 0; 1; 2; 3; 4 leads to a phase transitions for high-
Fig. 7. Threshold frequency of absorption ϖ 000 vs. applied magnetic field for ðaÞ GP model κ ¼ 1; ðbÞ PP model κ ¼ 0 and R ¼ 10nm; V0 ¼ 5meV in both the cases. Dashed curves for F ¼ 0 and Solid curves for F ¼ 600kV= cm:
lying states n > 0 in both the cases (κ ¼ 1; κ ¼ 0). In Fig. 7, we plot the GS TFA ϖ 000 in the presence and absence of the electric field for α ¼ 0; 2 vs external field B. In both the cases, the electric field lowers the GS TFA ϖ 000 as shown in Fig. 7(a and b). In Fig. 8, we present the GS threshold frequency of absorption (TFA) ϖ 000 of a QD as a function of QD radius (in units nm) for a particular value of field B ¼ 2T and QD potential V0 ¼ 5meV: It is seen from Fig. 8 that the GS TFA ϖ 000 increases when the QD size decreasing. In case of F 6¼ 0 and F ¼ 0, the TFA is same up to the certain value of the R then as QD R increases the TFA is decreasing more in case of F 6¼ 0 than in case of F ¼ 0: In the case of applied AB flux ΦAB creates a group of state transitions for α ¼ ΦΦAB0 ¼ 0; 1; 2; 3; 4: (here we have shown for α ¼ 0; 3
only). In Fig. 9, we plot the GS threshold frequency of absorption (TFA) ϖ 000 of a single electron QD as a function of the depth of the QD po tential (in units meV) for a particular value of field B ¼ 2T and QD size R ¼ 10nm: It is seen in Fig. 9 that the GS TFA ϖ 000 decreases when the QD potential increasing for α ¼ 0. But, the GS TFA ϖ 000 increases when the QD potential increasing for α ¼ 3. In the case of F 6¼ 0 and F ¼ 0, the TFA is shifting up to a certain value of V0 then as QD potential in creases, the TFA is same for F 6¼ 0 and. F ¼ 0:
Fig. 8. Threshold frequency of absorption ϖ 000 as a function of quantum dot size R for the GP model κ ¼ 1.
absence F ¼ 0 and presence of electric field F ¼ 600kVcm 1 : As it is seen, the GS energy shifts in the presence of an electric field. In Fig. 3(b), We have plotted the GS energy vs electric field for various values of the AB flux and for two values of B ¼ 0; 1T to study the effect of the electric field more rigorously. Initially, the GS energy decreases very slowly as 5
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Physica B: Physics of Condensed Matter 575 (2019) 411699
4. Conclusions
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
In the present work, we have studied the intersubband transitions in a quantum dot with the PP model and GP model in the presence of an applied magnetic, electric field and together with AB flux. We have calculated the light intersubband transition coefficient and threshold frequency of absorption by using the electron (hole) energy spectrum and their wave functions. Also, the thermodynamic properties of the present system can be studied by using this energy spectrum. We have studied the dependence of GS energy, magnetic moment, susceptibility and threshold frequency of absorption on the electric, magnetic fields and also the QD radius, potential. The GS energy increases as the field B is increased. For PPM, the threshold frequency of absorption is linear but it is non-linear for GPM. The energy levels make a shift by an amount ΔE ¼ ðe2 F 2 =2m* ω2 Þ in the presence of electric field and they are nondegenerate. It is also shown that the QD size and QD potential play important roles in GS energy and in absorption threshold frequency.
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
Acknowledgments
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Aalu Boda gratefully acknowledges the financial support from UGC, under the grant No: F./31-1/2017/PDFSS-2017-18-TEL-16652, India. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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Physica B: Physics of Condensed Matter journal homepage: http://www.elsevier.com/locate/physb
Intersubband optical absorption in Gaussian GaAs quantum dot in the presence of magnetic, electrical and AB flux fields Boda Aalu Department of Physics, Indian Institute of Technology Hyderabad, Kandi, 502285, Sangareddy, Telangana, India
A R T I C L E I N F O
A B S T R A C T
Keywords: Energetic spectrum Absorption threshold frequency Magnetic Electrical and AB flux fields Gaussian quantum dot
We theoretically study the spectral properties of single electron two-dimensional (2D) Gaussian quantum dot (GQD) in the presence of applied magnetic, electrical field and along with an Aharonov–Bohm (AB) flux field. We have calculated the exact solutions for the normalized wave functions and energy levels by using the Nikifor ov–Uvarov (NU) method within the effective-mass approximation and compared the results with parabolic po tential (PP) model. Based on the calculated energy spectrum and the wave function, we have obtained the intersubband light absorption coefficient ðK ðϖÞÞ and the value of absorption threshold frequency ðϖÞ. The main and important object of the present work is to study the effect of the GQD size ðRÞ and the strength of the po tential ðV0 Þ on the energetic spectrum and the absorption threshold frequency ðϖÞ. According to the present work results, the ground state (GS) energy and the ðϖÞ shows that the size of the QD, depth of the potential and electric field plays an important role.
1. Introduction The subject of low-dimensional structures (LDS) such as quantum dots (QDs) has recently been attracted more and theoretical efforts have been done to interpret their magnetic, electronic and optical properties. The unabated interest continued particularly in the last three decades, in the presence of electric field and magnetic field or so essentially for two reasons. Firstly, the applied magnetic field gives an extra potential to the system which can change the optical and transport properties of charge carries in QDs. Secondly, applying an electric field gives rise to charge carrier redistribution that makes the shift to the energy of quantum levels which experimentally control and change the intensity of opto electronic devices [1,2]. It is very necessary to investigate the influence of magnetic and electric fields on the charge carries in Gaussian quan tum dots (GQDs). Currently, the experimental research is made to calculate the linear, nonlinear optical and other properties of LDS for the purpose of fabrication and the proper working state of optical, electronic devices [3–14]. One of the most interesting electrical and optical properties is the ground state energy and the light intersubband absorption coefficient (LIAC). The previous works mainly focused on the LIAC with restricted geometries of spherical [15–18], parabolic, rectangular and cylindrical
QDs [19] and LDS such as quantum wires (QW), wells, antiwells, and antidots [20–22] in the absence and presence of magnetic field (B) [1,2]. To obtain more information about the optical properties of nano structures, the reader can also refer to Refs. [23–26]. Some recent experiments have indicated that the confining potential in a QD is not really harmonic but rather anharmonic and has a finite depth [27,28]. Recently Adamowski et al. [29] have proposed a Gaussian confining potential for the investigation of the properties of excess electrons in QD. This potential has a central minimum and a finite depth and in the neighbourhood of the dot, the centre would behave like a parabolic potential and would thus approximately satisfy the gener alized Kohn theorem. Furthermore, in contrast to the rectangular po tential well, it is continuous at the dot boundary and this makes it easier to handle mathematically. Also, the force experienced by the particles within this potential well is nonzero, which is again a desirable feature. The other advantages with the Gaussian confining potential vis-a-vis a parabolic potential are that the former can describe, in addition to the excitations, the ionization and tunnelling processes. Masumoto and Takagahara [30] have shown that for small QDs, the Gaussian potential is indeed a good approximation for the confining potential. The Gaussian potential has already been used by several authors as the model for confinement to study the electronic properties of a QD
E-mail address: [email protected]. https://doi.org/10.1016/j.physb.2019.411699 Received 18 July 2019; Received in revised form 5 September 2019; Accepted 14 September 2019 Available online 19 September 2019 0921-4526/© 2019 Elsevier B.V. All rights reserved.
A. Boda
Physica B: Physics of Condensed Matter 575 (2019) 411699
[31–47]. However, the effect of magnetic field on the GS energy and the light intersubband absorption of QDs are given considerable attention. But, the light intersubband absorption of Gaussian quantum dot (GQD) system under the influence of magnetic, electric and AB flux fields has not been reported theoretically so far in this direction. In this work, we study the GS energy and intersubband light ab sorption in a GQD system in the presence of magnetic, electric and AB flux fields. We investigated the wave functions and the light intersub band absorption coefficient to show the dependence of the GS energy and the threshold frequency of absorption on the strengths of applied magnetic, electric fields and also on size, the strength of the QD potential for different AB flux values. We also compared Gaussian potential model results with parabolic potential model results. The theoretical calcula tions are presented in Section. 2. The numerical discussion of results is presented in Section. 3. Finally, the conclusions are briefly given in Section. 4.
function of the harmonic oscillator (HO). The GPM problem now re duces to an effective PPM problem. The final confinement potential is taken the form � 1 1 Vðρ; zÞ ¼ m� ω2 r2 ¼ m� ω2 ρ2 þ z2 ; 2 2
(5)
The SE (1) with confinement potential (5) in cylindrical coordinates is given by � � � ℏ2 1 ∂ ∂2 1 ∂2 ∂2 iℏωc þ þ þ ψ ð ρ ; φ; zÞ 2m* ρ ∂ρ ∂ρ2 ρ2 ∂φ2 ∂z2 2 � � * iℏeΦAB 1 ∂ψ ðρ; φ; zÞ m 2 2 m* 2 2 e2 Φ2 1 e2 BΦAB þ þ þ Ω ρ þ ω z þ 2 AB 2πm* c ρ2 ∂φ 2 2 8π m* c2 ρ2 4πm* c2 � eF Z V0 � ψ ðρ; φ; zÞ ¼ Eψ ðρ; φ; zÞ; (6)
2. Theoretical model
where
The Schrodinger equation (SE) for a single electron GQD in the presence of magnetic, electrical and AB flux field in cylindrical co ordinates can be written as � � 1 � e �2 p þ A eF :Z þ VðrÞ ψ ðρ; φ; zÞ ¼ Eψ ðρ; φ; zÞ; (1) 2m* c
Ω2 ¼ ω2 þ
ω2 ¼ ð1
ω2c 4
;
κÞω20 þ 2κV0 Ω ℏ þ 2m* ΩR2
�
(7) 1
;
and, ωc ¼ meB* c is the cyclotron frequency, ψ ðρ; φ; zÞ is an electron wave function can be presented as
where p is the momentum of the charged electron in the applied mag netic field B ¼ r � A, A is the vector potential chosen in symmetric � � gauge as A ¼ Aρ ¼ Az ¼ 0; Aφ ¼ B2ρ The vector potential A can be
(8)
ψ ðρ; φ; zÞ ¼ R ðρ; φÞX ðZÞ;
and by separating variables in Eqn. (8), we obtain the equations that determine R ðρ; φÞ and X ðZÞ. � � � � � * ℏ2 1 ∂R ∂2 R 1 ∂2 R iℏωc iℏeΦAB 1 ∂R m 2 2 þ 2 þ 2 þ þ Ωρ * 2 * 2 2m ρ ∂ρ ∂ρ ρ ∂φ 2 2πm c ρ ∂φ 2 � e2 Φ2 1 e2 BΦAB R þ 2 AB þ * 2 8π m c ρ2 4πm* c2
represented as a sum of two terms, A ¼ A1 þ A2 so that r �A1 ¼ B; and r � A2 ¼ 0. Where B is the applied magnetic field in the z-direction and it is parallel to the two plane-parallel electrodes of infinite extent, and A2 gives the additional magnetic flux ΦAB generated by a solenoid inserted inside the QD. Therefore, the vector potentials have azimuthal compo nents, given by Refs. [48–50]. � � Bρ A1 ¼ A1ρ ¼ 0; A1φ ¼ ; A1z ¼ 0 ; 2 � � ΦAB ; A2z ¼ 0 ; A2 ¼ A2ρ ¼ 0; A2φ ¼ (2) 2πρ � � Bρ ΦAB A ¼ A1 þ A2 ¼ Aρ ¼ 0; Aφ ¼ ; A2 ¼ 0 ; þ 2 2πρ
¼ Eρ ðR Þ; (9) ℏ2 ∂ 2 X m* ω2 z2 þ X * 2 2m ∂z 2
eF ZX
V0 X ¼ Ez X ;
(10)
The solutions of Eqns. (9) and (10) are known and look like [51]. sffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffi � a ðjmjþαþ1Þ jmj þ α þ nρ imφ ρ22 jmjþα 4a � ρ R ðρ; φÞX ðZÞ ¼ pffiffiffiffiffi F nρ ; jmj e α jmjþ 2 jmj!nρ ! 2π � �2 � � �� �� � 1 z eF 2b2 2 2 * 2 ρ b eF m ω þ α þ 1; 2 � 1 pffiffiffiffiffiffiffiffiffiffiffie H nz z b ; m* ω2 2a π4 2nz nz !
b and now we as The electric field is applied in z-direction F ¼ F Z sume, the deviation of Gaussian potential (GP) from the parabolic po tential (PP) is small so that it can be treated as a PP plus a perturbation term. This is a reasonably good assumption for small r. For a QD, r will be generally small, then the Hamiltonian is now becoming a sum of a solvable harmonic 2D oscillator type (r-direction) and a shifted 1D oscillator (z-direction) type. It can be considered as a better approxi mation. The confinement potential is taken as 2 13 0
(11) qffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi where a ¼ mℏ* Ω ; b ¼ mℏ* ω ; α ¼ ΦΦAB0 ; Φ0 ¼ hce; and Φ0 is quantum flux.
where ω20 ¼ V0 =m* R2 ; V0 and R are respectively the strength and range of the confining potential of the QD and κ ¼ 0 for a parabolic potential model (PPM) and κ ¼ 1 for a Gaussian potential model (GPM). More effectively we write it as " # 2 2 V0 1 * 2 〈e r =2R 〉 2 κ 2 m ω0 V0 (4) r ; 〈r2 〉 〈r 〉 2
Fða; b; xÞ→ confluent hypergeometric function. nρ ; m are the radial and magnetic quantum numbers respectively. Hnz ½x�→ Hermit polynomial and nz is the quantum number. In order to solve Eqns. (9) and (10), we have used NU method [49,52] which is also recently used by Ref. [51]. We exactly follow the same treatment to get the final energy levels. "� � rffiffiffiffi2ffiffiffiffiffiffiffiffiffi !# ωc jmj þ α þ 1 Eðn; m; nz ; α; F Þ ¼ ℏω n þ þ4 2 ω2 �� � m þ α��ωc � 1 e2 F 2 þ ℏω þ nz þ V0 : (12) 2 2 ω 2m* ω2
� � where r2 is the expectation value of r2 with respect to GS wave
Eqns. (11) and (12), obtained for charge carriers wave functions and energy spectrum in a 2D-QD in the external magnetic and electrical
1 VðrÞ ¼ m* ω20 r2 2
V0
B 61 κ4 m* ω20 r2 þ V0 @e 2
r2 2R2
C7 1A5;
(3)
2
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Physica B: Physics of Condensed Matter 575 (2019) 411699
Fig. 1. In (a) Energy (b) Magnetic moment ðM ¼ ∂E=∂BÞ (c) Magnetic sus ceptibility ðS ¼ ∂M=∂BÞ as a function of the magnetic field for different AB flux fields α; n ¼ nz ¼ m ¼ 0: Dashed curves for the PP model and solid curves for the GP model.
Fig. 2. Energies as a function of the magnetic field for ðaÞ various quantum numbers nz ; n ¼ m ¼ 0: and ðbÞ different magnetic quantum numbers m in the absence of electric field (F ¼ 0Þ; n ¼ nz ¼ 0: Dashed curves for the PP model and solid curves for the GP model.
fields together with AB flux field allow us to calculate the direct inter subband light absorption coefficient K ðϖÞ and the threshold frequency ðϖÞ of absorption in the present system. We have an expression for the light absorption coefficient [13,14]. �2 � X X �� � K ðϖÞ ¼ N �ψ en;m;nz ðρ; φ; zÞψ hn’ ;m’ ;n’z ðρ; φ; zÞρdρ dφ dz� � δ Δ
0 1 � � sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � �m þ α� eB Eg jmj þ α þ 1 @ e2 B2 2 þ 4ωe A þ ω nz þ nþ þ ϖ¼ e 2 2 m* c ℏ 2 m* c2 1 0 � � � rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 e2 F 2 jm’ j þ α þ 1 @ e2 B2 ’ 2A þ þ n þ þ 4ωh 2 2 2 2ℏm* ω2e m*’ c2 � ’ � � � m þα eB 1 e2 F 2 V0 ’ þ ω þ 2 ; n þ h z *’ m c 2 2 2ℏm*’ ω2h ℏ (14)
n;m;nz n’ m’ n’z
e En;m;n z
� Enh’ ;m’ ;n’z ;
(13)
where N → is a quantity proportional to the square of dipole moment matrix element modulus. Δ ¼ ℏϖ E g; E g → is the width of the � forbidden energy gap, ϖ→ is incident light frequency, ψ en;m;nz ; � �� � ψ hn’ ;m’ ;n’ → is wave function of electron and (hole), Een;m;nz ; z � �� Ehn’ ;m’ ;n’ → is the energy of the electron and (hole). We finally
To obtain the ground state threshold frequency value of absorption, we have taken n ¼ nz ¼ m ¼ 0 in the above equation then � qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � � � � Eg αþ1 � e2 F 2 1 1 Ω2e þ 4ω2e ϖ 000 ¼ þ *’ 2 þ * 2 2 m ωe m ωh ℏ 2ℏ � ��qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � αþ1 α ω 2V0 e þ ωh 2 þ Ωh þ 4ω2h þ ðΩe þ Ωh Þ þ : (15) 2 2 2 ℏ
z
calculate the threshold frequency value of absorption is given by
3
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Physica B: Physics of Condensed Matter 575 (2019) 411699
Fig. 4. Energies vs. dot size R; for different AB flux fields α; n ¼ nz ¼ m ¼ F ¼ 0: Dashed curves for PP model and solid curves for the GP model.
function of the applied magnetic field for different values of the quan tum numbers nz in 2ðaÞ, and various values of the m in 2ðbÞ for the same set of parameter values of the QD ðR ¼ 10nm; V0 ¼ 5meV ; α ¼ 2Þ. In all the cases, the GS Eigen energies (in units meV) increase non-linearly as the applied magnetic field increases. The parabolic potential model (PPM) with κ ¼ 0 is overestimated the GS energies for higher values of nz and m particularly at a low magnetic field. But, at the high magnetic field, the GS energies are the same for GPM and PPM irrespective of the value of nz and m. As shown in Fig. 2(a) the GS energy ðn ¼ m ¼ 0Þ leads to a phase transition to high-lying states n > 0 as the nz quantum number varies in the ascending order. As shown in Fig. 2(b), The GS energies are enhanced as the applied magnetic field and the magnetic quantum number m increases when ωc > ω and we also found the crossing between m ¼ 1; m ¼ 1 when ωc < ω states. In Fig. 3(a), the GS energy ðnz ¼ n ¼ m ¼ 0Þ (in units meV) of an electron in a Gaussian potential model (GPM) κ ¼ 1 is plotted vs external magnetic field, B; for two different values of the AB flux field and in the
Fig. 3. ðaÞ Energies as a function of the magnetic field for α ¼ 0; 2; n ¼ nz ¼ m ¼ 0: ðbÞ Energies as a function of the electric field for various dashed curves for B ¼ 0T and solid curves for B ¼ 1T.
3. Numerical discussion of results The theoretical method discussed in section 2 is quite general and it can be applied to any quantum dot (QD). But, for the realness, we apply it to a GaAs QD for which the material parameters: m*h ¼ 0:09me (hole
mass), m*e ¼ 0:067me (electron mass) and E g ¼ 1:52eV (the width of the forbidden energy gap). In Fig. 1(a–c) We have plotted the ground state energy, magnetic moment and magnetic susceptibility, respectively as a function of the magnetic field for different values of the AB flux α: As shown in Fig. 1(a) the GS energy ðnz ¼ n ¼ m ¼ 0Þ leads to a phase transition to high-lying states ði:e: n > 0Þ in the presence of AB flux field. In Fig. 1(b) we plot the magnetic moment of an electron, in a GQD as a function of the B. It is clear from the curves of an electron system that they show diamagnetic behaviour. This figure also shows that the magnetic moment for an electron system decreases as the magnetic field increases. In Fig. 1(c), we also plot the magnetic susceptibility of an electron in a GQD as a function of the B: Firstly, we observe that in an electron system the magnetic susceptibility is diamagnetic. Secondly, we observe that it increase even for small values of the magnetic field. After the certain critical value of the magnetic field, the increase in the magnetic susceptibility cut down and finally, it saturates to the constant value. In Fig. 2(a–b) We have plotted the ground state (GS) energies as a
Fig. 5. Energies vs. potential strength V0 ; for different AB flux fields α; n ¼ nz ¼ m ¼ F ¼ 0: Dashed curves for PP model and solid curves for the GP model. 4
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Physica B: Physics of Condensed Matter 575 (2019) 411699
Fig. 6. Threshold frequency of absorption ϖ 000 vs. applied magnetic field for ðaÞ GP model κ ¼ 1; ðbÞ PP model κ ¼ 0 and R ¼ 10nm; V0 ¼ 5meV; F ¼ 0 in both the cases.
Fig. 9. Threshold frequency of absorption ϖ 000 as a function of potential strength V0 for the GP model κ ¼ 1. Dashed curves for F ¼ 0 and Solid curves for F ¼ 600kV=cm:
the electric field increases, after a certain value of the electric field the energy decreases very rapidly. With B ¼ 1T the GS energy increases because the applied magnetic field provides an extra potential to the system. In Fig. 4, the GS energy of a single electron QD is plotted vs the QD size, R. In Fig. 5, the GS energy of the same system is plotted vs the strength of the QD potential, V0: In these figures, the parameters are α ¼ 0; 1; 2; 3; 4; F ¼ 0; B ¼ 2T. According to the uncertainty principle, as the position coordinate of the electron decreases the momentum of the electron increases. As we see, for decreasing dot size the GS energy in creases. The well-known fact is that the GS energy decreases as the depth of the QD potential increases and it is true for α ¼ 0: But, for other values of the AB flux α ¼ 1; 2; 3; 4 as the QD potential, V0 increases the GS energy increases. In both cases, the PPM overestimates the GS energy. We plot the GS threshold frequency of absorption (TFA) ϖ 000 vs the magnetic field B; in Fig. 6. One can see from Fig. 6 (a) (Fig. 6 (b)) that the dependence of TFA ϖ 000 on the field, B is nonlinear for GPM and (linear) for PPM. The important behaviour in the application of AB flux field ΦAB leads to a group of the phase transition (PT) for the GS n ¼ 0, pre dominantly, α ¼ ΦΦAB0 ¼ 0; 1; 2; 3; 4 leads to a phase transitions for high-
Fig. 7. Threshold frequency of absorption ϖ 000 vs. applied magnetic field for ðaÞ GP model κ ¼ 1; ðbÞ PP model κ ¼ 0 and R ¼ 10nm; V0 ¼ 5meV in both the cases. Dashed curves for F ¼ 0 and Solid curves for F ¼ 600kV= cm:
lying states n > 0 in both the cases (κ ¼ 1; κ ¼ 0). In Fig. 7, we plot the GS TFA ϖ 000 in the presence and absence of the electric field for α ¼ 0; 2 vs external field B. In both the cases, the electric field lowers the GS TFA ϖ 000 as shown in Fig. 7(a and b). In Fig. 8, we present the GS threshold frequency of absorption (TFA) ϖ 000 of a QD as a function of QD radius (in units nm) for a particular value of field B ¼ 2T and QD potential V0 ¼ 5meV: It is seen from Fig. 8 that the GS TFA ϖ 000 increases when the QD size decreasing. In case of F 6¼ 0 and F ¼ 0, the TFA is same up to the certain value of the R then as QD R increases the TFA is decreasing more in case of F 6¼ 0 than in case of F ¼ 0: In the case of applied AB flux ΦAB creates a group of state transitions for α ¼ ΦΦAB0 ¼ 0; 1; 2; 3; 4: (here we have shown for α ¼ 0; 3
only). In Fig. 9, we plot the GS threshold frequency of absorption (TFA) ϖ 000 of a single electron QD as a function of the depth of the QD po tential (in units meV) for a particular value of field B ¼ 2T and QD size R ¼ 10nm: It is seen in Fig. 9 that the GS TFA ϖ 000 decreases when the QD potential increasing for α ¼ 0. But, the GS TFA ϖ 000 increases when the QD potential increasing for α ¼ 3. In the case of F 6¼ 0 and F ¼ 0, the TFA is shifting up to a certain value of V0 then as QD potential in creases, the TFA is same for F 6¼ 0 and. F ¼ 0:
Fig. 8. Threshold frequency of absorption ϖ 000 as a function of quantum dot size R for the GP model κ ¼ 1.
absence F ¼ 0 and presence of electric field F ¼ 600kVcm 1 : As it is seen, the GS energy shifts in the presence of an electric field. In Fig. 3(b), We have plotted the GS energy vs electric field for various values of the AB flux and for two values of B ¼ 0; 1T to study the effect of the electric field more rigorously. Initially, the GS energy decreases very slowly as 5
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Physica B: Physics of Condensed Matter 575 (2019) 411699
4. Conclusions
[15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
In the present work, we have studied the intersubband transitions in a quantum dot with the PP model and GP model in the presence of an applied magnetic, electric field and together with AB flux. We have calculated the light intersubband transition coefficient and threshold frequency of absorption by using the electron (hole) energy spectrum and their wave functions. Also, the thermodynamic properties of the present system can be studied by using this energy spectrum. We have studied the dependence of GS energy, magnetic moment, susceptibility and threshold frequency of absorption on the electric, magnetic fields and also the QD radius, potential. The GS energy increases as the field B is increased. For PPM, the threshold frequency of absorption is linear but it is non-linear for GPM. The energy levels make a shift by an amount ΔE ¼ ðe2 F 2 =2m* ω2 Þ in the presence of electric field and they are nondegenerate. It is also shown that the QD size and QD potential play important roles in GS energy and in absorption threshold frequency.
[25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
Acknowledgments
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