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Thermodynamic properties of electron gas in complex-shaped quantum well
Physica E 69 (2015) 24–26
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Thermodynamic properties of electron gas in complex-shaped quantum well S.R. Figarova a, G.N. Hasiyeva a, V.R. Figarov b,n a b
Department of Physics, Baku State University, 23 Z. Khalilov st., AZ 1148, Baku, Azerbaijan Institute of Physics, 131 H. Javid av., AZ 1143, Baku, Azerbaijan
art ic l e i nf o
a b s t r a c t
Article history: Received 21 October 2014 Accepted 6 January 2015 Available online 7 January 2015
Thermodynamic properties of degenerate two-dimensional electron gas in complex-shaped quantum well are studied. We determine the equation of state, chemical potential, entropy and heat capacity of the electron gas. An influence of profile and parameters of the quantum well on thermodynamic characteristics are investigated. & 2015 Elsevier B.V. All rights reserved.
Keywords: Quantum well Two-dimensional electron gas Pressure Entropy
1. Introduction In the last years, thermodynamics of two-dimensional electron gas (2DEG) in thin films has been quite intensively studied [1–4]. Thermodynamic characteristics of the films are dependent not only on the film thickness but also on profile and parameters of quantum well (QW). The rectangular-deep potential QW, aside to the parabolic one, is one of the most studied systems, however in reality we deal with QWs, firstly, of the finite height and, secondly, of complicated shape. At present, epitaxial growth techniques allows one to construct structures with arbitrary potential profiles. In this paper we investigate an influence of profile and parameters of QW on thermodynamic properties of degenerate 2DEG in complex-shaped QW. Based on the grand thermodynamic potential, we calculate general expressions for the equation of state, chemical potential, entropy and heat capacity of degenerate 2DEG. It has been shown that pressure monotonously depends on the QW parameter. For lower size-quantized levels pressure decreases, while for higher ones it increases. Entropy and heat capacity oscillate with the QW thickness reproducing the density of states. Thermodynamic characteristics are dictated by the ratio between the Fermi level, QW potential and first film energy level.
2. Thermodynamic properties Energy spectrum of charge carriers in QW has the appearance:
εn, k x, k y =
ℏ2k⊥2 + εn, 2m k⊥2
k x2
(1)
k y2.
here = We consider energy spectrum of 2DEG in QW + with the potential energy that, from Schrödinger equation solution, takes the form [5]:
⎛ εn = ε0⎜⎜1 + 2n + ⎝ 2 2
1+
U0 ε0
⎞2 ⎟⎟ , ⎠
(2)
2
here ε0 = ℏ π /2ma at n = 0 and U0 = 0, a is the QW width, U0 is the potential energy minimum, n is the quantum number. In the limit of large quantum numbers, expression (2) coincides with spectrum of infinitely-deep rectangular potential QW:
εn =
ℏ2π 2n2 . 2ma2
In the contrary limit – for small quantum numbers – this spectrum coincides with spectrum of the harmonic oscillatorparabolic QW:
εn = U0 + ℏω0(n + 1/2),
n
Corresponding author. E-mail addresses: fi[email protected] (S.R. Figarova), fi[email protected] (V.R. Figarov). http://dx.doi.org/10.1016/j.physe.2015.01.013 1386-9477/& 2015 Elsevier B.V. All rights reserved.
where ω0 = 2π 2U0/ma2 is the harmonic oscillator frequency. Thus, two model problems, frequently used in quantum mechanics, – the spectrum of particle energy in a rectangular potential well and the spectrum of harmonic oscillator energy – are particular cases of spectrum (2).
S.R. Figarova et al. / Physica E 69 (2015) 24–26
The wave function is given by 2 q /2
ψ = (1 + ξ )
F[q − s , q + s + 1, q + 1, (1 + ξ)/2],
(3)
where ψ = (1 + ξ 2)q /2F [q − s, q + s + 1, q + 1, (1 + ξ)/2] is the hypergeometric function, ξ = tg z /a ,q = 2mε a/ℏ, 2mU0a2/ℏ2=
(
s(s + 1), s = −1 −
)
1 + 8mU0a2/ℏ2 /2 .
For energy spectrum (2) the density of states of 2DEG can be written as
g (ε) =
m ∑ Θ(ε − εn), π ℏ2a n
(4)
where Θ(ε − εn) is the step-like Heaviside function. In order to determine thermodynamic quantities we proceed from the grand thermodynamic potential of 2DEG, which for energy spectrum (2) is
Ω=−
Vm ∑ Θ(ε − εn) πaℏ2 n
∫ε
∞
(ε − εn)f (ε)dε,
(5)
n
here f is the Fermi distribution function. In the 2D case energy is reckoned not from zero but from the first subband bottom of size quantization. From (5) for the concentration nel = − V −1(∂Ω /∂μ)T , V , equation of state P = − (∂Ω /∂V )μ, T and entropy S = − (∂Ω /∂T )V , μ of 2DEG in complex-shaped QW we have:
nel =
m ∑ Θ(ε − εn) πaℏ2 n
m P= ∑ Θ(ε − εn) 2 π aℏ n
∫ε
∫ε
∞ n
∞ n
⎛ ∂f ⎞ (ε − εn)⎜− ⎟dε, ⎝ ∂ε ⎠
By formula (9), the Fermi level was plotted as a function of the film thickness a (Fig. 1a) and QW parameter x = U0/ε0 (Fig. 1b). It is evident that the Fermi level decreases at small values of the film thickness, insignificantly increases at the large values, and monotonously grows with the QW parameter x . The Fermi level oscillates with the QW thickness. The minima are associated with reduction in the first film level energy (which is proportional to a−2) and the density of states (proportional to a−1). The maxima are determined by the QW thickness for which the Fermi level coincides with the next subband bottom. At large quantum numbers the Fermi level behavior is in good agreement with data of Ramos et al. [1]. In the general case, the equation of state for 2DEG is determined from two equations:
⎧ Ω = − PV ⎪ m ⎨ ∑ Θ(ε − εn) ⎪ nel = a π ℏ2 n ⎩
P=
m ∑ Θ(ε − εn) πaℏ2T n
∫ε
∞ n
+ 2xn0 +
(7)
+ where
⎪
+
3(n0 + 1) is
n0
n0 = 2mμ μF = εn .
a πℏ
−
1 2
the
−
1 2
1+
),
integer U0 ε0
(9) part
of
the
number
, which is found from the condition
a
ε02 ⎡ x2 1 2(1 + x) 3(n04 + + + + 5n03 + 12n02 + 11n0 2⎣ n n0 n0 3 3 μ 0
1 (1 + x) + + n03 + 4n02 + 7n0 2n 0 2n 0 3(k 0T )2 ⎛ 1 ⎞⎫ + 4) + x(8n02 + 24n0 + 24 + x)] + ⎜1 + ⎟⎬ , 2 n0 ⎠⎭ ⎝ μ
n0 + 1
(
⎤ x+1 + 4 1 + x (n0 + 2)⎥ n0 ⎦
(11)
⎡ 1 xε0 xε0n0 2ε 1 + x (n02 + 2n0 + 1) μF = ν⎢ + + + 0 ⎢⎣ n0 + 1 ν(n0 + 1) ν(n0 + 1) ν(n0 + 1)
2 ε02 + ε0U0 (n02 + 2n0 + 1) 4ε0 n03 + 3n02 + 2n0 + 1.5
x 1 + + n0 n0
here
nel π ℏ2a U0n0 U0 + + m(n0 + 1) n0 + 1 n0 + 1 +
(10)
+ 10) + 8 1 + x (0.1x +
(8)
Expressions (6)–(8) are just for any degree of EG degeneracy. In the case of degenerate EG in the first approximation with respect to degeneracy for the Fermi level from (6) we obtain:
μF =
⎛ ∂f ⎞ . (ε − εn)⎜− ⎟dε ⎝ ∂ε ⎠
⎡ ⎤2 xε0 2ε 4 ε0n0(n02 + 3n0 + 2)⎥ + n0(n0 + 2) 0 1 + x + ⎢⎣1 + n0 ⎦ 3ν ν ν ⎧ 4ε0 ⎡ 2 1 ⎨1 + − ⎣2n0 + + 6n0 + 7 n0 3μ ⎩
2
⎛ ∂f ⎞ (ε − εn)(ε − μ)⎜− ⎟dε. ⎝ ∂ε ⎠
∞ n
nel ν 2n 0
+
S=
∫ε
Taking into account (7), from (10) for the equation of state of degenerate 2DEG we get:
(6)
⎛ ∂f ⎞ (ε − εn) ⎜− ⎟dε, ⎝ ∂ε ⎠
25
4ε0(n03 + 3n02 + 2n0 + 1.5) ⎤ ⎥, ⎥⎦ 3ν(n0 + 1)
the following designations are used ν = nelπ ℏ2a/m, x = U0/ε0 . In Figs. 2 and 3 we show pressure of degenerate 2DEG, given by formula (11), against the QW thickness and the QW parameter. The numerical results were computed for m = 0.067 m0 , ε0 = 0.06 eV, a = 10 nm and n = 1025 m−3. As may be
b
Fig. 1. Fermi level vs. the film thickness a (a) and QW parameter x = U0/ε0 (b) for n = 1025 m−3, T = 4.2 K, m = 0.067 m0 .
26
S.R. Figarova et al. / Physica E 69 (2015) 24–26
Fig. 4. Entropy of degenerate 2DEG vs. the QW parameter. Fig. 2. Pressure of degenerate 2DEG vs. the QW thickness for n = 1025 m−3, T = 4.2 K , m = 0.067m0 and U0 = 0.18 eV .
The expression for g3D is given in [7]. From the formula it is obvious that 2DEG entropy is larger than that of 3DEG, which is associated with an increase in the density of states of 2DEG in comparison with the density of states of 3DEG. That takes place if the number of degrees of freedom of low-dimensional EG reduces, resulting in an increase in the concentration at the given chemical potential [8]. Dependence of entropy on the QW parameter is presented in Fig. 4; this dependence is step-wise and at the specific ratio between the QW parameter and first film level, entropy drastically approaches zero. Entropy drops since the density of states of 2DEG decreases with the Fermi level growth depending on the QW parameter x at the given concentration of electrons, as follows from the formula nel = g2Dμ and Fig. 1b.
3. Conclusions
Fig. 3. Pressure of degenerate 2DEG vs. the QW parameter.
inferred from Figs. 2 and 3, pressure, oscillating with the QW thickness grows. At the small thicknesses and lower film levels there are possible states with negative pressure. Pressure is nonmonotonic with the QW parameter. At small quantum number values (parabolic QW) pressure falls, whereas for large values (rectangular QW) pressure rises. As clear from formula (11), 2DEG pressure is more than an order of magnitude larger than 3DEG one. In the case of degenerate 2DEG in the first approximation with respect to degeneracy by the formula S = − (∂Ω /∂T )V , μ entropy acquires the form:
S=
π2 2 k 0 Tg (μF ), 3
(12)
where g(μF ) is the density of states at the Fermi level. A similar expression is also obtained for heat capacity of 2DEG. As seen, entropy and heat capacity of degenerate 2DEG oscillate with the thickness and reproduce the density of states behavior at the Fermi level [6], the ratio of 2DEG entropy to 3DEG one is defined by the ratio of their densities of states:
g S2D = 2D . S3D g3D
To summarize, we consider thermodynamic properties of degenerate 2DEG in complex-shaped QW. Taking the finite potential height of QW with complicated shape into consideration leads to nonmonotonic pressure, negative pressure at the very small QW thickness, and also to sharp entropy drop off. Thermodynamic characteristics reproduce the density of states behavior at the Fermi level. Dependence of thermodynamic characteristics on the concentration and QW parameters is determined by a relationship between the Fermi level, QW height and first film energy level. Features of thermodynamic characteristics are observed when the Fermi level crosses the size-quantized level. Entropy nonmonotonously depends on the QW height, at the specific ratio of the QW height to the first film level energy, entropy drastically drops. The entropy drop is due to the strong localization of electrons as the QW height becomes significantly larger than the film level energy and electrons pull into the well, therewithal ordering of the system grows.
(13)
References [1] [2] [3] [4] [5] [6] [7]
A.C.A. Ramos, G.A. Farias, N.S. Almeida, Physica E 43 (2011) 1878. D. Vagner, HIT J. Sci. Eng. A 3 (2006) 102. W. Zawadzki, R. Lassnig, Solid State Commun. 50 (1984) 537. D.N. Quang, N.H. Tung, Do Thi Hien, Anh Huy, Phys. Rev. B 75 (2007) 073305. A.B. Shartsburg, Adv. Phys. Sci. 43 (2000) 1201. E.N. Bogachek, A.G. Schebakov, Usi Landman, Phys. Rev. B 53 (1996) 13246. B.M. Askerov, S.R. Figarova, Thermodynamics, Gibbs method and Statistical Physics of Electron Gases, Springer-Verlag, Heidelberg, 2010. [8] D.A. Broido, T.L. Reinecke, Appl. Phys. Lett. 70 (1997) 2834.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Thermodynamic properties of electron gas in complex-shaped quantum well S.R. Figarova a, G.N. Hasiyeva a, V.R. Figarov b,n a b
Department of Physics, Baku State University, 23 Z. Khalilov st., AZ 1148, Baku, Azerbaijan Institute of Physics, 131 H. Javid av., AZ 1143, Baku, Azerbaijan
art ic l e i nf o
a b s t r a c t
Article history: Received 21 October 2014 Accepted 6 January 2015 Available online 7 January 2015
Thermodynamic properties of degenerate two-dimensional electron gas in complex-shaped quantum well are studied. We determine the equation of state, chemical potential, entropy and heat capacity of the electron gas. An influence of profile and parameters of the quantum well on thermodynamic characteristics are investigated. & 2015 Elsevier B.V. All rights reserved.
Keywords: Quantum well Two-dimensional electron gas Pressure Entropy
1. Introduction In the last years, thermodynamics of two-dimensional electron gas (2DEG) in thin films has been quite intensively studied [1–4]. Thermodynamic characteristics of the films are dependent not only on the film thickness but also on profile and parameters of quantum well (QW). The rectangular-deep potential QW, aside to the parabolic one, is one of the most studied systems, however in reality we deal with QWs, firstly, of the finite height and, secondly, of complicated shape. At present, epitaxial growth techniques allows one to construct structures with arbitrary potential profiles. In this paper we investigate an influence of profile and parameters of QW on thermodynamic properties of degenerate 2DEG in complex-shaped QW. Based on the grand thermodynamic potential, we calculate general expressions for the equation of state, chemical potential, entropy and heat capacity of degenerate 2DEG. It has been shown that pressure monotonously depends on the QW parameter. For lower size-quantized levels pressure decreases, while for higher ones it increases. Entropy and heat capacity oscillate with the QW thickness reproducing the density of states. Thermodynamic characteristics are dictated by the ratio between the Fermi level, QW potential and first film energy level.
2. Thermodynamic properties Energy spectrum of charge carriers in QW has the appearance:
εn, k x, k y =
ℏ2k⊥2 + εn, 2m k⊥2
k x2
(1)
k y2.
here = We consider energy spectrum of 2DEG in QW + with the potential energy that, from Schrödinger equation solution, takes the form [5]:
⎛ εn = ε0⎜⎜1 + 2n + ⎝ 2 2
1+
U0 ε0
⎞2 ⎟⎟ , ⎠
(2)
2
here ε0 = ℏ π /2ma at n = 0 and U0 = 0, a is the QW width, U0 is the potential energy minimum, n is the quantum number. In the limit of large quantum numbers, expression (2) coincides with spectrum of infinitely-deep rectangular potential QW:
εn =
ℏ2π 2n2 . 2ma2
In the contrary limit – for small quantum numbers – this spectrum coincides with spectrum of the harmonic oscillatorparabolic QW:
εn = U0 + ℏω0(n + 1/2),
n
Corresponding author. E-mail addresses: fi[email protected] (S.R. Figarova), fi[email protected] (V.R. Figarov). http://dx.doi.org/10.1016/j.physe.2015.01.013 1386-9477/& 2015 Elsevier B.V. All rights reserved.
where ω0 = 2π 2U0/ma2 is the harmonic oscillator frequency. Thus, two model problems, frequently used in quantum mechanics, – the spectrum of particle energy in a rectangular potential well and the spectrum of harmonic oscillator energy – are particular cases of spectrum (2).
S.R. Figarova et al. / Physica E 69 (2015) 24–26
The wave function is given by 2 q /2
ψ = (1 + ξ )
F[q − s , q + s + 1, q + 1, (1 + ξ)/2],
(3)
where ψ = (1 + ξ 2)q /2F [q − s, q + s + 1, q + 1, (1 + ξ)/2] is the hypergeometric function, ξ = tg z /a ,q = 2mε a/ℏ, 2mU0a2/ℏ2=
(
s(s + 1), s = −1 −
)
1 + 8mU0a2/ℏ2 /2 .
For energy spectrum (2) the density of states of 2DEG can be written as
g (ε) =
m ∑ Θ(ε − εn), π ℏ2a n
(4)
where Θ(ε − εn) is the step-like Heaviside function. In order to determine thermodynamic quantities we proceed from the grand thermodynamic potential of 2DEG, which for energy spectrum (2) is
Ω=−
Vm ∑ Θ(ε − εn) πaℏ2 n
∫ε
∞
(ε − εn)f (ε)dε,
(5)
n
here f is the Fermi distribution function. In the 2D case energy is reckoned not from zero but from the first subband bottom of size quantization. From (5) for the concentration nel = − V −1(∂Ω /∂μ)T , V , equation of state P = − (∂Ω /∂V )μ, T and entropy S = − (∂Ω /∂T )V , μ of 2DEG in complex-shaped QW we have:
nel =
m ∑ Θ(ε − εn) πaℏ2 n
m P= ∑ Θ(ε − εn) 2 π aℏ n
∫ε
∫ε
∞ n
∞ n
⎛ ∂f ⎞ (ε − εn)⎜− ⎟dε, ⎝ ∂ε ⎠
By formula (9), the Fermi level was plotted as a function of the film thickness a (Fig. 1a) and QW parameter x = U0/ε0 (Fig. 1b). It is evident that the Fermi level decreases at small values of the film thickness, insignificantly increases at the large values, and monotonously grows with the QW parameter x . The Fermi level oscillates with the QW thickness. The minima are associated with reduction in the first film level energy (which is proportional to a−2) and the density of states (proportional to a−1). The maxima are determined by the QW thickness for which the Fermi level coincides with the next subband bottom. At large quantum numbers the Fermi level behavior is in good agreement with data of Ramos et al. [1]. In the general case, the equation of state for 2DEG is determined from two equations:
⎧ Ω = − PV ⎪ m ⎨ ∑ Θ(ε − εn) ⎪ nel = a π ℏ2 n ⎩
P=
m ∑ Θ(ε − εn) πaℏ2T n
∫ε
∞ n
+ 2xn0 +
(7)
+ where
⎪
+
3(n0 + 1) is
n0
n0 = 2mμ μF = εn .
a πℏ
−
1 2
the
−
1 2
1+
),
integer U0 ε0
(9) part
of
the
number
, which is found from the condition
a
ε02 ⎡ x2 1 2(1 + x) 3(n04 + + + + 5n03 + 12n02 + 11n0 2⎣ n n0 n0 3 3 μ 0
1 (1 + x) + + n03 + 4n02 + 7n0 2n 0 2n 0 3(k 0T )2 ⎛ 1 ⎞⎫ + 4) + x(8n02 + 24n0 + 24 + x)] + ⎜1 + ⎟⎬ , 2 n0 ⎠⎭ ⎝ μ
n0 + 1
(
⎤ x+1 + 4 1 + x (n0 + 2)⎥ n0 ⎦
(11)
⎡ 1 xε0 xε0n0 2ε 1 + x (n02 + 2n0 + 1) μF = ν⎢ + + + 0 ⎢⎣ n0 + 1 ν(n0 + 1) ν(n0 + 1) ν(n0 + 1)
2 ε02 + ε0U0 (n02 + 2n0 + 1) 4ε0 n03 + 3n02 + 2n0 + 1.5
x 1 + + n0 n0
here
nel π ℏ2a U0n0 U0 + + m(n0 + 1) n0 + 1 n0 + 1 +
(10)
+ 10) + 8 1 + x (0.1x +
(8)
Expressions (6)–(8) are just for any degree of EG degeneracy. In the case of degenerate EG in the first approximation with respect to degeneracy for the Fermi level from (6) we obtain:
μF =
⎛ ∂f ⎞ . (ε − εn)⎜− ⎟dε ⎝ ∂ε ⎠
⎡ ⎤2 xε0 2ε 4 ε0n0(n02 + 3n0 + 2)⎥ + n0(n0 + 2) 0 1 + x + ⎢⎣1 + n0 ⎦ 3ν ν ν ⎧ 4ε0 ⎡ 2 1 ⎨1 + − ⎣2n0 + + 6n0 + 7 n0 3μ ⎩
2
⎛ ∂f ⎞ (ε − εn)(ε − μ)⎜− ⎟dε. ⎝ ∂ε ⎠
∞ n
nel ν 2n 0
+
S=
∫ε
Taking into account (7), from (10) for the equation of state of degenerate 2DEG we get:
(6)
⎛ ∂f ⎞ (ε − εn) ⎜− ⎟dε, ⎝ ∂ε ⎠
25
4ε0(n03 + 3n02 + 2n0 + 1.5) ⎤ ⎥, ⎥⎦ 3ν(n0 + 1)
the following designations are used ν = nelπ ℏ2a/m, x = U0/ε0 . In Figs. 2 and 3 we show pressure of degenerate 2DEG, given by formula (11), against the QW thickness and the QW parameter. The numerical results were computed for m = 0.067 m0 , ε0 = 0.06 eV, a = 10 nm and n = 1025 m−3. As may be
b
Fig. 1. Fermi level vs. the film thickness a (a) and QW parameter x = U0/ε0 (b) for n = 1025 m−3, T = 4.2 K, m = 0.067 m0 .
26
S.R. Figarova et al. / Physica E 69 (2015) 24–26
Fig. 4. Entropy of degenerate 2DEG vs. the QW parameter. Fig. 2. Pressure of degenerate 2DEG vs. the QW thickness for n = 1025 m−3, T = 4.2 K , m = 0.067m0 and U0 = 0.18 eV .
The expression for g3D is given in [7]. From the formula it is obvious that 2DEG entropy is larger than that of 3DEG, which is associated with an increase in the density of states of 2DEG in comparison with the density of states of 3DEG. That takes place if the number of degrees of freedom of low-dimensional EG reduces, resulting in an increase in the concentration at the given chemical potential [8]. Dependence of entropy on the QW parameter is presented in Fig. 4; this dependence is step-wise and at the specific ratio between the QW parameter and first film level, entropy drastically approaches zero. Entropy drops since the density of states of 2DEG decreases with the Fermi level growth depending on the QW parameter x at the given concentration of electrons, as follows from the formula nel = g2Dμ and Fig. 1b.
3. Conclusions
Fig. 3. Pressure of degenerate 2DEG vs. the QW parameter.
inferred from Figs. 2 and 3, pressure, oscillating with the QW thickness grows. At the small thicknesses and lower film levels there are possible states with negative pressure. Pressure is nonmonotonic with the QW parameter. At small quantum number values (parabolic QW) pressure falls, whereas for large values (rectangular QW) pressure rises. As clear from formula (11), 2DEG pressure is more than an order of magnitude larger than 3DEG one. In the case of degenerate 2DEG in the first approximation with respect to degeneracy by the formula S = − (∂Ω /∂T )V , μ entropy acquires the form:
S=
π2 2 k 0 Tg (μF ), 3
(12)
where g(μF ) is the density of states at the Fermi level. A similar expression is also obtained for heat capacity of 2DEG. As seen, entropy and heat capacity of degenerate 2DEG oscillate with the thickness and reproduce the density of states behavior at the Fermi level [6], the ratio of 2DEG entropy to 3DEG one is defined by the ratio of their densities of states:
g S2D = 2D . S3D g3D
To summarize, we consider thermodynamic properties of degenerate 2DEG in complex-shaped QW. Taking the finite potential height of QW with complicated shape into consideration leads to nonmonotonic pressure, negative pressure at the very small QW thickness, and also to sharp entropy drop off. Thermodynamic characteristics reproduce the density of states behavior at the Fermi level. Dependence of thermodynamic characteristics on the concentration and QW parameters is determined by a relationship between the Fermi level, QW height and first film energy level. Features of thermodynamic characteristics are observed when the Fermi level crosses the size-quantized level. Entropy nonmonotonously depends on the QW height, at the specific ratio of the QW height to the first film level energy, entropy drastically drops. The entropy drop is due to the strong localization of electrons as the QW height becomes significantly larger than the film level energy and electrons pull into the well, therewithal ordering of the system grows.
(13)
References [1] [2] [3] [4] [5] [6] [7]
A.C.A. Ramos, G.A. Farias, N.S. Almeida, Physica E 43 (2011) 1878. D. Vagner, HIT J. Sci. Eng. A 3 (2006) 102. W. Zawadzki, R. Lassnig, Solid State Commun. 50 (1984) 537. D.N. Quang, N.H. Tung, Do Thi Hien, Anh Huy, Phys. Rev. B 75 (2007) 073305. A.B. Shartsburg, Adv. Phys. Sci. 43 (2000) 1201. E.N. Bogachek, A.G. Schebakov, Usi Landman, Phys. Rev. B 53 (1996) 13246. B.M. Askerov, S.R. Figarova, Thermodynamics, Gibbs method and Statistical Physics of Electron Gases, Springer-Verlag, Heidelberg, 2010. [8] D.A. Broido, T.L. Reinecke, Appl. Phys. Lett. 70 (1997) 2834.