Scattering in a varying mass PT symmetric double heterojunction
aa r X i v : . [ qu a n t - ph ] J un Scattering in a varying mass
P T symmetric doubleheterojunction
Anjana Sinha ∗ Department of Instrumentation Science,Jadavpur University, Kolkata - 700 032, INDIAand
R. Roychoudhury
Advanced Centre for Nonlinear and Complex Phenomena1175 Survey Park, Kolkata - 700075, INDIA
Abstract
We observe that the reflection and transmission coefficients of a particle within a double, PT symmetric het-erojunction with spatially varying mass, show interesting features, depending on the degree of non Hermiticity,although there is no spontaneous breakdown of PT symmetry. The potential profile in the intermediate layer isconsidered such that it has a non vanishing imaginary part near the heterojunctions. Exact analytical solutionsfor the wave function are obtained, and the reflection and transmission coefficients are plotted as a function ofenergy, for both left as well as right incidence. As expected, the spatial dependence on mass changes the na-ture of the scattering solutions within the heterojunctions, and the space-time ( PT ) symmetry is responsible forthe left-right asymmetry in the reflection and transmission coefficients. However, the non vanishing imaginarycomponent of the potential near the heterojunctions gives new and interesting results. Key words :
Position-dependent-effective-mass; Rosen-Morse II; PT -symmetry; Semiconductor Heterojunction;Scattering PACS numbers : ∗ e-mail : anjana23@rediffmail.com; [email protected] e-mail : rajdaju@rediffmail.com; [email protected] : +91 33 24146321; phone : +91 33 25753020 I. INTRODUCTION
Position dependent effective mass (PDEM) for-malism is extremely important in describing theelectronic and transport properties of quantumwells and quantum dots, impurities in crystals,He-clusters, quantum liquids, semiconductor het-erostructures, etc. [1-6]. In semiconductor het-erostructures (e.g., say, Al x Ga − x As), the spatialdependence on the mass of the charge carrier (elec-tron or hole) occurs due to its interaction with anensemble of particles within the device, as the par-ticle propagates along the z direction, because ofthe varying doping concentration or the mole frac-tion x along the z -axis. On the other hand, itis established fact that non Hermitian quantumsystems with PT symmetry, open up a fascinat-ing world, unknown to conventional Hermitian sys-tems [7-9]. PT synthetic novel optical devices havebeen engineered to exhibit several intriguing fea-tures [10-19]. These are structures with balancedgain and loss such that the parity-time ( PT ) sym-metry of the entire system is preserved. Thesematerials depict altogether new behavior unknownto Hermitian optical systems — e.g., double re-fraction, power oscillations, unidirectional invisi- bility, left right asymmetry, non reciprocal diffrac-tion patterns, etc.In this work we study a special form of semi-conductor device consisting of a thin layer of PT symmetric material sandwiched between two nor-mal semiconductors, such that the effective massof the charge carrier (electron or hole) varies withposition within the heterojunctions, but is con-stant outside. The mass m ( z ) and the real partof the potential V R ( z ) are taken to be continuousthroughout the device. The form of V ( z ) (where V ( z ) = V R ( z ) + V I ( z )) is such that V I ( z ) does notvanish near the heterojunctions. It is this non van-ishing imaginary component that gives rise to someinteresting results. We obtain the exact analyticalsolutions for the bound and scattering states ofa particle inside such a semiconductor device andalso obtain the reflection and transmission ampli-tudes, R and T respectively. Our primary aim hereis to look for any anomaly in | R | and/or | T | , withincrease in the magnitude of the imaginary com-ponent V I ( z ).The article is organized as follows : For thesake of completeness, the position-dependent-massSchr¨odinger equation is briefly introduced in Sec-tion 2, to show the procedure for obtaining the1xact analytical solutions. In Section 3, we studyan explicit non Hermitian PT symmetric doubleheterojunction, the potential profile of which hasa non vanishing imaginary part near the hetero-junctions. The potential and mass functions stud-ied here are shown graphically, as function of z ,in Fig. 1. The exact analytical solutions for thescattering states of a particle in such a device areplotted in Fig. 2. A complete understanding of anymodel requires knowledge of both bound as well asscattering states. Bearing this in mind, we plotthe first three bound states in Fig. 3, for the sameparameter values as in Figures 1 and 2. The mainstress in this work is on the behaviour of the trans-mission and reflection coefficients, with respect tothe real and imaginary part of the potential, andthe mass functions. To explore the phenomenonof left-right asymmetry, typical of non Hermitianquantum systems, a series of graphs showing thetransmission coefficient | T | and reflection coeffi-cient | R | , for left and right incidence, are plottedin Figures 4 to 8. Section 4 is kept for Conclusionsand Discussions. II. THEORY
Within the heterojuntions a < z < a , where theparticle mass varies with position, the Hermitiankinetic energy term T EM is given by [20, 21] T EM = 14 (cid:0) m α pm β pm γ + m γ pm β pm α (cid:1) = 12 p (cid:18) m (cid:19) p (1)where p = − i ~ ddz is the momentum operator. Theambiguity parameters α , β , γ obey the von Roosconstraint [20] α + β + γ = − ~ = c = 1, and use prime to denote dif-ferentiation w.r.t. z . Furthermore, for continuityconditions at the abrupt interfaces, well behavedground state energy [22, 23], and the best fit toexperimental results [24], we shall restrict the am-biguity parameters to satisfy the BenDaniel-Dukechoice, viz., α = γ = 0 , β = −
1. Thus, in theintermediate layer a < z < a , the Hamiltonianfor the particle with PDEM assumes the form [25] H = − m ( z ) d dz − (cid:18) m ( z ) (cid:19) ′ ddz + V R ( z )+ iV I ( z )(3) whereas, outside the well, z < a and z > a , theparticle obeys the conventional Schr¨odinger equa-tion : (cid:26) − m , d dz + V , (cid:27) ψ ( z ) = Eψ ( z ) (4)having plane wave solutions. In case we considera wave incident from left, the solutions in the tworegions are ψ L ( z ) = e ik z + Re − ik z , −∞ < z < a ψ R ( z ) = T e ik z , a < z < ∞ (5)where R and T denote the reflection and transmis-sion amplitudes, and k , = q m , ( E − V , ) (6)The important point worth noting here is that forPDEM systems, the solutions ψ ( z ) obey modifiedboundary conditions [26, 27] — the functions ψ ( z )and 1 m ( z ) dψ ( z ) dz are continuous at each hetero-junction a and a . These are used to calculate R and T .In case one uses the following transformations [28] ψ in ( z ) = { m ( z ) } / φ ( ρ ) , ρ = Z p m ( z ) dz (7)then the Schr¨odinger equation for PDEM in theregion a < z < a , viz., eq. (3), reduces to onefor constant mass viz., − d φdρ + n e V ( ρ ) − E o φ = 0 (8)with e V ( ρ ) = V ( z ) + 732 m ′ m − m ′′ m (9)Evidently, eq. (8) can be solved analytically forsome particular cases of V ( z ) and m ( z ) only. Inref. [29], we had given the exact analytical so-lutions for one such case, which shows the phe-nomenon of spontaneous PT symmetry breaking,and admits a spectral singularity. In this work, weshall study a second case, which neither has any ex-ceptional point in the bound state spectrum, nora spectral singularity in the continuous spectrum,but nevertheless, shows some interesting results.Additionally, contrary to our previous model, themost important contribution of the imaginary partof the present potential is near the heterojunctions.Thus, this present study is distinctively differentfrom the work done in ref. [29].2 II. EXPLICIT MODEL : PT SYMMETRICPOTENTIAL WELL WITH POSITION DE-PENDENT EFFECTIVE MASS
We assume the real part of the intermediate layerto be a diffused quantum well, similar to our earlierwork [29]. However, contrary to our earlier study,the main contribution from the imaginary part ofthe potential is near the heterojunctions. To beprecise, we consider the following ansatz for thepotential V ( z ) and mass m ( z ) : V ( z ) = − µ z + i µ z √ z , | z | < a − µ a = V , | z | > a (10) m ( z ) = g z ) , | z | < a g a ) = m , | z | > a (11)where µ , µ , g are some constant parameters.Fig. 1 shows the mass dependence m ( z ) and thepotential V ( z ) in the entire semiconductor device,as a function of z , for a suitable set of parametervalues, viz., g = 1 . , µ = 4 , µ = . , a = 2 . m H z L m m V V Re V H z L Im V H z L a - a Black (dotted) line ® m (z) Red (dashed) line ® Re V ( z )Blue (solid) line ® Im V(z) - - - - - - FIG. 1: Colour online : Plot showing m ( z ) and V ( z )w.r.t. z For the spatial mass dependence given by eq. (11),eq. (7) transforms the coordinate z to ρ = β sinh − z (12)so that after some straightforward algebra e V ( ρ ) ineq. (9) reduces to e V ( ρ ) = 14 g − (cid:18) µ − g (cid:19) sech ρg + iµ g tanh ρg (13)Thus equation (8) may be written as d φd ¯ ρ + (cid:8) κ + s ( s + 1) sech ¯ ρ − iλ tanh ¯ ρ (cid:9) φ = 0(14)where κ = Eg − , ¯ ρ = ρg (15)and the new parameters s and λ are expressed interms of the constants µ , µ and g , as λ = 12 µ g , s = − ± g √ µ (16)One can check from eq. (13) that the diffused po-tential well [eq. (10)] within the intermediate layer | z | < a , with spatially varying mass m ( z ), reducesto the PT symmetric Rosen Morse II potential forconstant mass [30, 31]. One may note that thestandard PT symmetric Rosen Morse II potentialhas the following unique characteristics :(i) absence of quasi-parity,(ii) only real energy due to the absence of sponta-neous breakdown of PT symmetry(iii) switching of the bound state energies switchfrom negative to positive values, with increase inthe magnitude of the non Hermiticity parameter.Our aim in this work is two fold —(i) to find the bound states of the system, check forexceptional points, and observe the effect of µ , ifany(ii) to see the effect of µ on the behaviour of thereflection and transmission coefficients.To obtain the solution of (14), we introduce anew variable y = 1 − i tanh ¯ ρ φ = y α/ (1 − y ) β/ χ ( y ) (18)After some straightforward algebra, equation (14)reduces to the hypergeometric equation [32] y (1 − y ) d χdy + { α + 1 − ( α + β + 2) y } dχdy − ((cid:18) α + β + 12 (cid:19) − µ g ) χ = 0 (19)3here α and β are determined from the expressions α + κ − iλ = 0 , β + κ + 2 iλ = 0 (20)Now, (19) has complete solution [32] χ = P F ( a, b, c ; y )+ Qy − c F (1 + a − c, b − c, − c ; y )(21)where P and Q are constants, and the parameters a , b and c are as defined below : a = α + β + 12 + g √ µ b = α + β + 12 − g √ µ c = α + 1 (22)It is known from literature [30] that for boundstates Re ( α ) > Re ( α ) < φ + (¯ ρ )and φ − (¯ ρ ), respectively. Additionally, regularityof the solution demands Re ( β ) >
0. So, for boundstates, we shall restrict ourselves to φ + (¯ ρ ) only. a - a - - - - - z Ψ FIG. 2: Colour online : A plot of Re ψ ( z ) vs z ; Dashed(black) lines show the abrupt heterojunctions at ± a After some straightforward algebra, the final so-lution to the PDEM Schr¨odinger equation withinthe potential well | z | < a , is obtained as ψ in ( z ) = (2 m ) / y α/ (1 − y ) β/ { P F ( a, b, c ; y )+ Qy − c F (1 + a − c, b − c, − c ; y ) (cid:9) (23)Outside the well ( | z | > a ), the scattering solutionsare given by eq (5), with k = k = k (say), viz., ψ L ( z ) = e ikz + Re − ikz , −∞ < z < a ψ R ( z ) = T e ikz , a < z < ∞ while the bound states are given by ψ ( b ) L ( z ) = A e k b z , −∞ < z < a ψ ( b ) R ( z ) = A e − k b z , a < z < ∞ k b = 2 m √ V − E (24)To obtain the solution in the entire region, weneed the expressions for the different coefficients — A , A , P , Q , R and T . These are determined byapplying the modified boundary conditions at eachheterojunction ± a , as mentioned earlier. Further-more, we consider the various properties of the Hy-pergeometric functions F ( a, b, c ; y ) [32], and takethe help of Mathematica. The scattering solutionin the entire region is plotted in Fig. 2, for thesame set of parameter values as in Fig. 1, viz., g = 1 . , µ = 4 , µ = 0 . , a = 2 .
5, for E = 44.Analogous to our previous studies on non Hermi-tian [29] and Hermitian [33] models, we again ob-serve that the dependence of position on the massof the particle in the intermediate layer, changesthe nature of the otherwise plane wave solution. a - a z Ψ H z L Red (dashed) line ® Re Ψ Black (dot dashed) line ® Re Ψ Blue (solid) line ® Re Ψ - - - FIG. 3: Colour online : A plot of Re ψ ( z ) vs z , for thefirst 3 bound states. Notice the kinks in the solutionsat the abrupt heterojunctions at ± a To have a full understanding of this model, wealso calculate the bound state energy and eigen-functions. For the same set of parameter val-ues as in Figures 1 and 2, viz., g = 1 . , µ =4 , µ = 0 . , a = 2 .
5, we obtain the ground stateat E = − .
82, first excited state at E = − . E = − .
7. The corre-sponding wave functions are plotted in Fig. 3. Weobserve a very interesting phenomenon — the stan-dard PT symmetric Rosen Morse II potential forconstant mass and the varying mass diffused quan-tum well sandwiched between two heterojunctions,have some similar features :(i) The bound state energy is always real, hencethere is no spontaneous breakdown of PT symme-try.4ii) In this particular case too, the bound state en-ergy switches from negative to positive value, de-pending on the width of the potential well 2 a , andthe relative strengths of µ and µ .However, the bound state energies of Rosen MorseII and the present model are significantly different.Additionally, for µ = 4, while the ground stateenergy switches from negative to positive value at µ = 34 . µ = 65 .
87, for g = 1 . , µ = 4 , a = 2 .
5. In-creasing the value of a decreases this value of µ for energy switching.As stated earlier, the main purpose of this workis to study the behaviour of reflection and trans-mission amplitudes. For this purpose, we plota series of graphs showing | R | and | T | , in dif-ferent regimes. The system clearly depicts left-right asymmetry, typical of non Hermitian sys-tems. While the transmission amplitude comes outto be the same for both left and right incidence | T L | = | T R | = | T | (say), the case is quite differentfor the reflection amplitude | R R | 6 = | R L | . Fig. 4shows these values for µ = 4 and a low value of thenon Hermiticity parameter, viz., µ = 0 . | R L | isnormal ( | R L | <
1) when the particle enters fromleft — the absorptive side ( V I ( z ) < | R R | >
1) when the particle enters from right— the emissive side ( V I ( z ) >
0) [29, 34, 35]. R R R L T Energy0.20.40.60.81.0
FIG. 4: Colour online : A plot of | T | , | R R | and | R L | for different Energies, for µ = 4 , µ = 0 . , g =1 . , a = 2 . As the non Hermiticity parameter µ increases,the behaviour of | R L | , | R R | and | T | changesabruptly. For low values of µ , for particle en-tering the device from left or right, | T | increaseswith increasing energy, finally reaching unity —total transmission. This observation is similar tothat given in our earlier study [29]. However, as µ increases, | T | first decreases, reaches a minimum,and then increases to reach a saturation value.Once again, the pattern is identical for left andright incidence. This peculiar behaviour is shownin the 3D plot of Fig. 5. This abrupt change ofbehaviour occurs at a particular value of µ , andthe trend continues for all values of µ greater thanthis value. Similarly, if one draws the 3D plots for | R L | and | R R | , with respect to energy and µ , as shown inFig. 6 and Fig. 7 respectively, once again thereis an abrupt change in their behaviour at and be-yond some critical value of µ . The qualitativebehaviour of | T | , | R L | and | R R | for large valuesof µ , is shown in Fig. 8 (for µ = 3). This is insharp contrast to their behaviour at low values of µ , as shown in Fig. 4. However, the interestingpoint to note here is that the scattering coefficientsremain finite everywhere, so the system does notexhibit spectral singularity. Thus, in spite of theabsence of spectral singularity, the non Hermiticityparameter µ plays a crucial role in the behaviourof the scattering amplitudes, similar to its role indeciding the sign of bound state (negative or pos-itive). These are the most important findings ofthe present study.Calculating the coefficients | T | , | R L | and | R R | in the limit µ →
0, gives back the Hermitian re-sults for these coefficients — viz., | R L | = | R R | , and | T | + | R | = 1. IV. CONCLUSIONS AND DISCUSSIONS
To conclude, the special form of semiconductordevice studied in this work displays some uniquecharacteristics. The particular model consideredhere does not undergo spontaneous breakdown of PT symmetry, nor it does not exhibit spectral sin-gularity. Rather, the highlight of this PDEM de-vice is the non vanishing imaginary part of the po-tential near the heterojunctions, within the inter-mediate layer.The series of graphs plotted in the paper showthe potential and mass functions (Fig. 1), the ex-act analytical scattering solutions in the entire de-vice (Fig. 2), and also the first three bound statesolutions (Fig. 3). The behaviour of the scatteringamplitudes are shown in Figures 4 to 8. While theeffect of the PDEM is to introduce a non linear-ity in the otherwise plane wave solutions (see Fig.2), Fig. 4 shows the kinks at the heterojunctions.Numerical calculations show that the bound stateenergy switches from negative to positive value as µ increases. Figures 4 to 8 give credence to the im-portant role played by the non Hermiticity param-eter µ , in determining the scattering amplitudes.For low values of µ , the nature of the reflectionand transmission coefficients as shown in Fig. 4, isanalogous to the observation in our previous study[29]. However, as µ increases beyond a certainvalue, the qualitative picture of these coefficientschanges abruptly. This is a new observation, hith-erto unnoticed in earlier studies. At the same timethis cannot be called a spectral singularity (ss),5
10 20 30 40Energy 0.5 1.0 1.5 2.0 Μ T FIG. 5: Colour online : A 3D plot of | T | , with respectto Energy and µ , for µ = 4 , g = 1 . , a = 2 . Μ R L FIG. 6: Colour online : A 3D plot of | R L | , with re-spect to Energy and µ , for µ = 4 , g = 1 . , a = 2 . Μ R R FIG. 7: Colour online : A 3D plot of | R R | , with re-spect to Energy and µ , for µ = 4 , g = 1 . , a = 2 . T R L R R
20 40 60 80 100 Energy0.51.01.5
FIG. 8: Colour online : A plot of | T | , | R L | and | R R | with respect to Energy, for large µ , viz, µ = 3 for the transmission and reflection coefficients viz., | T | , | R L | and | R R | , blow up at a ss [29, 34, 36].In this particular case, all the three coefficients re-main finite. Additionally, their behaviour dependson whether the particle is entering the device fromthe left or from the right — i.e., this non Her-mitian system too possesses left-right asymmetry,despite the particle having PDEM in the regionwithin the heterojunctions. For the particle enter-ing the semiconductor device from the absorptiveside ( V I ( z ) < | R L | < V I ( z ) >
0) the reflection is anoma-lous ( | R R | > | T | + | R | = 1.In a fairly recent work it has been shown that forthe PT symmetric Scarf II potential, in a partic-ular regime, | T | + | R L || R R | = 1 [37]. However,it did not consider spatially varying mass, nor anyabrupt heterojunction. In our present study of a PT symmetric heterojunction in the form of a dif-fused quantum well with PDEM, this conjecture isnot valid.With increase in artificial PT symmetric artifi-cial optical structures, and semiconductor deviceswith position dependent mass heterojunctions, itis anticipated that this work provides some valu-able insight into the transport properties of sucha device, when a particle enters the material fromone end and leaves from the other. V. ACKNOWLEDGEMENT
The authors thank B. Roy and B. Midya forsome useful comments. One of the authors (AS)acknowledges financial assistance from the Depart-ment of Science and Technology, Govt. of India,through its grant SR/WOS-A/PS-11/2012.6
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