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Algebraic approximations for transcendentalequations with applications in nanophysics

Victor Barsan

To cite this article: Victor Barsan (2015) Algebraic approximations for transcendentalequations with applications in nanophysics, Philosophical Magazine, 95:27, 3023-3038, DOI:10.1080/14786435.2015.1081425

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Philosophical Magazine, 2015Vol. 95, No. 27, 3023–3038, http://dx.doi.org/10.1080/14786435.2015.1081425

Algebraic approximations for transcendental equations withapplications in nanophysics

Victor Barsan∗

Department of Theoretical Physics, Horia Hulubei National Institute for Physics and NuclearEngineering, Str. Reactorului 30, Magurele 077125, Romania

(Received 30 April 2015; accepted 5 August 2015)

Using algebraic approximations of trigonometric or hyperbolic functions, a classof transcendental equations can be transformed in tractable, algebraic equations.Studying transcendental equations this way gives the eigenvalues of Sturm–Liouville problems associated to wave equation, mainly to Schroedinger equation;these algebraic approximations provide approximate analytical expressions forthe energy of electrons and phonons in quantum wells, quantum dots (QDs)and quantum wires, in the frame of one-particle models of such systems. Theadvantage of this approach, compared to the numerical calculations, is that thefinal result preserves the functional dependence on the physical parameters ofthe problem. The errors of this method, situated between some few percentagesand 10−5, are carefully analysed. Several applications, for quantum wells, QDsand quantum wires, are presented.

Keywords: nanostructures; quantum mechanical calculation; quantum dots;quantum wells; quantum wires

1. Introduction

Quantum dots (QDs), quantum wells (QWs) and quantum wires are essential components ofa huge number of electronic and optoelectronic devices. Their understanding, manufacturingand control depend, to a large extent, on the precise knowledge of the energy of electrons,holes, excitons, etc. moving in these devices. The one-particle models give satisfactoryresults for a large class of nanostructures. In these cases, the main task of the theory is tosolve the Schroedinger equation for the envelope function or wave function for a simplepotential, in general, described by a piecewise constant function. The wave function hasa simple form – linear combination of exponentials with imaginary or real exponents, forplanar heterostructures, or spherical Bessel functions multiplied by spherical harmonics,both of low order, for devices with spherical symmetry.

However, the eigenvalue equations, which determine the energy, are transcendentalequations, whose analytical solutions are difficult to obtain. Of course, they can be calculatednumerically, with high precision, but their dependence on the physical parameters of theproblem is totally lost. The main goal of this paper is to obtain analytical approximate

∗Email: vbarsan@theory.nipne.ro

© 2015 Taylor & Francis

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solutions for such equations. The results will be applied at various types of QWs, QDs andquantum wires, as illustrations of this approach.

The structure of this paper is as follows. Section 2 is a short presentation of tran-scendental equations. In Section 3, we shall expose the physical motivation of this study,mentioning some applications in quantum mechanics and nanophysics. Section 4 is devotedto the approximate solutions of the equations sin ζ/ζ = ±p, cos ξ/ξ = ±p, giving theeigenvalues of the symmetric and slightly asymmetric finite square wells. In Section 5, theequations tan x = ±ax±1 are studied, using the de Alcantara-Bonfim–Griffiths (dABG)algebraization [1]. Physical applications for QDs and Kronig–Penny model are exposed inSections 6 and 7. Section 8 is devoted to a transcendental equation involving trigonometricand hyperbolic functions, with applications to asymmetric infinite square wells and to aclass of core-shell QDs. The conclusions are presented in Section 9.

2. Transcendental equations – an overview

Transcendental equations have ubiquitous presence in theoretical physics. In many cases,they represent eigenvalue equations of Sturm–Liouville problems, from quantum mechan-ics, electromagnetism or elasticity. In this paper, we shall mainly focus on transcendentalequations which give the energy eigenvalue of several quantum mechanical and nanophys-ical systems.

Among the most popular transcendental equations, we mention the following:

ζeζ = p,sin ζ

ζ= ±p,

cos ζ

ζ= ±p (1)

Similar equations may contain other trigonometric functions (tan, cot), hyperbolic func-tions (sinh, cosh, tanh) or more complicated algebraic expressions. To solve the aforemen-tioned equations means to find the explicit form of the function ζ (p), which is equivalentto obtaining the inverse of each of the functions p = p (ζ ), defined by (1).

There are at least three general methods of solving such equations. The first is to applythe Lagrange–Burmann [2] inversion theorem. This method can be conveniently used infew cases, when the derivative of the expressions given in the l.h.s. of Equations (1), or ofsimilar transcendental equations, has a simple form. This is the case with the first equationof (1), when the solution ζ (p) is the Lambert function W (p), i.e. ζ (p) = W (p) . Its seriesexpansion can be easily obtained using the inversion theorem.

The second general method is based on a domain of the theory of complex functions –the theory of singular integral equations [3] – and it was systematically applied by Siewertand his co-workers (see, for instance [4]). Unfortunately, the results obtained this way aregiven in an implicit form, and their use in practical cases is extremely difficult.

The third method, applicable to an important class of transcendental equations, consistsin writing them in differential form, and constructing, from the differential equation obtainedthis way, the series expansion of the desired function. This method, invented and re-inventedby several authors [5,6], has been recently applied (see [6]) in order to obtain the first 16terms of the power series of the solutions of the second and third equation from (1).

As the exact solutions of transcendental equations are, in general, of limited practical use,it is important to find analytical approximations of such solutions. An attractive approachis to approximate the trigonometric or hyperbolic functions by algebraic functions andto transform the transcendental equation into an algebraic equation. For instance, if we

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approximate with cubic polynomials the restrictions of the functions sin ζ/ζ, cos ζ/ζ ontheir intervals of monotony, the respective equations in (1) will be approximately trans-formed in third-order algebraic equations [7,8]. This is why we shall refer to this approachas “cubic approximation”. However, this method does not work for the first root of thethird equation in (1), as the function cos ζ/ζ cannot be conveniently approximated by apolynomial, for small ζ.

More efficient than the cubic approximation is the approach which has been proposed byde Alcantara Bonfin and Griffiths [1], with a quite precise “algebraization” of trigonometricand hyperbolic functions, for instance:

cos x � f (s, c; x) = 1 − (2x/π)2(1 + cx2

)s , −π/2 < x < π/2 (2)

tan x � f̃ (x) = 0.45x

1 − (2x/π), 0 < x < π/2 (3)

where s in (2) is 1/2 or 1, and the parameter c depends on the choice of s. They can beextended to the entire domain of definition of cos and tan using the parity and periodicityproperties of the respective functions. Also, they can be used in order to invert thesefunctions, for instance:

arctan x � x

0.45 + (2/π) x

The algebraic approximations of the trigonometric functions, mentioned in this section,are obtained by a two-step procedure. In the first step, one finds a simple algebraic function,which shares some evident similarities with the trigonometric one; for instance, in (2), thepolynomial in the denominator describes qualitatively the bump of the cos function. Inthe second step, a corrective factor is added; it has a simple form, compatible with thesymmetry of the exact function, and contains one or more numerical constants, obtainedafter a numerical optimization; to refer to the same example, Equation (2), the nominatoris an even function, the value of s is chosen to give a simple factor (s = 1/2 or 1), and thevalue of c is given by a least-squares fit. For the approximation given in Equation (2),in the first step, one simulates the singularity of the tan function with the expression1/ (1 − (2x/π)); the number 0.45 appearing at the denominator is given by a nonlinearcurve fit [1], done in the second step. Of course, the result must be simple enough in orderto transform the transcendental equation in a tractable algebraic one.

Functional dependencies in the solutions ζ (p) of Equation (1), or of similar ones, canbe obtained by solving numerically the transcendental equation for a set of values of p, andfrom the obtained set of data, find an expression that gives the best fit for it. This way, onecan obtain, for instance, a polynomial approximation, ζ (p)� ∑

anζn . The disadvantages

of this “blind” method, compared to the “algebraization”, are that the result is not intuitiveand that we have to keep a large number of terms in the polynomial in order to get a goodprecision. The “algebraization” reduces the number of numerical constants entering in theapproximate solution and gives an intuitive idea about its behaviour.

The main goal of this paper is to use algebraic approximations of trigonometric andhyperbolic functions in order to obtain reasonably accurate analytical approximations forthe solutions of several transcendental equations relevant for quantum mechanics andnanophysics.

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3. Transcendental equations for QWs, QDs and quantum wires

We shall examine some systems for which we can find the energy of electrons, holes orphonons using approximate solutions of the transcendental equations studied in the presentpaper.

A thin GaAs layer sandwiched between thick Alx Ga1−x As layers is a typical exampleof a QW and is the most common heterostructure [9]. The electron envelope function (weshall call it “wave function”, for sake of simplicity) satisfies the Schroedinger equation witha symmetric finite square well potential. A thin layer sandwiched between two thick layersof different composition can be described by an asymmetric finite square well.

The radial wave function of an electron in a binary core-shell spherical QD satisfies theSchroedinger equation with a “radial asymmetric square well” (the quotation marks intend todiminish the oxymoronic effect of “radial ... square”). The problem can be simplified furtherby assuming infinitely high barriers at the semiconductor/matrix interface, which is a goodapproximation for nanocrystals embedded in silica or similar dielectrics. In the oppositedirection, more complex structures, so-called quantum dots–quantum wells (QDQWs), withthe architecture core A, thin shell B, thin shell A, thin shell B and outer thin shell A, canbe described with more complex potential stepping [10]. The energy of an electron in suchpotentials can be conveniently obtained applying the Dalgarno–Lewis perturbation theoryto the energy levels of simpler square potentials [19,20].

An interesting class of nanowires, consisting in a succession of small, cylindrical QDsof InAs alternating with GaAs (vertically stacked self-assembled QDs), was obtained in1995 [21]. This nanosystem is the first realistic illustration of the Kronig–Penny model,considered until recently a toy model for the existence of energy bands in solids, and isimportant both from fundamental and applicative perspective.

But the interest for piecewise constant potentials is not restricted to nanophysics.A rectangular potential hole in the middle of a large square well (the case of an impenetrablewell corresponds to the problem 26 of [11]) exhibits the level rearrangement (Zeldovicheffect), which is very pronounced if the distance between walls is much larger than thedimension of the hole [12]. Few particle systems contained in such a potential (withimpenetrable outer walls) have a negative heat capacity [13]. Several variants of rectangularpotentials are discussed in detail in [14,22,23].

So, there are serious reasons for investigating the transcendental equations satisfied bythe eigenvalues of simple quantum mechanical problems.

4. The equations sin ζ/ζ = ± p, cos ξ/ξ = ± p and the finite square wells

4.1. The symmetric finite square well

In order to make more transparent the physical meaning of these equations, let us mentionthat the solutions ζ, ξ of

sin ζ = ±ζ p, cos ξ = ±ξp (4)

are related to the eigenenergies of a particle in a symmetric finite square well (the signalternation will be not discussed here; the issue is not trivial, see, for instance [6]). Thesymmetric finite square well is defined by the potential:

V (z) = V0

[1 − θ

(a

2− |z|

)](5)

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a

2

a

2

U 1

U

U

Figure 1. The asymmetric finite square well (see (15)).

It corresponds to the potential plotted in Figure 1, if η = U ′ = 0, U = V0. The energylevels of a particle of mass m, moving in the potential (5) and having the same value insideand outside the well, is given by the expressions:

E2n−1 = 2�2

ma2[ξn (p)]

2 , E2n = 2�2

ma2[ζn (p)]

2 , n = 1, 2, . . . (6)

Actually, the quantities ξn, ζn are proportional to the wave vectors:

ξn = 1

2ak2n−1, ζn = 1

2ak2n (7)

In this context, the parameter p is:

p = 2�

a√

2mV0(8)

and it characterizes globally the well (a, V0) and the particle (m); a deep (and/or large) wellcorresponds to a small p and to a large number of roots ξn, ζn (to a large nmax). Otherforms of the eigenvalue equations, equivalent to (4), are:

tan ξ =√(1/p)2 − ξ2

ξ, tan ζ = − ζ√

(1/p)2 − ζ 2(9)

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or: nπ

2− Xn = arcsin pXn, X2n−1 = ξn, X2n = ζn, n = 1, 2, . . . (10)

It is easy to check (for instance, comparing the plots of functions f (s, c; x) / cos x , forall s and c, with the constant function C (x) = 1) that the most precise “algebraization” ofcos is given by f (1, 0.1010164; x); however, as noticed in [1], the function f (1/2, c; x)has the advantage of transforming the equation for ξ1,

cos ξ1

ξ1= p, 0 � ξ1 � π

2(11)

in a second-order algebraic equation in p2. Its solution can be approximated as ξ1 �f(

12 , 1 − 8

π2 ; p)

(see Equation (2)). It has the correct behaviour in the limit of very shallowwells, which corresponds to p → ∞:

ξ1 ∼ 1

p(12)

The function ξ1 � f(

12 , 0.2120126; p

)has a similar expression and an identical

asymptotic behaviour. It represents an even better approximation for ξ1 (p) , especiallyfor 0 < p < 1; for p � 2, the error is about 10−4, and decreases strongly for larger values

of p, for both functions ξ1 � f(

12 , c; p

)(see [1]). The fact that the dABG approximation

for ξ1, (12) is so precise is important because the shallow wells play a significant rolein practical applications. As previously explained, it cannot be obtained with the cubicapproximation, used in [6,7].

The QW model for semiconductor heterojunction is largely used in order to interpretthe absorption and photoluminiscence data. Several authors [25,28–30], obtained approxi-mate analytical formulas for several quantities of interest using the Barker approximation[27], which is obtained keeping the first two terms of the exact series expansions of the“wavevectors” ξn (p) , ζn (p), [6,8]. The use of the approximations just described wouldincrease the precision of their results. The same remark is valid for the analysis of resonanttunnelling diodes done in [31].

In many realistic cases of semiconductor physics, the effective mass of the electronhas different values, outside and inside the well. The quantum mechanics of particleswith position-dependent mass can be treated within a general frame (see, for instance[26] and references given there), but if the position dependence can be described by astep function, an elementary approach can be followed. More precisely, in this situation,the usual boundary conditions of quantum mechanics (the continuity of the wavefunctionand of its derivative) are replaced with the BenDaniel–Duke boundary conditions, and theeigenvalue equations for the wavevector are slightly modified. Let us adapt the previousapproach to the heterostructure already mentioned in Section 3, when a thin GaAs layer(material B) is sandwiched between thicker Al0.3Ga0.7As layers (material A). We shallconsider that the growth direction is Oz, so the layer A intersects the Oz axis on the interval(−a/2, a/2), and the layer B intersects the same axis outside this interval. The conductionband edge of this heterostructure can be obtained from Figure 1, for the particular case whenη = 0, E (A)c = U, E (B)c = U ′, U − U ′ = V0, so the band offset is E (A)c − E (B)c = V0(see also [9], Fig. 5.5, p. 67). We shall obtain an analytical approximate expression forthe energy E of the electronic bound states, E (A)c > E > E (B)c . With the usual notation

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β = m B/m A, the Equations (4), valid in the case of usual boundary conditions, are replaced,for BenDaniel–Duke boundary conditions, by:

cos x

x= ± p√

β + (1 − β) p2x2,

sin x

x= ± p√

β + (1 − β) p2x2(13)

with

x = 1

2akA, p = 2�

a√

2m AV0(14)

The sign rule is similar to that corresponding to usual boundary conditions (β = 1) .The first root of each equation can be obtained with the dABG algebraization; the otherroots using a variant of the parabolic approximation for the l.h.s. of (13) [8]. For β > 1, theequations have a finite number of roots.

The physics of heterostructures using the BenDaniel–Duke boundary conditions hasbeen analysed in detail by Singh and co-workers [32,33]. They obtain analytic approximateresults using low-order series expansions of trigonometric functions. In these cases, thealgebraization would produce much more precise results, valid on all range of parametersp and β.

4.2. The asymmetric square well

Let us consider an asymmetric well, described by the potential (see Figure 1):

V (x) = V · (1 + η) · θ(−a

2− x

)+ V · θ

(x − a

2

), η � 0 (15)

The parameter η measures the well asymmetry; the case η = 0 corresponds to asymmetric well. As previously mentioned, it represents a good model for a heterostructurecomposed of a layer sandwiched between two thick layers of different composition. If thebound state energy E is written, as usual, in terms of the wavevector k, E = (

�2k2)/2m,

the eigenvalue equation satisfied by k is:

nπ − ka = arcsinka

2P+ arcsin

ka

2P√

1 + η(16)

As expected, for η = 0, the Equation (16) becomes identical to (10).

4.2.1. The ground state energy

For n = 1 and ka < π/2, the transcendental equation (16) can be transformed in a quarticalgebraic equation using the following approximation for the cos function:

cos x � 1 − (2x/π)2√1 + (

1 − (8/π2

))x2, −π

2� x � π

2(17)

proposed by dABG [1]. We shall not write here the solution of this quartic equation, but itis however important that an elementary expression, giving a very precise approximationof the ground state energy, exists.

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4.2.2. Slightly asymmetric well

If the asymmetry parameter η is small,

η 1,

the potential can describe a heterostructure composed of a thin layer sandwiched betweentwo thicker, slightly different layers, with a small band offset.

In this case, we obtain, instead of (16), the equation:

sin

(nπ

2− ka

2

)= ka

2P

(1 − η

4

)(18)

which is a variant of the Equation (10) of the symmetric well, corresponding to a somewhatstronger potential. So, in the first order in the asymmetry parameter η, the presence of aslightly higher wall of the well has the effect of the replacement:

P → P

1 − η4

� P(

1 + η

4

)(19)

All the energy levels are consequently given by formulas obtained from (6), making thereplacement (19).

Like in the previous subsections, the results (18) and (19) can be easily extended toenvelope functions in the conduction bands at heterointerfaces, where the usual quantummechanical boundary conditions are replaced by BenDaniel–Duke ones [9,15].

5. The equations tan x = ±ax±1

In order to evaluate the advantages and disadvantages of the dABG algebraization of thetangent function (Equation (3)), let us firstly study the equation:

tan x = ax, a > 0, x > 0 (20)

It gives, for instance, the bound state energies of a particle moving in a “cavity” boundedby a rigid wall and a δ function (problem 27, [11]), or in an infinite square well, with anegative δ potential in the middle (problem 19, [11]), or the normal modes of acousticphonons in a spherical QD [16], etc.

It has an infinity of roots, corresponding to the intersections of the line y = ax withthe branches of the function y = tan x . We can reduce the equation to the first quadrant,noticing that, if x0n is the nth non-trivial root of (20) and X0n is the first quadrant solutionof the equation:

tan x = ax + nπ (21)

they satisfy the condition:x0n = X0n + nπ (22)

We can see how the errors in the evaluation of roots of (18) are generated by theapproximation (3), considering a particular case, namely a = 1:

tan x = x, x > 0 (23)

Plotting the functions tan x, f̃ (x) = 0.45x/ (1 − (2x/π)) (see Equation (3)) andy (x) = x , for 0 � x < π/2, it is easy to see that the algebraization produces a spurious

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2 5 10 20 50Log n

5 10 5

1 10 4

2 10 4

5 10 4

0.001

Logx

x

Figure 2. (colour online) Log–log plot of the absolute value of relative errors of the approximateroots of Equation (23) obtained through the algebraization (3).

root in this interval. Denoting by xn (a = 1) ≡ xn (1), n � 1, the exact roots of (27)corresponding to the intersection of the first bisectrix with the (n + 1)th branch of thetangent, and comparing them with the roots xappr

n (1) of the equation obtained from (21)through algebraization of tan function, according to (3):

f̃ (x) = 0.45x

1 − (2x/π)= x + nπ (24)

we can evaluate the errors

xn (1)− xapprn (1)

xn (1)=(x

x

)n. (25)

The plot of the value of the relative errors, as a function of n (Figure 2), shows a decreaseof (x/x)n when n increases, from about 10−2 for n = 1 at about 10−5 for n = 50.

The apparently unexpectedly small values of (x/x)2, (x/x)3, (x/x)4 are due tothe fact that tan x = f̃ (x) at x = 1.40111, while x2 (1) = 1.442 + 2π. For n � 6, theerror decreases exponentially, for large n. So, we can conclude that the roots xn (1) , n � 1are given by algebraization with reasonable accuracy, and this accuracy increases while nincreases asymptotically.

Concerning the root of (20) from the first quadrant for 0 < a < 1.2, it can be obtainedquite precisely replacing the tangent with its Taylor series near x = 0, cut at the 5th or7th power; this “algebraization” gives a quadratic or cubic equation in x2. The dABGalgebraization becomes semi-quantitatively applicable for somewhat larger values of a; itgives an error which decreases from 4%, if a = 2, to less than 1%, if a = 10.

A more subtle algebraization, also proposed by dABG (note 24 of [1]),

tan x � f (x) =x(

1 −(

1 − 8π2

)x2)

1 − (2x/π)2(26)

gives, for tan x = ax , a third-order equation, which does not produce any spurious root.

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This “more subtle” approximation is better in the domain where the “simpler” one isunsatisfactory, 0 � x � 1.1, but it fails for larger arguments; it is not a global solution forthe algebraization of tan, but merely a complementary variant. So, this case shows that amore sophisticated approximation is not necessarily better than a simpler one.

It is useful to extend the dABG algebraization of tangent for an arbitrary value of theargument. We get:

tan x = 0.45π (x − nπ)

2x − (2n − 1) π,

(n − 1

2

)π < x < nπ (27)

tan x = 0.45π (x − nπ)

(2n + 1) π − 2x, nπ < x <

(n + 1

2

)π (28)

The formulas (26)–(28) can be extended to negative arguments using the fact that tan x is anodd function. For x < 0, the “algebraization” does not produce, in this case, any spuriousroots. They are also absent for the equations tan x = ±a/x .

Another application of the “algebraization” (3) is connected to the fact that the intervalsof monotony of the functions sin x/x, cos x/x , important for a precise description of thesolutions ζn, ξn (7), are given by the roots rsn, rcn of the equations:

tan x = x, tan x = − 1

x(29)

respectively. The advantage of the approximate roots, obtained with (3), compared tothe “exact” numerical roots, consists in the fact that the approximate ones are explicitlyn-dependent, n being the order of the root. Let us also mention that the roots of the equationtan x = x are also the roots of the spherical Bessel function j1 (x), and the roots of

tan x = 3x

3 − x2,

which can be transformed, through “algebraization”, in a quadratic or cubic equation, arethe roots of j2 (x).

6. Physical applications: electrons and phonons in a spherical QD

Let us see now how these results can be applied in the study of QDs. In spite of the complexityof semiconductor QDs, the one-particle approach is satisfactory in many relevant cases. Forwide-gap semiconductors, such as CdSe, the confined electron and hole levels may be treatedindependently [16]. For electrons near the bottom of the conduction band, it is sufficientto consider a single parabolic band. For very high barriers at the semiconductor/matrixinterface, the radial wave function can be assimilated with the solution of a Schroedingerequation for an asymmetric infinite square well.

These situations lead to simple expressions for the radial envelope wave function for theelectrons in a QD, which is the solution of a Schrodinger-type equation with a rectangularpotential:

ψ(−→r ) ∼ jl

(ξln

Rr

)Ylm (θ, ϕ) (30)

where R is the radius of the QD, and ξln are the roots of the spherical Bessel function jl .The radial envelope function for holes is more complicated (but still a combination of

spherical Bessel functions and spherical harmonics of low order (l � 2), and their radial

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argument contains, instead of ξln , a parameter χ , which is the first root of the equation[16,17]:

j0 (bχ) j2 (χ)+ j0 (χ) j2 (bχ) = 0 (31)

Here, b is a dimensionless parameter, (b2 is a mass ratio – light hole/heavy hole) and thequantity to be found is χ. In the particular case mentioned by Efros [17], b2 = 0.115, onefinds the value χ = 5.18961 (actually, Efros gives a somewhat smaller value, χ = 5.21).It can be put in the form:

2bχ

tan bχ+ b2 χ

tan χ+(

b2(χ2 − 1

)− 2

)= 0 (32)

As, in this case, 1.5π < χ = 5.19 < 2π, π/2 < bχ = 1. 76 < π , the algebraization oftangent functions gives, according to (27) and (28):

tan bx = 0.45π (bx − π)

2bx − π, tan x = 0.45π (x − 2π)

2x − 3π(33)

Replacing in the exact equation, we get a 4th-order algebraic equation; its solution (wedo not give explicitly its solution, which is elementary, but cumbersome), for b = √

0.115,is χ = 5.14, quite close to the exact value 5.19 (so, with an error of about10−2). We canobtain a third-order equation, after an algebraization of (32), only when one of the argumentsof the tan functions belongs to the first or fourth quadrant. Knowing the value of b, in termsof the parameters of the problem, we know the analytic form of holes enveloping function.

The phonons are also essential ingredients for understanding the exciton and polaronphysics, which finally gives the optical properties of the QDs [16]. The quantized frequenciesof the acoustic phonons in a spherical QD are the roots of the equations for Lamb’s l p = 0modes for a free-standing sphere:

4

(cT

cL

)2

j1 (k R)− k R j0 (k R) = 0 (34)

Putting

a = 4

(cT

cL

)2

, k R = x (35)

one obtains the following equation:

tan x = ax

a − x2(36)

Through the algebraization of tangent, it becomes a quadratic or cubic equation in x .

7. Vertically stacked self-assembled QDs and the Kronig–Penny model

A simple model describing the 1D motion of an electron in a periodic potential is generatedby the translation of a rectangular well of depth V0 and length b, with the lattice constanta, over the real axis (the well-known Kronig–Penny model [11,18]). In the limit

b → 0, V0 → ∞, V0b = const (37)

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5 10 15

0.2

0.4

0.6

0.8

1.0

Figure 3. (colour online) Graphical solutions of Equation (39).

the wells become δ functions, and the potential takes the form:

V (x) = �2

m

∞∑n=−∞

δ(x + na) (38)

The model provides a simple way of understanding the formation of energy bands in aperiodic solid and of the evaluation of their edges. According to standard books of quantummechanics (see, for instance [11]), the values of the wave vector k, which give the bandedges, are obtained from the equation (see Figure 3):∣∣∣∣cos

(ka − arctan

k

)∣∣∣∣ = 1√1 + ( /k)2

(39)

where ka = K , and a = ω. Denoting by K1, K2, . . . the non-zero roots of Equation(39), written in ascending order, we can see that the even indices correspond to trivial roots,determined by intersections of the square root (the function in the r.h.s. of (39)) with theascending part of the bumps of the |cos| function:

K2 = π, K4 = 2π, K6 = 3π, K8 = 4π, ... (40)

The odd indices correspond to the non-trivial ones, K4p+1, K4p+3, determined similarlyby the intersections of the same function with the descendent part of the bumps, and given,respectively, by the equations:

tanka

2= a

ka, tan

ka

2= − ka

a(41)

By algebraization of the tan function, we obtain:

K4p+1 = 2πp + 1

0.45π

(ω

p− 1

0.9π2

ω2

p3+ ...

), p > 0 (42)

K4p+3 = (2p + 1) π + π

4

2 + 0.45ω

p+ O

(ω2

p3

)(43)

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Also,

K1 = 20

9

(−ωπ

+√(ω

π

)2 + 0.9ω

)(44)

The width of the allowed bands, in terms of the “dimensionless wavevector” K is:

(a)p (ω) = K4p+2 − K4p+1 = π − 1

0.45π

ω

p+ O

(ω2

p3

)(45)

Similarly, for the forbidden ones:

( f )p (ω) = K4p+3 − K4p+2 = π

4

1 + 0.45ω

p+ O

(ω2

p3

)(46)

The Kronig–Penny model has been considered until recently a “toy model” in theabsence of any 1D (or quasi-1D) physical system it could describe: more exactly, until1995, when a nanowire, composed by pieces of InAs alternating with GaAs, has beenproduced [21]. The incorporation of BenDaniel–Duke boundary conditions in the modelhas been worked out in [24] and will be not repeated here. The algebraization allows adeeper understanding of the physical significance of the Kronig–Penny model results.

8. Equations involving trigonometric and hyperbolic functions

Such an equation appears in the elementary problem of an asymmetric infinite square well(see, for instance [14], problem 13), defined by the potential:

V (x < 0) = ∞, V (0 < x < a) = V1,

V (a < x < b) = V2 > V1, V (x > b) = ∞ (47)

(see Figure 4).As already mentioned, it is relevant for a class of QDs embedded in a matrix with a high

energy gap. Presuming that0 < E < V2 − V1

and denoting by

k =√

2m

�2 (E − V1), K =√

2m

�2 (V2 − E), k20 = 2m

�2 (V2 − V1) = k2 + K 2

b − a = εb, k0b = p, kb = x (48)

we can write the eigenvalue equation for the energy in the form:

x cot x (1 − ε) = −√

p2 − x2 coth√

p2 − x2 (49)

One could try to solve this equation using an algebraization of the hyperbolic functionsproposed in dABG (Equation (45) of [1]), which gives:

coth x � F (x) = 0.0572x2 + 0.286x + 1

x (1 − 0.0572x)(50)

Just comparing the plots of the function coth x/F (x) and of the constant functionC (x) = 1, it is clear that the previous approximation is a poor one. However, a key

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0 a b

V2

V1

Figure 4. The asymmetric infinite square well potential (see (47)).

2 4 6 8 10 12 14

0.4

0.2

0.2

0.4

Figure 5. (colour online) Plot of the two sides of (49), for ε = 0.1, p = 14.

for solving the Equation (49) emerges from noticing the very elementary fact that thetrigonometric (hyperbolic) function has a rapid (slow) variation for a large range of valuesof physical parameters entering in the respective relations. For instance, let us consider(Figure 5) the realistic case ε = 0.1, corresponding to a spherical QD, where the shellrepresents 10% of the total radius (this is the only place where we use the fact that ε isa small quantity), p = 14; 1/p is the potential strength of height V2 − V1 and range a,corresponds to a mass m (see Figure 5).Asimple numerical exercise shows that replacing the

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function in the r.h.s. of (49) with a conveniently chosen segment of its tangent, we obtainthe roots of this equation with an error smaller than 10−3. So, considering the simplestalgebraization of the r.h.s. as

an (ε, p) x + bn (ε, p) (51)

and the dABG algebraization of the tangent, we obtain the approximate roots of (49) froma third-order algebraic equation, with errors decreasing from 5% (for the smallest root) to1.6% (for the largest one).

9. Conclusions

Using algebraic approximations of trigonometric or hyperbolic functions, it is possible totransform a class of transcendental equations in approximate, tractable algebraic equations.As the algebraization used in this paper is, to a certain extent, an ad hoc procedure, thisapproximation must be used with a certain caution in order to avoid the appearance ofspurious roots or of roots with too large errors. A more sophisticated approximation is notnecessarily better than a simpler one. The conclusions based on “eye naked” examinationsof “indistinguishable curves” may be tricky. The frequent check of the results by numericalexamples or by graphical methods is highly recommended. Such issues have been studiedin detail in Section 5, for the equation tan x = ax .

The transcendental equations solved this way are, in general, eigenvalue equationsof the energy, with relevance for quantum mechanics and nanophysics; they are part ofSturm–Liouville problems, a theory covering a huge area of research, from optoelectronicsto elasticity or quantum graph problems [34]. They can be used in a large number ofmodels for heterojunctions, QDs and quantum wires, when the one-particle descriptionis applicable, in terms of wave functions or enveloping functions. These models containintrinsic approximations (for instance, the “spherical” QDs are not perfectly spherical),producing errors of the order of a few percentages; consequently, an exact expression forthe eigenvalues produced by such a model would not present a practical interest. So, ifthe advantage of an exact solution, compared to an approximate one, obtained throughalgebraization is decisive for a mathematician, it is of secondary interest for a researcherworking in applied physics. Also, an important benefit of approximate analytical solutionsis the fact that they provide a semi-quantitatively precise image of the dependence of theresults on the physical parameters of the problem (effective masses, potential barriers, etc.)

Several applications of this approach to multiple heterostructures, core-shell sphericalQDs, Kronig–Penny model and quantum wires are worked out, as an illustration of themethod. Similar approaches can be used to many other problems of nanophysics andquantum mechanics.

Disclosure statementNo potential conflict of interest was reported by the author.

FundingThis work was supported by Autoritatea Nationala pentru Cercetare Stiintifica [PN 09 37 01 02/2009], [grant number 04-4-1121-2015/2017]; HOPE Network [2013-3710_540130-LLP-1-2013-1-FR-ERASMUS-ENW].

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