Optical properties in a two-dimensional quantum ring: Confinement potential and Aharonov-Bohm effect
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Optical properties in a two-dimensional quantum ring:Confinement potential and Aharonov-Bohm effect
Shijun Liang ∗ , Wenfang Xie ∗ ( ∗ Department of Physics, College of Physics and Electronic Engineering,Guangzhou University, Guangzhou 510006, P.R. China)
Abstract
Optical properties of a two-dimensional quantum ring with pseudopotentialin the presence of an external magnetic field and magnetic flux have beentheoretically investigated. Our results show that both of the pseudopotentialand magnetic field can affect the third nonlinear susceptibility and oscillatorstrength. In addition, we found that the oscillator strength and the absolutevalue of the resonant peak of the linear, non-linear and total absorption coef-ficient vary periodically with magnetic flux, while the resonant peak value ofthe linear, non-linear and total refractive index changes decreases as magneticflux increases.
Keywords:
Pseudopotential, Magnetic field, Magnetic flux, Opticalproperties, Oscillator strength
1. Introduction
Since 1970s Scientific research into electronic structure had been de-voted to two-dimensional structure — quantum wells [1,2], the new andunusual properties of quasi-two-dimensional systems, which promise applica-tions mostly in electronics and opto-electronics, have attracted the attentionof many researchers [3-10]. This in turn has resulted in a rapid developmentof production technology and extensive research. This rapid progress in tech-nology made it possible to create the quantum wire and quantum dot. Due tomuch more particular properties of electron confined in a quantum dot, a lot E-mail: shijun [email protected]
Preprint submitted to Optics Communications June 11, 2018 f studies on quantum dot have been done experimentally and theoretically[11-20]. One of the subjects concerned with quantum dots is to investigatetheir optical properties, especially nonlinear optical properties. Over the lastdecades, researchers have reported the linear and nonlinear optical propertiesof semiconductor quantum dots [21-30]. For instance, G. Rezaei, M. R. K.Vahdani and B. Vaseghi [31] studied nonlinear optical properties of a hydro-genic impurity in an ellipsoidal finite potential quantum dot, their resultsshow that the light intensity, size and geometry of the dot and aluminiumconcentration have a great influence on the absorption coefficient and refrac-tive index changes of the dot. H. A. Sarkisyan et al [32] presented indirecttransitions in thin films due to the Coulomb interactions between electronsand the frequency dependence as well as dependence on the concentration ofconductivity electrons and thickness of the film has been obtained. Karab-ulut [33] reported laser field effect on the nonlinear optical properties of asquare quantum well under the applied electric field, the results show thatthe laser field considerably affects the confining potential of the quantumwell and the nonlinear optical properties.As we know, modern electronic and optoelectronic devices can be nano-metric dimensions where microscopic details can not be treated in an ef-fective way, atomistic approaches become necessary for modelling struc-tural, electronic and optical properties of such nanostructured devices, andthe pseudopotential plays a important role in the studies of semiconduc-tor low-dimensional structures. Researchers often use the Pseudopotentialapproaches to theoretically obtain some important information about elec-tronic structures in semiconductor [34-43]. In addition, the pseudopotentialwas applied to interpreting some results from experiments with great success[44-46]. And it is well-known that Aharonov-Bohm effect actually refers tothe quantum mechanical phase of the wave function which is not a physi-cal observable. We can acquire the AB phase by a charged carrier whichtraverses a region where magnetic field doesn’t exist. Since Aharonov andBohm [47] provided the interpretation for Aharonov-Bohm effect in 1959,a lot of studies [48-57] have been reported about this topic. Those paperscovered many properties of low-dimensional semiconductor structures, someof which is focused on the optical properties of low-dimensional semiconduc-tors [58-61]. And most researchers focused their studies on the quantumring and results from calculations or experiments indicate that optical prop-erties of quantum ring are strongly affected by the Aharonov-Bohm effect.Recently, some new nanostructures, such as antidots, have attracted much2ttention. And researchers have reported in a lot of literatures [62-65] aboutthese structures. It would be very interesting if we can investigate a quantumring with pseudopotential. For this purpose, we will focus on studying effectsof an external magnetic field, magnetic flux quantum, pseudopotential on theoptical properties of a two-dimensional quantum ring in the present paper.The paper is organized as follows: in section 2 we describe the model andtheoretical framework, section 3 is dedicated to the results and discussions,and finally, our conclusions are given in section 4.
2. Model and calculations
Consider a two-dimensional quantum ring with pseudopotential, totalHamiltonian of the system with a uniform magnetic field B and AB fieldapplied simultaneously in the z-direction can be written as H = 12 m ∗ e h p + ec A i + V ( r ) , (1)In Eq.(1), m ∗ e is electronic effective mass, e is the electron charge, c is thespeed of light, A is a sum of two terms, A = A + A such that ∇ × A = B and ∇ × A = 0 for r = 0, where B denotes magnetic field. V ( r ) is thepseudopotential given as follows [65] V ( r ) = V ( rr − r r ) , (2)where V denotes the confinement strength on the two-dimensional elec-tron gas and r is the zero point of the potential. With the gauge A ϕ = Br and A ϕ = Φ AB πr , this is, A = (0 , Br + Φ AB πr , o dinger equation incylindrical coordinates can be written as − ~ m ∗ e [ 1 r ∂∂r ( r ∂∂r )+ 1 r ( ∂∂ϕ + i ΦΦ ) ]Ψ+ i ~ ω c ∂∂ϕ + i Φ AB Φ )Ψ+ m ∗ e ω c r V ( r )Ψ = E Ψ , where ω c = eB/m ∗ e c is the cyclotron frequency, Φ = hce is magnetic fluxquantum. The wave functions and the energy spectrum of an electron con-fined in a quantum ring can be obtained by solving the Schr¨ o dinger equation.Ψ( r, ϕ ) = N r | ξ | e − δr F ( − n, | ξ | + 1; δr ) e imϕ , (3)3 nm ( β ) = ~ ( n + | ξ | + 12 ) s ω c + 8 V m ∗ e r + ( m + α ) ~ ω c − V . (4)With ξ = r ( m + α ) + 2 m ∗ e V r ~ , (5)and δ = s e B ~ c + 2 m ∗ e V ~ r . (6)Where N is the normalization constant. F ( a, b ; x ) is the confluent hypergeo-metric function. n = 0 , , · · · , is main quantum number, m = 0 , ± , ± , · · · , is magnetic quantum number. α = Φ AB Φ is dimensionless measure of magneticflux Φ AB which is created by a solenoid inserted into center of the quantumring. Calculation of the linear and the third-order nonlinear opticalabsorption coefficient and refractive index changes
We employ compact-density approach to calculate the absorption coeffi-cient and the changes of the refractive index for a two-dimensional quantumring structure. Suppose the system is excited by an electromagnetic field as E ( t ) = E cos( ωt ) = e Ee iωt + e Ee − iωt . (7)The electronic polarization P ( t ) and susceptibility χ ( t ) are defined by thedipole operator M , and the density matrix ρ , respectively P ( t ) = ǫ χ ( ω ) e Ee iωt + ǫ χ ( − ω ) e Ee − iωt = 1 V T r ( ρM ) . (8)Where ǫ is the permittivity of free space, V denotes the volume of thesystem. T r denotes the trace or summation over the diagonal elements ofthe matrix ρM . We can obtain the analytic expressions [67] of the linear andthe third-order nonlinear susceptibilities.For the linear term ǫ χ (1) ( ω ) = ρ | M fi | E fi − ~ ω − i ~ Γ if , (9)4or the nonlinear term ǫ χ (3) ( ω ) = − ρ | M fi | e E E fi − ~ ω − i ~ Γ if [ 4 | M fi | ( E fi − ~ ω ) + ( ~ Γ if ) − ( M ff − M ii ) ( E fi − i ~ Γ if )( E fi − ~ ω − i ~ Γ if ) ] , where ρ denotes the carrier density. E fi = E f − E i is the energy interval ofthe two level system. M fi = e < Ψ f | x | Ψ i > is the electric dipole momentof the transition from the Ψ i state to Ψ f state. Γ is the phenomenologicaloperator. Non-diagonal matrix element Γ if ( i = f ) of operator Γ, which iscalled as relaxation rate of f th state, is the inverse of the relaxation time T if ,In our calculations Γ if = 1 /T if = 1 / . ps [68]. The susceptibility χ ( ω ) isrelated to the changes in the refractive index △ n ( ω ) /n r and the absorptioncoefficient α ( ω ) as follows △ n ( ω ) n r = Re ( χ ( ω )2 n r ) , (10) α ( ω ) = ω r µǫ R Im ( ǫ χ ( ω )) , (11)where µ is the permeability of the material, ǫ R = n r ǫ ( n r is the refractiveindex) is the real part of the permittivity.The linear and third-order nonlinear absorption coefficients are obtainedas follows α (1) ( ω ) = ω r µǫ R ρ s | M fi | ~ Γ if ( E fi − ~ ω ) + ( ~ Γ if ) (12) α (3) ( ω, I ) = − ω r µǫ R ( I ǫ n r c ) × ρ s | M fi | ~ Γ if ( E fi − ~ ω ) + ( ~ Γ if ) [4 | M fi | −| M ff − M ii | [3 E fi − E fi ~ ω + ~ ( ω − Γ if )] E fi + ( ~ Γ if ) ] . (13)Here I is is the intensity of incident radiation. So, the total absorptioncoefficient α ( ω, I ) is given by α ( ω, I ) = α (1) ( ω ) + α (3) ( ω, I ) . (14)The linear and the third-order nonlinear refractive index changes are ob-tained as follows △ n (1) ( ω ) n r = ρ s | M fi | n r ǫ E fi − ~ ω ( E fi − ~ ω ) + ( ~ Γ if ) , (15)5nd △ n (3) ( ω ) n r = − ρ s | M fi | n r ǫ µcI [( E fi − ~ ω ) + ( ~ Γ if ) ] × [4( E fi − ~ ω ) | M fi | − | M ff − M ii | ( E fi ) + ( ~ ω ) (( E fi − ~ ω )[ E fi ( E fi − ~ ω ) − ( ~ Γ if ) ] − ( ~ Γ if ) (2 E fi − ~ Γ))] . (16)Therefore, the total refractive index change △ n ( ω ) /n r can be written as △ n ( ω ) n r = △ n (1) ( ω ) n r + △ n (3) ( ω ) n r . (17) The oscillator strength of a transition is a dimensionless number whichis useful for comparing different transitions. And it is a very importantphysical quantity in the study of the optical properties which are related tothe electronic dipole-allowed transitions. Generally, the oscillator strength P fi is defined as P fi = 2 m ∗ e ~ E fi | M fi | . (18)
3. Results and Discussions
Our calculations are performed for
GaAa/Al x Ga − x As quantum dot. Theparameters chosen in this work are the followings: m ∗ e = (0 .
067 + 0 . x ) m ,where m is the free electron mass and x = 0 . ρ s = 5 × cm − . Inthis section, we set magnetic flux to zero. The third-order susceptibility as afunction of the photon energy with r = 4 nm and V = 350 meV for three dif-ferent magnetic field values, is shown in Fig. 1. From this figure, we can findthat the resonance peak decreases as the magnetic field increases, and thatthe peak position moves towards higher energies. Also, we can observe thatthe external magnetic field has a weak effect on the third-order susceptibilityin two-dimensional quantum ring. In order to give an explanation for thesebehaviors, energy differences and the product of geometric factor as a func-tion of the magnetic field with r = 4 nm and V = 350 meV , are plotted in6ig. 2 and Fig. 3. It can be seen that the energy intervals slightly increaseand the product of geometric factor, α α α α , considerably decreaseswhen increasing the external magnetic field. Therefore, both two trends cancontribute to reducing the third-order susceptibility. Moreover, in physicalstatement, By the increasing of the field the localization increases, resultingto the increase of the overlap integral. That is why the peak value of thethird-order susceptibility decreases. In Fig. 4, we plot the third-order sus-ceptibility as a function of the photon energy with B = 1 T and V = 350 meV for three different r values. We can clearly observe that the resonance peakof third-order susceptibility increases dramatically as r increases. Also, itis found that a red shift occurs in the resonance peak. The physical originsare shown in Fig. 5 and Fig. 6. As can be seen from Fig. 5 and Fig. 6, theenergy differences considerably decrease for smaller r and the product geo-metric factor α α α α , sharply increases as r increases, both of whichenhance the resonance peak value. So, we can draw a conclusion that thethird-order susceptibility is strongly dependent on the r . The third-ordersusceptibility as a function of the photon energy with B = 1 T and r = 4 nm for three different V values, has been presented in Fig. 7, to study the ef-fect of V on the third-order susceptibility. This figure clearly exhibits thatincreasing V leads to the increment in resonance peak of third-order sus-ceptibility. Meanwhile, we can also see that the peak position shifts towardshigher energies with increased V . Next, we illustrate these behaviors byFig. 8 and Fig. 9. We can find that from Fig. 8 energy intervals increasewith V . And in Fig. 9, product of the geometric factor is also enhancedwith V . However, the magnitude of the increment in the product of thegeometric factor is much bigger than that in energy differences. Hence, thethird-order susceptibility is enhanced. In Fig. 10, we demonstrate the third-order susceptibility of two-dimensional Al x Ga − x As pseudodot system as afunction of the photon energy with B = 1 T , r = 4 nm and V = 350 meV forthree different aluminium concentration x values. From this figure, we canobserve that aluminium concentration plays an important role in the third-order susceptibility of two-dimensional Al x Ga − x As pseudodot system, thepeak value increases as aluminium concentration x increases. In addition, itcan be seen that the resonance peak moves to lower energies. Finally, in Fig.11, we show the third-order susceptibility as a function of the photon energywith B = 1 T , r = 4 nm and V = 350 meV for three different τ (relaxationtime) values. The effect of relaxation time on the third-order susceptibilityis obvious. The longer the relaxation time is, the bigger the peak value of7hird-order susceptibility is. So the relaxation time has a strong influence onthe third-order susceptibility. In this section, we will discuss effecs of pseudopotential and magneticflux quantum on the oscillator strength. The Aharonov-Bohm effect is, quitegenerally, a non-local effect in which a physical object travels along a closedloop through a gauge field- free region and thereby undergoes a physicalchange, but we set the magnetic field value to 5 T in this section, which hasno influence on behavior that Oscillator strength versus magnetic flux α . InFig. 12, we presented the oscillator strength as a function of magnetic flux α ,magnetic field B , zero point of the pseudoharmonic potential r and potentialstrength of two-dimensional electron gas V . From the Fig. 12, we can seethat the oscillator strength has a continuous increase until the magnetic fluxcomes up to 0.3 where the oscillator strength reaches the maximum value, asthe magnetic flux increases. In Fig. 12a, we find that all three curves withdifferent magnetic field overlap. So the magnetic field has no influence on thecurve of oscillator strength with magnetic flux. But in Fig. 12b and Fig. 12cthe increasing r and V enhance the magnitude of oscillator strength. Andwe should note that the varying r has more effect on the curve curvature ofoscillator strength with magnetic flux than V does. We can interpret thisbehavior as follows. Confinement potential is enhanced by increasing r and V , increasing confinement potential leads to localization of wave functionand reduces the transition probability between the initial state and the finalstate. In addition, we also observe that the magnitude of oscillator strengthis very small. This indicates the transition probability is also very small. Soit is very difficult to observe this transition under this confinement. From previous literatures [62,66], we know that the smaller magneticfield can greatly affect Linear and nonlinear optical absorption coefficientand refractive index changes of two-dimensional quantum system with pseu-dopotential. In this section we will report magnetic flux effect on Linear andnonlinear Optical absorption coefficient and refractive index changes. It iswell-known that the magnetic flux influences the behavior of carrier wavefunction [47]. As we predict, in Fig. 13, the resonant peak value of linear,8on-linear and total absorption coefficient show the tendency to vary periodi-cally with magnetic flux. Also, the resonant peak is moved to higher energiesdue to increasing magnetic flux. This phenomenon occurs when the phase ofcarrier wave function is periodically changed by magnetic flux. In order tofurther study the effect of magnetic flux on the liner and non-linear opticalproperties of two-dimensional quantum ring, in Fig. 14, we plotted the curveof linear, non-linear and total refractive index changes versus photon energywith different magnetic flux value. the resonant peak value of the linear,non-linear and total refractive index changes, however, doesn’t vary period-ically as we expect. It is found that the increasing magnetic flux causes acontinuous decrease in the magnitude of resonant peak.
4. Summary
We have investigated the effects of an external magnetic field, magneticflux and confinement potential on optical properties of a two-dimensionalquantum ring. Our results shown: (i) The resonance peak of the third-ordersusceptibility decreases as the magnetic field increases, and that the peakposition moves towards higher energies. Also, the external magnetic fieldhas a weak influence on the third-order susceptibility in two-dimensionalquantum ring. (ii) The resonance peak of third-order susceptibility increasesdramatically as r increases. It is also found that the resonance peak hasa red shift. (iii) Increasing the potential V leads to the increment in reso-nance peak of third-order susceptibility, meanwhile, we can also see that thepeak position shifts towards higher energies as V increases. (iv) Aluminiumconcentration plays an important role in the third-order susceptibility of two-dimensional Al x Ga − x As pseudodot system. (v) The relaxation time has astrong influence on the third-order susceptibility. (vi) Unlike magnetic field,the magnetic flux, potential V , zero point r , have a great influence on theoscillator strength. (vii) Resonant peak value of the linear, non-linear andtotal absorption coefficient varies periodically with magnetic flux, while notfor refractive index changes.In conclusion, optical properties of two-dimensional quantum ring arestrongly affected by the external magnetic field, confinement potential, mag-netic flux, aluminium concentration and relaxation time. Especially for theeffect of magnetic flux on the optical properties, the researcher should takeinto account in designing optical devices. Finally we hope our research can9ontribute to understanding the two-dimensional quantum ring with pseu-dopotential better.
5. Acknowledgement
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The product of geometric factor as a function of the geometricsize of dot r with B = 1 T and V = 350 meV .Fig. 7. The third-order susceptibility as a function of the photon energywith B = 1 T and r = 4 nm for three different V values.Fig. 8. The energy differences as a function of chemical potential V with B = 1 T and r = 4 nm .Fig. 9. The product of geometric factor as a function of chemical potential V with B = 1 T and r = 4 nm .Fig. 10. The third-order susceptibility of two-dimensional Al x Ga − x As quantum system as a function of the photon energy with B = 1 T , r = 4 nm and V = 350 meV for three different aluminium concentration x values.Fig. 11. The third-order susceptibility as a function of the photon energywith B = 1 T , r = 4 nm and V = 350 meV for three different τ (relaxationtime) values.Fig. 12. (a) The oscillator strength versus magnetic flux with r = 2 nm and V = 100 meV for three different magnetic field values B . (b) Theoscillator strength versus magnetic flux with r = 2 nm and B = 5 T forthree different chemical potential values V . (c) The oscillator strength versusmagnetic flux with V = 100 meV and B = 5 T for three different zero pointvalues r .Fig. 13. The linear, non-linear and total absorption coefficient as a func-tion of magnetic flux with r = 2 nm and V = 100 meV and I = 0 . M W/cm .Fig. 14. The linear, non-linear and total refractive index changes as a func-tion of magnetic flux with r = 2 nm , V = 100 meV and I = 0 . M W/cm .16
35 440 445 450 455050100150200 | ( ) | ( - e s u . un it s ) ( meV ) r = = 350 meV Figure 1: The third-order susceptibility as a function of the photon energy with r = 4 nm and V = 350 meV for three different magnetic field values. E ne r g y d i ff e r en c e c hange ( e s u . un i t s ) ( ) r = = 350 meV Figure 2: The energy differences as a function of the external magnetic field with r = 4 nm and V = 350 meV . ( -- e s u . un it s ) ( ) r = = 350 meV Figure 3: The product of geometric factor as a function of the external magnetic field with r = 4 nm and V = 350 meV .
80 300 320 340 360 380 400 420 440 46002004006008001000120014001600 V = 350 meVB=1T ( meV ) | ( ) | ( - e s u . un it s ) r =4 nm r =5 nm r =6 nm Figure 4: Third-order susceptibility as a function of the photon energy with B = 1 T and V = 350 meV for three different r values. V = 350 meVB = 1 T E ne r g y d i ff e r en c e c hange ( e s u . un i t s ) r0 (nm ) Figure 5: The energy differences as a function of the geometric size of dot r with B = 1 T and V = 350 meV . ( - - e s u . un it s ) r nmB = 1TV0 = 350 meV Figure 6: The product of geometric factor as a function of the geometric size of dot r with B = 1 T and V = 350 meV .
00 410 420 430 440 450 460050100150200 ( meV ) | ( ) | ( - e s u . un it s ) V = 300 meV V = 325 meV V = 350 meVr = 1 nmB=1T Figure 7: The third-order susceptibility as a function of the photon energy with B = 1 T and r = 4 nm for three different V values.
00 150 200 250 300 3500.40.60.81.01.21.41.61.82.02.2
B = 1T r = E ne r g y d i ff e r en c e c hange ( e s u . un i t s ) V0 (meV ) Figure 8: The energy differences as a function of chemical potential V with B = 1 T and r = 4 nm .
00 150 200 250 300 35056789101112 ( -- e s u . un it s ) V meV B = 1T r = Figure 9: The product of geometric factor as a function of chemical potential V with B = 1 T and r = 4 nm .
70 380 390 400 410 420 430050100150200250 | ( ) | ( - e s u . un it s ) ( meV ) x=0.1 x=0.2 x=0.3 r = = 350 meV Figure 10: The third-order susceptibility of two-dimensional Al x Ga − x As pseudodot sys-tem as a function of the photon energy with B = 1 T , r = 4 nm and V = 350 meV forthree different aluminium concentration x values.
40 445 450 455050100150200250300350400450500550600650700750 | ( ) | ( - e s u . un it s ) ( meV ) ps ps ps Figure 11: The third-order susceptibility as a function of the photon energy with B = 1 T , r = 4 nm and V = 350 meV for three different τ (relaxation time) values. .0 0.2 0.4 0.6 0.8 1.04.04.55.05.56.06.5 0.0 0.2 0.4 0.6 0.8 1.01.52.02.53.03.54.04.55.05.56.06.50.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.02.53.03.54.04.55.05.56.06.5 V =100meV O sc ill a t o r s t r eng t h ( - )
5T 10T 15T (a) r =2nm O sc ill a t o r s t r eng t h ( - ) r =2nm V =100meV V =200meV V =300meV (b) O sc ill a t o r s t r eng t h ( - ) V =100meV r =2nm r =4nm r =6nm (c) Figure 12: (a) The oscillator strength versus magnetic flux with r = 2 nm and V =100 meV for three different magnetic field values B . (b) The oscillator strength versusmagnetic flux with r = 2 nm and B = 5 T for three different chemical potential values V .(c) The oscillator strength versus magnetic flux with V = 100 meV and B = 5 T for threedifferent zero point values r . Photon energy (meV) L i n ea r a b s o r p ti on c o e ff i c i e n t ( m - ) T o t a l a b s o r p ti on c o e ff i c i e n t ( m - ) Photon energy (meV) N on - li n ea r a b s o r p ti on c o e ff i c i e n t ( m - ) Photon energy (meV)
Figure 13: The linear, non-linear and total absorption coefficient as a function of magneticflux with r = 2 nm and V = 100 meV and I = 0 . M W/cm . Photon energy (meV) L i nea r r e f r a c t i v e i nde x c hange s Photon energy (meV) N on - li nea r r e f r a c t i v e i nde x c hange s Photon energy (meV) T o t a l r e f r a c t i v e i nde x c hange s Figure 14: The linear, non-linear and total refractive index changes as a function ofmagnetic flux with r = 2 nm , V = 100 meV and I = 0 . M W/cm ..