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Using a model anharmonic oscillator with asymptotically decreasing effective mass to study the effect of compositional grading on the quantum mechanical properties of a semiconductor heterostructure, we determine the exact bound states and spectral values of the system. Furthermore, we show that ordering ambiguity only brings about a spectral shift on the quantum anharmonic oscillator with spatially varying effective mass. A study of thermodynamic properties of the system reveals a resonance condition dependent on the magnitude of the anharmonicity parameter. This resonance condition is seen to set a critical value on the said parameter beyond which a complex valued entropy which is discussed, emerges.

In material science, the electron effective mass m * is the mass, it appears to have when probed in the periodic potential of a crystal lattice. The quantum mechanical description of this phenomenon is furnished by the effective mass approximation which is the Schrödinger equation given by:

[ − ℏ 2 2 m * ∇ 2 + U ( x ) ] ϕ ( x ) = [ E − E c ( r ) ] ϕ ( x ) , (1)

wherein E c ( x ) is the variable conduction band edge energy, ϕ ( r ) is the envelope function and U ( r ) some potential energy function which might be due for instance, to an impurity in the crystal. Describing impurities in crystals was the initial purpose of the Effective Mass Theory at its birth in the 1940s [_{δ}Ga_{1-δ}/GaAs/Al_{δ}Ga_{1-δ}As quantum well ( δ being the mole fraction of the Al constituent in the growth direction of the structure), the effective mass becomes position-dependent and the need to do the replacement m * → m ( x ) arises. However, the quantum kinetic energy operator cannot be expressed as p 2 / 2 m ( x ) since this structure is obviously non-Hermitian because of the non-vanishing commutator of the momentum and the effective mass operators. Notwithstanding the Hermiticity necessity, an all around characterized quantum kinetic energy operator is relied upon to satisfy various exigencies, for example, Be Galilee Invariant, not prompt infringement of Heisenberg uncertainty rule [

Ordering these two operators cannot be done in a unique way for a consistent quantum theory with variable mass, reason why one finds a good number of proposals in the literature. O von Roos [

T = 1 4 [ m α p m b p m γ + m γ p m b p m α ] , (2)

with the parameters α , b , γ (referred to as ordering ambiguity parameters) constraint by the relation α + b + γ = − 1 . One can derive most of the operators proposed in the literature [

With this plethora of kinetic energy operators in the literature, the conflict as to which one should be preferred is a long-standing and unresolved one. In an attempt to resolve this issue, Dutra and Almeida [

The study of thermodynamic properties of physical systems permits for instance to determine system parameters that allow minimum thermodynamic frequency instability such as in quartz resonators [

It is note worthy that not all kinetic energy operators seen in the literature are derivable from the von Roos operator. In [

T = 1 4 ( a + 1 ) [ a [ 1 m p 2 + 2 p 1 m p + p 2 1 m ] + m α p m b p m γ + m γ p m b p m α ] . (3)

Here, the parameter a ∈ { 0,1 } and the ambiguity parameters are mutually exclusive. Considering the particle in an arbitrary potential V ( x ) , one can construct the Hamiltonian H = T + V ( x ) which can be cast in the form:

H = 1 2 m p 2 + i ℏ 2 m ′ m 2 p − ℏ 2 4 ( a + 1 ) [ ( α + γ − a ) m ″ m 2 + 2 ( a − α γ − α − γ ) m ′ 2 m 3 ] + V ( x ) , (4)

where the primes represent differentiation with respect to x. The time-independent Schrödinger equation resulting from the effective Hamiltonian H expressed as H Ψ ( x ) = E Ψ ( x ) can be re-written with the substitution Ψ ( x ) = m 1 / 2 ψ ( x ) as

− ℏ 2 2 m d 2 d x 2 ψ ( x ) − ℏ 2 4 ( a + 1 ) [ ( α + γ + 1 ) m ″ m 2 + 2 ( − α γ − α − γ + a − 3 4 ) m ′ 2 m 3 ] + { V ( x ) − E } ψ ( x ) = 0. (5)

Equation (5) can be simplified further to

− ℏ 2 2 m d 2 d x 2 ψ ( x ) + { [ η 1 m ″ m 2 − η 2 m ′ 2 m 3 ] + V ( x ) − E } ψ ( x ) = 0,

− ℏ 2 2 m d 2 d x 2 ψ ( x ) + { m ′ − η 1 m 2 − η 2 [ m ′ η 1 m η 2 ] + V ( x ) − E } ψ ( x ) = 0, (6)

where we have used

η 1 = − ℏ 2 4 ( a + 1 ) ( α + γ + 1 ) , η 2 = ℏ 2 2 ( a + 1 ) ( − α γ − α − γ + a − 3 4 ) . (7)

Now, for a particle with asymptotically decreasing effective mass

m ( x ) = m 0 1 + λ x , (8)

in a parabolic confinement potential

V ( x ) = V 0 x 2 , (9)

Equation (6) takes the form

ψ ( x ) ( 6 a λ 2 m 0 − 4 b λ 2 m 0 − E n + V 0 x 2 ) − ℏ 2 ( λ x + 1 ) 2 ψ ″ ( x ) 2 m 0 = 0. (10)

With the transformation

y → 2 m 0 V 0 λ 2 ℏ ( 1 + λ x ) ,

ψ ( x ) → ( 1 + λ x ) 2 m 0 V 0 λ 2 ℏ − n e − 2 m 0 V 0 λ 2 ℏ ( 1 + λ x ) ξ ( y ) , (11)

Equation (1) can be transformed to the familiar form

y d 2 d y 2 ξ + ( 1 − n − y ) d d y ξ + 2 n ξ = 0, (12)

after setting

− 2 m 0 ( E n − ρ ) ℏ 2 + 2 m 0 V 0 ( 2 n + 1 ) ℏ − λ 2 n ( n + 1 ) = 0. (13)

Comparing Equation 13 with the confluent hypergeometric differential equation

z Y ″ + ( p − z ) Y + q Y = 0, (14)

that has the confluent hypergeometric functions of the first kind as solutions, i.e.

Y = A 1 F 1 ( q , p , z ) + B U ( q , p , z ) , (15)

it follows that the solutions to Equation (12) are

ξ ( z ) = A 1 F 1 ( − 2 n , 1 − n , y ) + B U ( − 2 n , 1 − n , y ) , (16)

in the domain 0 ≤ y < ∞ corresponding to − 1 λ ≤ x < ∞ , where A and B are integral constants. For A = 0 , the solution reduces to

ψ n ( x ) = B n ( 1 + λ x ) 2 m 0 V 0 ℏ λ 2 − n e − 2 m 0 V 0 λ 2 ℏ ( 1 + λ x ) × U [ − 2 n ,1 − n , 2 m 0 V 0 λ 2 ℏ ( 1 + λ x ) ] , (17)

hence

Ψ n ( x ) = B n m 0 ( 1 + λ x ) 2 m 0 V 0 ℏ λ 2 − n + 1 2 e − 2 m 0 V 0 λ 2 ℏ ( 1 + λ x ) × U [ − 2 n ,1 − n , 2 m 0 V 0 λ 2 ℏ ( 1 + λ x ) ] . (18)

The energy spectrum is obtained from Equation (13) as

E n = ( ρ + e 0 ) + ( 2 e 0 − δ ) n − δ n 2 , (19)

where

ρ = λ 2 m 0 ( 6 η 1 − 4 η 2 ) ; e 0 = ℏ 2 V 0 m 0 ; δ = ℏ 2 λ 2 2 m 0 . (20)

For this exactly solved model, the spectral values are real for all possible values of the ambiguity parameters. Therefore one cannot rely on the admissibility test in [

To get an appraisal of the thermodynamic properties of the system, the obvious starting point is the partition function. In the present case, it is given by

Z = ∑ n = 0 ∞ exp ( − β E n ) = ∑ n = 0 ∞ exp ( − β ( ρ + e 0 ) + β n 2 δ + n ( β δ − 2 β e 0 ) ) = e − β ( ρ + e 0 ) ∑ i = 0 ∞ ∑ n = 0 ∞ e − n ( 2 β e 0 − β δ ) ( β δ n 2 ) i i ! . (21)

In the last line of Equation (21), we have converted the exponential with argument linear in n to a sum with summation index i. Here β = K B T where K B is the Boltzmann constant and T the absolute temperature of the system. To evaluate the sum with respect to n, we apply the formula

∑ m = 0 ∞ ( c m 2 ) g g ! e − d m = Φ ( e − d , − 2 m ,0 ) c g g ! , (22)

where Φ ( ) is the Lerch transcendent function [

Z = e − β ( ρ + e 0 ) ∑ i = 0 ∞ ( β v ) i Φ ( e β ( δ − 2 e 0 ) , − 2 i ,0 ) i ! . (23)

For small δ , we neglect terms in δ 2 and higher to obtain

Z = e − β ( ρ + e 0 ) 1 − e β ( δ − 2 e 0 ) + β δ e − β ( ρ + e 0 ) ( e β ( δ − 2 e 0 ) + e 2 β ( δ − 2 e 0 ) ) ( 1 − e β ( δ − 2 e 0 ) ) 3 . (24)

It can be seen from Equation (24) that with the substitution of λ = 0 , which translates to ρ = δ = 0 , the partition function reduces to that of the linear harmonic oscillator, i.e.

Z λ = 0 = e − β e 0 1 − e − 2 β e 0 = 1 2 csch ( β e 0 ) . (25)

The Free energy of the system is given by

F = − K B T log Z = − K B T log ( β δ e − β ( ρ + e 0 ) ( e β ( δ − 2 e 0 ) + e 2 β ( δ − 2 e 0 ) ) ( 1 − e β ( δ − 2 e 0 ) ) 3 + e − β ( ρ + e 0 ) 1 − e β ( δ − 2 e 0 ) ) , (26)

its entropy reads

S = − d F d T = 6 ( δ − 2 e 0 ) e δ K B T e δ T K B − e e 0 K B T + e 0 T + K B log ( e − e 0 2 K B T ( K B T ( e δ − 2 e 0 K B T − 1 ) 2 + δ ( e δ − 2 e 0 K B T + e 2 ( δ − 2 e 0 ) T K B ) ) K B T ( 1 − e δ − 2 e 0 T K B ) 3 ) + e δ + e 0 K B T ( K B T ( δ − 2 e 0 ) + δ e 0 ) + e 2 δ K B T ( K B T ( 2 e 0 − 3 δ ) + 2 δ e 0 ) T ( ( δ − 2 K B T ) e δ + e 0 K B T + e 2 δ K B T ( K B T + δ ) + K B T e 2 e 0 T K B ) , (27)

and has the shape shown in

It is easily verified that the entropy reduces to that of the linear harmonic oscillator

S = e 0 T coth ( e 0 K B T ) + K B log [ 1 2 csch ( e 0 T K B ) ] , (28)

when λ = 0 .

Thermodynamic ResonanceFrom the result Equation (27), we observe a resonance condition given by

δ = 2 e 0 , (29)

from where a critical value for the anharmonicity parameter can be defined as

λ c = 2 m 0 e 0 ℏ , (30)

for which the entropy reads

S = e 0 T + K B log [ ∞ ] − K B . (31)

For δ < 2 e 0 , it is observed that the entropy remains real and tends to increase more rapidly with increase in strength of the anharmonicity as shown in

Above the critical value, i.e. δ > 2 e 0 , we observe that the argument of the logarithm in the expression of the entropy becomes negative. The logarithm therefore becomes complex valued. We can separate the real and imaginary parts of the logarithm as follows:

K B log ( … ) = i π K B θ [ δ − 2 e 0 ] + K B log ( | … | ) , (32)

where θ is the Heaviside theta function. As such, this leads the entropy to be expressible as S = ℜ ( S ) + i ℑ ( S ) where

ℑ ( S ) = π K B θ [ δ − 2 e 0 ] . (33)

Complex entropy can arise as an extension of the Shannon entropy to classical and nonclassical components of generalized entropy/information descriptors of molecular states in which the real and imaginary parts are provided by the system complex electronic wave-function (quantum probability amplitude) [

Equation (19) portrays an intricate relationship between the ordering ambiguity and anharmonicity introduced by the position dependence of the effective mass. In fact setting the deformation parameter λ to zero automatically eliminates ordering ambiguity terms from the energy spectrum, reducing the spectrum to the harmonic oscillator spectrum. On the other hand, using the most unambiguous form for the kinetic energy operator in which α = γ = − 1 / 4 , b = − 1 / 2 which returns η 1 = η 2 = 0 , the spectrum reduces to that of the harmonic oscillator shifted by an infinite square well like term ℏ 2 λ 2 n 2 / 2 m 0 . Here it is apparent that anharmonicity persists in the absence of ordering ambiguity.

We have shown that the effect of position dependence in the effective mass in this oscillator model is the induction of a sort of thermodynamic resonance during which the entropy of the system becomes infinitely large. The resonance condition establishes a threshold on the anharmonicity strength δ c below which regular evolution of entropy is observed and above which one observes the onset of complex valued entropy. In the latter case, the imaginary part of the entropy has been given the interpretation of the entropy transfered from the system to the environment. This entropy transferred turns out to be a constant π K B .

Our model exhibits a singularity in the entropy at T = 0 . This temperature is of course not physically attainable. If instead of T, one looks at the results in terms of β = 1 / K B T where heat always flows from a small β (albeit negative) to a bigger one, it turns out that in such an anharmonic oscillator, the system becomes an efficient emitter of heat once the critical anharmonicity λ c is exceeded.

Starting with the most general form of the kinetic energy operator for quantum systems with position dependent effective masses, we have shown that the generation of a real spectrum cannot be used to discard some kinetic energy operators from the literature. Using an exactly solved model, of an asymptotically decreasing effective mass system in a parabolic confinement potential, we have shown that the prevalence of ordering ambiguity simply introduces a constant shift in the spectral values of the system. Our results show that anharmonicity introduces a thermodynamic resonance condition in the system, with the onset of a critical value for the anharmonicity parameter beyond which the emitting nature of the system is enhanced. This observation gives a valuable insight for designing semiconductor materials with desired thermodynamic properties

The authors declare no conflicts of interest regarding the publication of this paper.

Vubangsi, M., Migueu, F.B., Kamsu, B.F., L.S.Y., Tchapda, Tchoffo, M. and Fai, L.C. (2021) A Model Effective Mass Quantum Anharmonic Oscillator and Its Thermodynamic Characterization. Journal of Applied Mathematics and Physics, 9, 306-316. https://doi.org/10.4236/jamp.2021.92022