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Scattering in a spatially varying massPT -symmetric double heterojunction

Anjana Sinha and R. Roychoudhury

EPL, 103 (2013) 50007

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EPL, 103 (2013) 50007 www.epljournal.org

doi: 10.1209/0295-5075/103/50007

Scattering in a spatially varying mass PT -symmetric double

heterojunction

Anjana Sinha1(a) and R. Roychoudhury2,3(b)

1 Department of Instrumentation Science, Jadavpur University - Kolkata 700 032, India2 Department of Mathematics, Visva Bharati University - Santiniketan 731 235, West Bengal, India3 Advanced Centre for Nonlinear and Complex Phenomena - 1175 Survey Park, Kolkata - 700075, India

received 12 June 2013; accepted in final form 4 September 2013published online 27 September 2013

PACS 03.65.Nk – Scattering theoryPACS 03.65.-w – Quantum mechanics

Abstract – We observe that the reflection and transmission coefficients of a particle within adouble, PT -symmetric heterojunction with spatially varying mass and non-vanishing imaginarypart near the edges, show interesting features, depending on the degree of non-Hermiticity, despitethe absence of exceptional point and spectral singularity. Exact analytical solutions for the wavefunction for both bound and scattering states are obtained, and the reflection and transmissioncoefficients are plotted as a function of energy, for both left as well as right incidence. As expected,the spatial dependence on mass changes the nature of the scattering solutions within the hetero-junctions, whereas the space-time (PT ) symmetry is responsible for the left-right asymmetry inthe scattering coefficients. However, the non-vanishing imaginary component of the potential nearthe heterojunction edges gives new and interesting results, hitherto unnoticed.

Copyright c© EPLA, 2013

Introduction. – Position-dependent effective mass(PDEM) formalism is extremely important in describ-ing the electronic and transport properties of quan-tum wells and quantum dots, impurities in crystals,He-clusters, quantum liquids, semiconductor heterostruc-tures, etc. [1–6]. In semiconductor heterostructures (e.g.,AlxGa1−xAs), the varying doping concentration x alongthe z-axis makes the mass of the charge carrier (elec-tron or hole) dependent on its position. On the otherhand, it is an established fact that non-Hermitian quan-tum systems with PT symmetry, open up a fascinatingworld, unknown to conventional Hermitian systems [7–9].PT synthetic novel optical devices, with balanced gainand loss, have been engineered to exhibit several intrigu-ing features unknown to Hermitian optical systems —e.g.,double refraction, power oscillations, unidirectional invisi-bility, left-right asymmetry, non-reciprocal diffraction pat-terns, etc. [10–19].

In this work we study a special form of semiconduc-tor device consisting of a thin layer of PT -symmetricmaterial sandwiched between two normal semiconductors,such that the effective mass of the charge carrier (electron

(a)E-mail: anjana23@rediffmail.com; sinha.anjana@gmail.com(b)E-mail: rajdaju@rediffmail.com; rroychoudhury123@gmail.com

or hole) varies with the position within the heterojunc-tions, but is constant outside. The mass m(z) and thereal part of the potential VR(z) are taken to be contin-uous throughout the device. The form of V (z) (whereV (z) = VR(z) + VI(z)) is such that VI(z) does not vanishnear the heterojunctions. It is this non-vanishing imagi-nary component that gives rise to some interesting results.We obtain the exact analytical solutions for the bound andscattering states of a particle inside such a semiconductordevice and also the reflection and transmission amplitudes,R and T , respectively. Our primary aim here is to lookfor any anomaly in |R| and/or |T |, with increase in themagnitude of the imaginary component VI(z).

The paper is organized as follows: For the sake of com-pleteness, the PDEM Schrodinger equation is briefly in-troduced in the second section to show the procedure forobtaining the exact analytical solutions. In the third sec-tion, we study an explicit non-Hermitian PT -symmetricdouble heterojunction, with the properties described ear-lier. Though we pay sufficient attention to obtaining theexact analytical solutions and bound-state energies, themain stress in this work is on the behaviour of the trans-mission and reflection coefficients, |T |2 and |R|2, respec-tively, with respect to the relative strengths of the realand imaginary parts of the potential, and also the side of

50007-p1

Anjana Sinha and R. Roychoudhury

entry of the particle. A series of graphs plotted in figs. 1to 8 illustrate our observations. The fourth section is keptfor conclusions and discussions.

Theory. – Within the heterojunctions a1 < z < a2,where the particle mass varies with position, the Hermi-tian kinetic energy term TEM is given by [4,20]

TEM =1

4

(mαpmβpmγ + mγpmβpmα

), (1)

where p=−ih ddz is the momentum operator. The ambigu-

ity parameters α, β, γ obey the von Roos constraint [4,20]

α + β + γ = −1 (2)

but have no unique definition. For simplicity of calcula-tions, we shall work in units h = c = 1, and use primeto denote differentiation with respect to z. For continuityconditions at the abrupt interfaces, well-behaved ground-state energy, and the best fit to experimental results [21],we shall restrict the ambiguity parameters to satisfy theBenDaniel-Duke choice, viz, α = γ = 0, β = −1 [22].Hence, in the intermediate layer a1 < z < a2, the kineticenergy term in eq. (1) reduces to

T =1

2p

(1

m

)p (3)

so that the Hamiltonian for the particle with PDEM be-comes [23]

H = − 1

2m(z)

d2

dz2−

(1

2m(z)

)′

d

dz+ VR(z) + iVI(z), (4)

whereas, outside the well, z < a1 and z > a2, the particleobeys the conventional Schrodinger equation

{− 1

2m1,2

d2

dz2+ V01,02

}ψ(z) = Eψ(z) (5)

having plane-wave solutions. In case we consider a waveincident from the left, the solutions in the two regions are

ψL(z) = eik1z + Re−ik1z, −∞ < z < a1,

ψR(z) = Teik2z, a2 < z < ∞,

(6)

where R and T denote the reflection and transmission am-plitudes, and

k1,2 =√

2m1,2 (E − V01,02). (7)

The important point worth remembering here is that forPDEM systems, the solutions ψ(z) obey modified bound-

ary conditions [22,24] —the functions ψ(z) and 1m(z)

dψ(z)dz

are continuous at each heterojunction a1 and a2. Theseare used to calculate R, T and the bound-state energies.

Using the transformations [25]

ψin(z) = {2m(z)}1/4φ(ρ), ρ =

∫ √2m(z)dz, (8)

the Schrodinger equation (4) for PDEM in the region a1 <

z < a2, reduces to one for constant mass viz,

− d2φ

dρ2+

{V (ρ) − E

}φ = 0 (9)

with

V (ρ) = V (z) +7

32

m′2

m3− m′′

8m2. (10)

Evidently, eq. (9) can be solved analytically for someparticular cases of V (z) and m(z) only. In ref. [21],we had given the exact analytical solutions for one suchcase, with vanishing imaginary part near the heterojunc-tions, which shows the phenomenon of spontaneous PT -symmetry breaking, and admits a spectral singularity. Inthis work, we shall study a second case, where the mostimportant contribution of the imaginary part of the poten-tial is near the heterojunctions. Thus, this present studyis distinctively different from the work done in ref. [21].

Explicit model: PT -symmetric diffused potential

well with PDEM. – We assume the real part of theintermediate layer to be a diffused quantum well, with thefollowing ansatz for the potential V (z) and mass m(z):

V (z) =

⎧⎪⎪⎨⎪⎪⎩

− µ1

1 + z2+ i

µ2z√1 + z2

, |z| < a0,

− µ1

1 + a20

= V0, |z| > a0;(11)

m(z) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

g2

2 (1 + z2), |z| < a0,

g2

2 (1 + a20)

= m0, |z| > a0.

(12)

Here µ1, µ2, g are some constant parameters.

Figure 1 shows the position-dependent effective massm(z) and the potential V (z) in the entire semiconductordevice, as a function of z, for a suitable set of parametervalues, viz, g = 1.5, µ1 = 4, µ2 = .3: (a) a0 = 2.5;(b) a0 = 3.2. While the imaginary part of the potentialVI(z) may appear to be the same as in ref. [21] for a verythin intermediate layer, the difference between the twomodels becomes more prominent as the thickness of theintermediate layer increases —while VI(z) tapers off as onegoes farther away from the centre in ref. [21], it reaches aconstant value in the model considered in this work.

For the spatial mass dependence given by eq. (12),eq. (8) transforms the coordinate z to

ρ = g sinh−1 z, (13)

so that, after some straightforward algebra, V (ρ) ineq. (10) reduces to

V (ρ) =1

4g2−

(µ1 −

1

g2

)sech2 ρ

g+ iµ2g

2 tanhρ

g. (14)

50007-p2

Scattering in a spatially varying mass PT -symmetric double heterojunction

10 5 0 5 10

2.0

1.5

1.0

0.5

0.0

0.5

1.0

10 5 0 5 10

2.0

1.5

1.0

0.5

0.0

0.5

1.0

m0 m0 m0m0

V0V0 V0V0

VR zVR zVI zVI z

m zm z

(a) (b)

Fig. 1: (Colour on-line) Plot showing m(z) (black dotted line),VR(z) (red dashed line) and VI(z) (blue solid line) with respectto z. Solid (black) vertical lines show the heterojunctions.

Thus, eq. (9) may be written as

d2φ

dρ2+

{κ2 + s (s + 1) sech2ρ − 2iλ tanh ρ

}φ = 0, (15)

where κ2 = Eg2 − 1

4, ρ =

ρ

g(16)

and the new parameters s and λ are expressed in terms ofthe constants µ1, µ2 and g, as

λ =1

2µ2g

2, s = −1

2± g

õ1. (17)

One can check from eq. (14) that the diffused potentialwell (eq. (11)) within the intermediate layer |z| < a0,with spatially varying mass m(z), reduces to the stan-dard PT -symmetric Rosen-Morse II potential for constantmass [26,27]. Now, the standard PT -symmetric Rosen-Morse II potential has the following unique characteris-tics: i) absence of quasi-parity; ii) only real energy due tothe absence of spontaneous breakdown of PT symmetry;iii) switching of the bound-state energies from negativeto positive values, with increase in the magnitude of thenon-Hermiticity parameter.

Additionally, the soliton solution of the correspondingnon-linear Schrodinger equation is always unstable [27],unlike that in the Scarf II case.

These points of difference from the PT -symmetric ScarfII model studied in ref. [21] provide the main motivationfor studying this particular case.

Our aim in this work is basically two fold: i) to find thebound states of the system, check for exceptional points,and observe the effect of µ2, if any; ii) to see the effectof µ2 on the behaviour of the reflection and transmissioncoefficients.

To obtain the solution of (15), we introduce a newvariable

y =1 − i tanh ρ

2. (18)

and write the solution as

φ = yδ/2(1 − y)ν/2 χ(y). (19)

After some straightforward algebra, eq. (15) reduces to

a0–a0 z

Red (dashed) line → Re Ψ0 (z)

Black (dot dashed) line →Re Ψ1 (z)

Blue (solid) line → Re Ψ2 (z)

–a0–a0

–10 –5 0 5 10

–1.5

–1.0

–0.5

0.0

0.5

1.0

1.5

z

Ψ

Inset blue shows scattering Ψ (z)

Ψ (z)

Inset red shows bound Re Ψ1 (z)

for Re ∆ < 0, Re Ν < 0

a0 shows the heterojunctions

–10 –5 0 5 10

–0.10.00.10.20.30.40.50.6

x

–10 –5 0 5 10

–0.5

0.5

Fig. 2: (Colour on-line) A plot of Re ψ(z) vs. z, for the first3 bound states, for Re δ > 0, Re ν > 0. Notice the kinksin the solutions at the abrupt heterojunctions at ±a0. Theinset blue plot on the top right shows the scattering state. Theinset red plot on the bottom left shows bound Re ψ1(z) forδ = −1.51243 − 0.111575i, ν = −1.51243 + 0.111575i.

the hypergeometric equation [28]

y(1 − y)d2χ

dy2+ {δ + 1 − (δ + ν + 2) y} dχ

dy

−{(

δ + ν + 1

2

)2

− µ1g2

}χ = 0, (20)

where δ and ν are determined from the expressions

δ2 + κ2 − 2iλ = 0, ν2 + κ2 + 2iλ = 0. (21)

Now, (20) has complete solution [28]

χ = P 2F1 (a, b, c; y)

+Qy1−c2F1 (1 + a − c, 1 + b − c, 2 − c; y) , (22)

where P and Q are constants, and the parameters a, b andc are as defined below

a =δ + ν + 1

2+ g

õ1,

b =δ + ν + 1

2− g

õ1,

c = δ + 1.

(23)

It is known from the literature [26] that for bound statesof the standard PT -symmetric Rosen-Morse II potential,Re(δ) > 0 and Re(δ) < 0 are two mutually exclusivecases. Additionally, the regularity of the solution demandsRe(ν) > 0. These conditions follow from the asymptoticform of the Rosen-Morse II wave functions. However, sincethe Rosen-Morse II solutions are restricted to the centralregion only in the present model, these conditions are nolonger relevant. This allows for a wider variety of solu-tions, although the qualitative picture remains unchanged,as we shall see in fig. 2.

50007-p3

Anjana Sinha and R. Roychoudhury

a0 2.6

a0 2.2

a0 2.4

a0 2.5

PT sym Rosen Morse II pot

2 3 4 5 6

20

30

40

50

60

70

80

µ1

µ2

Contour Plot for Zero ground state energy

Fig. 3: (Colour on-line) Contour plot of the zero-energy groundstate for the PT -symmetric Rosen-Morse potential with con-stant mass and the model in this study for different µ1, µ2, forg = 1.5.

After some straightforward algebra, the final solution tothe PDEM Schrodinger equation within the potential well|z| < a0, reduces to

ψin(z) = (2m)1/4

yδ/2(1 − y)ν/2 {P 2F1 (a, b, c; y)

+ Qy1−c2F1 (1 + a − c, 1 + b − c, 2 − c; y)

}.

(24)

Outside the well (|z| > a0), the scattering solutions aregiven by eq. (6), with k1 = k2 = k (say), viz,

ψL(z) = eikz + Re−ikz, −∞ < z < a1,

ψR(z) = Teikz, a2 < z < ∞,

while the bound states are given by

ψ(b)L (z) = A1e

kbz, −∞ < z < a1,

ψ(b)R (z) = A2e

−kbz, a2 < z < ∞,

kb = 2m0

√V0 − E.

(25)

To obtain the solution in the entire region, A1, A2, P , Q,R and T are determined by applying the modified bound-ary conditions at each heterojunction ±a0, and the prop-erties of the hypergeometric functions 2F1(a, b, c; y) [28],and taking the help of Mathematica. Analogous to ourprevious studies on non-Hermitian [21] and Hermitian [29]models, the scattering solution (plotted for the set of pa-rameter values g = 1.5, µ1 = 4, µ2 = 0.3, a0 = 2.5, forE = 44) depicts the change in the nature of the other-wise plane-wave solutions due to the dependence of masson the position of the particle in the intermediate layer.For the above parameter values, bound-state energies are

T2

RL

2

RR

2

Energy

0.2

0.4

0.6

0.8

1.0

Fig. 4: (Colour on-line) A plot of |T |2, |RR|2 and |RL|

2 fordifferent energies, for µ1 = 4, µ2 = 0.3, g = 1.5, a0 = 2.5.

obtained at: ground state E0 = −8.82, first excited stateE1 = −2.57, second excited state E2 = −0.9, etc. Thecorresponding wave functions are plotted in fig. 2, forRe δ > 0, Re ν > 0. The inset curve on the top rightshows the scattering state solution, and that on the bot-tom left shows the fist excited state for Re δ < 0, Re ν < 0.Positive and negative values of Re(δ) and Re(ν) leave thequalitative nature of the wave functions unchanged.

We observe a very interesting phenomenon —the stan-dard PT -symmetric Rosen-Morse II potential for constantmass (I) and the diffused quantum well with PDEM sand-wiched between two heterojunctions (II), share some sim-ilar features: i) the bound-state energy is always real,hence there is no spontaneous breakdown of PT symme-try; ii) the bound-state energy switches from a negative toa positive value, depending on the width of the well 2a0,and relative strengths of µ1, µ2.

However, the bound-state energies of I and II are sig-nificantly different. Additionally, for µ1 = 4, while theground-state energy switches from a negative to a positivevalue at µ2 = 34.7222 in case of the former, for the latterit occurs at µ2 = 65.87, for g = 1.5, µ1 = 4, a0 = 2.5. Thisis shown in the contour plot for zero ground-state energy,in fig. 3. If the value of a0 is decreased, energy switchingfrom a negative to a positive value occurs at a lower valueof µ2.

Since the main emphasis of this work is on the behaviourof reflection and transmission amplitudes, we plot a seriesof graphs showing |R|2 and |T |2, in different regimes. Thesystem clearly depicts left-right asymmetry, typical of non-Hermitian systems. While |T |2 is the same for both leftand right incidence, the case is quite different for the reflec-tion amplitude: |RR| �= |RL|. Figure 4 shows these valuesfor µ1 = 4 and a low value of the non-Hermiticity param-eter, viz, µ2 = 0.3. |RL| is normal (|RL| < 1) when theparticle enters from left —the absorptive side (VI(z) < 0),and anomalous (|RR| > 1) when the particle enters fromright —the emissive side (VI(z) > 0), analogous to obser-vations in refs. [21,30,31].

As µ2 increases, the behaviour of |RL|, |RR| and |T |changes abruptly. For low values of µ2, for particleentering the device from left or right, |T | increases withincreasing energy, finally reaching unity —total transmis-sion. This observation is similar to that given in our earlierstudy [21]. However, as µ2 increases, |T | first decreases,

50007-p4

Scattering in a spatially varying mass PT -symmetric double heterojunction

0

10

20

30

40

Energy0.5

1.0

1.5

2.0

μ2

0.0

0.5

1.0

T2

Fig. 5: (Colour on-line) A 3D plot of |T |2, with respect toenergy and µ2, for µ1 = 4, g = 1.5, a0 = 2.5.

5

10

15

20

Energy0.5

1.0

1.5

2.0

2

0

1

2

R L2

μ

Fig. 6: (Colour on-line) A 3D plot of |RL|2, with respect to

energy and µ2, for µ1 = 4, g = 1.5, a0 = 2.5.

reaches a minimum, and then increases to reach a satu-ration value. Once again, the pattern is identical for leftand right incidence. This peculiar behaviour is shown inthe 3D plot of fig. 5. This abrupt change of behaviouroccurs at a particular value of µ2, and the trend continuesfor all values of µ2 greater than this value.

Similarly, if one draws the 3D plots for |RL|2 and |RR|2,with respect to energy and µ2, as shown in fig. 6 andfig. 7, respectively, once again there is an abrupt changein their behaviour at and beyond some critical value ofµ2. The qualitative behaviour of |T |2, |RL|2 and |RR|2for large values of µ2, is shown in fig. 8 (for µ2 = 3).This is in sharp contrast to their behaviour at low valuesof µ2, as shown in fig. 4. However, the interesting pointto note here is that the scattering coefficients remain fi-nite everywhere, so the system does not exhibit spectralsingularity. Thus, in spite of the absence of spectral sin-gularity, the non-Hermiticity parameter µ2 plays a crucialrole in the behaviour of the scattering amplitudes, similarto its role in deciding the sign of bound state (negativeor positive). These are the most important findings ofthe present study. In this respect, the present work givesresults quite different from ref. [21]. In the limit µ2 → 0,we get back the Hermitian result: |RL| = |RR| = |R|, and|T |2 + |R|2 = 1.

0

10

20

30

40

Energy0.5

1.0

1.5

2.0

2

0.0

0.5

1.0

R R2

μ

Fig. 7: (Colour on-line) A 3D plot of |RR|2, with respect to

energy and µ2, for µ1 = 4, g = 1.5, a0 = 2.5.

RL

2

T2

RR

2

20 40 60 80 100Energy

0.5

1.0

1.5

Fig. 8: (Colour on-line) A plot of |T |2, |RL|2 and |RR|

2 withrespect to energy, for large µ2, viz, µ2 = 3.

Conclusions and discussions. – To conclude, in thepresent work we obtain the exact analytical solutions forthe bound and scattering states of a particle (charge car-rier) when a thin layer of non-Hermitian PT material withspatially varying mass is sandwiched between two layersof a normal semiconductor, thus forming a double het-erojunction. The intermediate layer described by a dif-fused quantum well with spatially varying mass, reduces tothe PT -symmetric Rosen-Morse II potential with constantmass, after a suitable transformation. Despite the absenceof spontaneous breakdown of PT symmetry and spectralsingularity (ss), this particular model displays someunique characteristics, possibly due to the non-vanishingimaginary part of the potential near the heterojunctions.This is in stark contrast to the spontaneous breakdown ofPT symmetry and ss observed in our earlier work [21].

The series of graphs plotted in the paper show the po-tential and mass functions in fig. 1, and the first threebound-state solutions in the semiconductor device in fig. 2,for Re δ > 0, Re ν > 0. Additionally, in fig. 2, the in-set graph on the top right shows a typical exact analyti-cal scattering state solution, while the inset graph on thebottom left shows the first excited state for negative valuesof Re δ and Re ν. The truncation of the imaginary poten-tial at the bounding walls at ±a0 allows for all possiblecombinations of δ, ν. While the effect of the PDEM is to

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Anjana Sinha and R. Roychoudhury

change the nature of the otherwise plane-wave scatteringsolutions in the intermediate layer, bound-state solutionsshow kinks at the heterojunctions. Numerical calculationsshow that the bound-state energy switches from a neg-ative to a positive value with the increase in the valueof the non-Hermiticity parameter µ2, similar to the stan-dard PT -symmetric Rosen-Morse II potential for constantmass, but quite unlike the model considered in ref. [21].The relative strengths of µ1, µ2, where this switching oc-curs for the ground-state energy, are shown in the contourplot of fig. 3. The behaviour of the scattering amplitudes,as shown in figs. 4 to 8, gives credence to the importantrole played by µ2. For low values of µ2, the nature of thereflection and transmission coefficients, as shown in fig. 4,is analogous to the observation in our previous study [21].However, as µ2 increases beyond a certain value, the qual-itative picture of these coefficients changes abruptly. Nev-ertheless, this cannot be called a spectral singularity (ss),since contrary to the blowing up of these coefficients viz,|T |2, |RL|2 and |RR|2, at ss [21,30,32], they remain finite.This observation is totally different from that observed inour earlier study done in ref. [21]. The behaviour of |R|2and |T |2 depends on whether the particle is entering thedevice from the left or from the right —this non-Hermitiansystem too possesses left-right asymmetry, despite the par-ticle having PDEM in the region within the heterojunc-tions. For the particle entering the semiconductor devicefrom the absorptive side (VI(z) < 0), reflection is nor-mal (|RL| < 1), while for the particle entering the devicefrom the emissive side (VI(z) > 0), reflection is anoma-lous (|RR| > 1). At the same time |T |2 + |R|2 �= 1.In a fairly recent work it has been shown that for thePT -symmetric Scarf II potential, in a particular regime,|T |2 + |RL||RR| = 1 [33]. However, it did not considerspatially varying mass, nor any abrupt heterojunction. Inour present study of a PT -symmetric heterojunction inthe form of a diffused quantum well with PDEM, this con-jecture is not valid.

Since artificial PT -symmetric optical structures are nowa reality, as are semiconductor devices with position-dependent effective mass, it is anticipated that the ob-servations made in this work will provide some significantinsight in the study of electron transport in semiconductorheterostructures.

∗ ∗ ∗

One of the authors (AS) acknowledges financial assis-tance from the Department of Science and Technology,Government of India, through its grant SR/WOS-A/PS-11/2012. The authors thank the unknown referee for someinvaluable comments. Thanks are also due to B. Roy andB. Midya for some interesting discussions.

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