Size-dependent electron chemical potential in nanostructures derived from statistical configuration

: We rederived the fermion distribution function by considering the effect of assembly size. We did not use Stirling approximation to avoid the deviation generated by this approximation for a small number of constituents and small assembly size. Furthermore, we identified that in small systems, the chemical potential should also depend on the assembly size. We also rederived a general expression for the size-dependent chemical potential from a statistical configuration and showed that it is consistent with the results from previously reported theoretical or simulation methods. Finally, we applied the model to derive a size-dependent thermoelectric power factor of nanostructured materials. One important finding is that the power factor initially increases when reducing the particle size; however, it then reduces to approach zero when further reducing the material size, due to a dramatic change in the material behaviors.


Introduction
Chemical potential plays an essential role in materials physics since it controls several important properties. For example, thermoelectricity, the property of materials that are able to convert a temperature gradient into electric power, a potential approach for improvement of renewable energy resources [1,2], depends on the chemical potential, where the thermoelectric power increases with the chemical potential [3][4][5][6][7]. Takeuchi et al. [3] showed that the fine electronic structure near the chemical potential plays an essential role in realizing unusual behavior in thermoelectric power [8][9][10][11]. Cook and Dickerson explained in a simple way how to understand the chemical potential, especially in small systems [12], starting from a definition of the chemical potential as µ = ∂U/∂N ] s,v , where U is the system internal energy, N is the number of particles in the system, S is the entropy, and V is the system volume. This relation states that the chemical potential is equal to the change in energy when one particle is added to the system while maintaining the entropy. Indeed, this is not the only relation to define the chemical potential.
Instead of the presence of a band gap such as that in the bulk state, in a nanostructure with a limited number of particles we only have a stack of discrete lines spreading in an energy range similar to the bandwidth.
The distance between the discrete lines decreases as the particle size increases [13][14][15][16][17]. For small particle diameters, the energy spacing in the group is clearly separated [18]. For example, for a spherical particle that is assumed to have a spherical potential well, the energy of each discrete state is given by ε n,ℓ = 2ℏ 2 χ 2 nℓ /m * e D 2 , where n and ℓ are quantum numbers, ℏ is Planck's constant, χ nℓ is the root of a Bessel function, m * e is the * Correspondence: mikrajuddin@gmail.com This work is licensed under a Creative Commons Attribution 4.0 International License.
effective mass of the electron or the hole, and D is the particle diameter [15,16]. It is clear that if a particle is added to the system (electron or hole), to maintain the entropy, the system energy must be reduced by at least −∆ε n,ℓ ∝ −1/D 2 . Therefore, we can conclude that the chemical potential will depend on the particle size as There are many reports that the chemical potential of nanostructures depends on the size. Champbell and Mao proposed an increment of chemical potential relative to that for an infinitely large particle as ∆µ(D) = [(3γ m −E adh )(1+D 0 /D)(2v m /D) , where γ m is the surface energy of the bulk metal, E adh is the adhesion energy of the bulk metal/oxide interface, v m is the molar volume of the bulk metal, and D 0 is a constant ( ≈1.5 nm for a transition metal) [19]. Nagaev provided a compressive review of the size dependence of the chemical potential [20]. For example, an increase in the chemical potential dependence on the particle size is approximated as ∆µ ∝ γA/N , where A is the surface area. From this we obtain ∆µ = κv m γ/D where κ is a constant. This equation is similar to that previously proposed [21]. Simulation by Croitoru et al. showed that for small nanoparticles, the chemical potential can be approximated by µ(D) = µ(∞) + 2ℏ/m * e α 2 iℓ D 2 [18], with α iℓ the i th zero of a spherical Bessel function of the first kind, j l (x).
Kiyonaga et al. reported an increase in the Fermi energy in TiO 2 -supported Au particles of sizes ranging between 3.0 and 13 nm, where the Fermi energy increased with diameter [22]. In addition, there have been several theoretical works and simulations regarding the dependence of the Fermi energy or chemical potential on size [23][24][25][26]. For example, Sodha and Dubey [24] applied the potential equation from Seitz [27] for a thin film of thickness d deposited on a dielectric substrate as a potential well and obtained an equation relating the Fermi energy with the film thickness in a complicated form [24]. They also derived an equation for spherical particles and obtained the same complicated relation for the Fermi energy and the particle diameter.
Sieperman also derived equations explaining the dependence of the excess chemical potential on the number of particles forming an assembly [26]. Korotun theoretically showed that the Fermi energy of Al and Au nanoparticles increases with decreasing particle diameter, although the increases are not continuous, and on average, the dependence satisfies the empirical function ε F (D) = ε F (∞)(1 + p/D δ ), where p and δ are positive constants and ε F (∞) is the Fermi energy in the bulk state [28]. Using a simple quantum well model, Seki also showed that the chemical potential of an electron in a thin slab depends on the slab thickness [29].
From all those reports arise the following questions: How can we prove fundamentally that the chemical potential depends on the assembly size? Can we show from a derivation of the distribution function that the fermion chemical potential depends on the assembly size? The purpose of this work is to show from the initial consideration, i.e. from the derivation of the distribution function, that the fermion chemical potential depends on the assembly size as well as the size dependence of the thermoelectric power factor.
In this work, we rederive the distribution functions for an assembly consisting of an arbitrary particle number, including very small particle numbers. We will derive the distribution function for fermion assemblies using a method similar to that discussed by Turoff [30]. Using this approach, we are free to apply the method for arbitrary assembly sizes (arbitrary number of particles).

The dependence of the distribution function on the assembly size
We consider an assembly consisting of N identical noninteracting fermions with volume V and temperature T , with the energies of the assembly divided into groups. Each group has a large number of discrete states. However, although a group consists of a large number of states, we further consider that the ith group has an average energy ε i and a number of states g i and is occupied by n i particles, as illustrated in Figure 1. As explained in standard statistical physics books [31][32][33], the number of different ways of arranging the fermions into all the groups in Figure 1a is If an incremental external energy dU is supplied to the assembly, it results in one particle from a state in the p th group jumping to a state in the q th group (Figure 1b). The energy in the p th group is ε p , and the energy in the q th group is ε q . Hence, the energy increment satisfies dU = ε q − ε p . This jumping causes the number of particles in the p th ( q th) group to change to n p − 1(n q + 1) , so by referring to Equation (1) .
We assume that even the number of particles forming the assembly is finite, and the following condition is always fulfilled: n p , n q >> 1. Based on this condition, Equation (2) can be approximated as .
We then use the definition of entropy as S = klnW and S ′ = klnW ′ . The change in entropy due to a particle jumping is dS = S ′ − S . Based on this definition, Equation (3) becomes .
We have a thermodynamic equation relating the energy, entropy, and temperature as T = ∂U/∂S or dU ≈ T dS . If this definition is applied to Equation (4), we obtain which can be rearranged as Figure 1. (a) The energies of the particles forming the assembly are assumed to be discrete, and several nearest energies are grouped. The s th group has g s states and an average energy of g s , and it is occupied by ns particles. (b) One particle in a state in the p th group jumps to a state in the q th group, implying that the number of particles in the p th group decreases by one while the number of particles in the q th group increases by one.
From Equation (6), we obtain the general expression where ψ depends only on constant parameters belonging to the assembly. In equilibrium, the constant parameters are temperature and volume.
Equation (8) describes the dependence of the distribution function on the assembly size merely because of the dependence of the chemical potential on size. The form of the distribution function itself does not change.

General formula for the size dependence of the chemical potential
The dependence of the chemical potential on particle size can also be derived statistically as follows. If one additional fermion is added to the system, the configuration in Equation (1) will change. To maintain the entropy, the number of fermions in each group must reorganize. Some groups will lose fermions, and other groups will receive new fermions. Suppose the change in the number of fermions in the sth group is δn s , which might be a positive or negative integer or zero and satisfies |δn s | << n 2 . The new number of configurations becomes Since S = klnW and S + = klnW + , to conserve entropy, the following equality must be satisfied: The equality is thoroughly satisfied if for all k . From Equation.
(11), we have g s − n s − δn s + k = n s + k , resulting in The change in energy when the entropy is constant is simply the chemical potential. Therefore, the chemical potential in the present case is where ⟨ε⟩ is the average specific energy.
The energy of each group should be multiplied by the energy spacing between the discrete states. For a confined system, this is proportional to the inverse of the particle diameter power and some positive value. We can then approximate where θ s is real and σ is a positive number. In most cases, σ = 2 [34][35][36][37][38][39][40].
The average energy depends on the particle size. We can estimate the average energy based on data of the specific heat capacity. Since the heat capacity is C v = d⟨ε⟩/dT , we can assume that the dependence of the specific energy on the particle dimension is precisely the same as the dependence of the heat capacity on the dimension. There are many proposals for the dependence of heat capacity on particle size, such as those by Singh et al. [41] and Patel et al. [42], ] to explain the size dependence of the melting temperature, where D 0 is a constant [43]. Meanwhile, Xiong et al. proposed with R being the gas constant, and the Debye temperature satisfies is the Debye temperature in the bulk state and θ is a constant [44]. The electronic heat capacity of metals satisfies C v = bT , where T is temperature and b is a factor independent of temperature [33]. Therefore, for nanometer-sized materials, the heat capacity has the relation

and the average energy is given by ⟨ε⟩
showing that the size dependence of the average energy is similar to that of the heat capacity. Using the aforementioned arguments, we can approximate with ϕ (D) being a function of D , based on Singh et al. [41], Zhang et al. [43], Xiong et al. [44], or others. In each case, it always satisfies ϕ(∞) = 1 . A further approximation is found by considering ∑ s ε s θ s ≈ C as a constant. Therefore, we obtain the following approximate expression: In the case of D → σ , we haveµ(∞) ≈ −A. Consequently, Eq. (16) can be further explicitly expressed as Equation (17) is the general equation for the size dependence of the chemical potential. Different materials might have slightly different forms such as different σ or a different function ϕ(D) . However, they must still satisfy ϕ(∞) = 1 , and σ should not be far from 2. In general, Equation (17) states that the chemical potential increases or decreases as the size decreases, depending on the function ϕ(D) . How far it increases depends on the function ϕ(D) , which originates from the material properties and the effect of the surrounding material.
The chemical potential in an intrinsic semiconductor can be written as [45] where m * k is the effective mass of the hole. However, the band gap of a semiconductor has been known to depend on size, as proposed by Brus [46]: where m ef f is the reduced mass and ε is the dielectric constant of the nanoparticles. Using a finite-depth square-well effective mass approximation to provide a quantitative description of the quantum confinement effect, Ferreira et al. obtained a new analytical expression for the size dependence of colloidal semiconductor nanocrystals as [47] Based on tight-binding molecular dynamic simulations, Takai et al. [48] confirmed that the band gap of Si, 3C-SiC, and C quantum dots can be fitted with In other reports, the band gap of most semiconductors satisfies the approximate formula where δ varies between 1 and 1.4 [34][35][36]. Examples of the reported δ values are 1.16 [37], 1.3 [38], and 1.37 [39]. However, a large power has been observed, such as for undoped ZnO and aluminum-doped ZnO films, and the band gap opening as a function of grain size has been well fitted using δ = 1.65 [40]. From these findings, the general expression for the chemical potential inside an intrinsic semiconductor reads as This equation also satisfies Equation (17) when choosing σ = δ and C ′ ≈ c/2µ f (∞), and ϕ = 1 . Therefore, we conclude that the general formula can explain all the observed data regarding the effect of size on the chemical potential.

Size-dependent thermoelectric power
In thermoelectricity, one important parameter related to the conversion figure of merit is the thermoelectric power factor defined as P F = S 2 c, with S the Seebeck coefficient and c the electrical conductivity of the material. It was predicted by Hicks and Dresselhaus that the thermoelectric power factor increases when the material dimension is decreased [49,50] and this prediction has challenged many researchers to find the optimal dimension where the conversion is as high as possible. In the last two decades, there were many investigations to deeply understanding how the thermoelectric power changes with material dimension. Both theoretical and experimental investigations have reached the same conclusion that the thermoelectric power changes with material dimension, in accordance with the Hicks and Dresselhaus prediction [49,50]. In the calculation, they determined r = +0.5, r = 0 , and r = −0.5 for 1d, 2d, and 3d, respectively, to fit the experimental data. Strangely, by selecting these parameters, it all produced the optimal η = 0 to mean that the chemical potential was precisely the same as the bottom of the conduction band or the top of the valence band. They compared their prediction with some experimental data from others and there was no evidence that the experimental data were obtained under such conditions (η = 0) . Based on our finding on the size dependence of chemical potential, we will rederive the size dependence of the thermoelectric power factor with a slightly different approach. We start with the definition of the thermoelectric power factor based on the Mott equation for electrical conductivity and the Seebeck coefficient [52]: and where q is the electron charge, c(E) is the energy-dependent electrical conductivity, and f is the Fermi-Dirac distribution function. From this very basic equation, some more detailed formulas for different materials (metals and semiconductors) have been derived. For example, if the equations are applied to semiconductor materials, one obtains for electron transport located at the conduction band. The symbol c refers to the conduction band. Similar equations are applied for holes at the valence band. At present, we focus only on electron transport at the conduction band and assume the hole transport behaves similarly. The parameter a c originates from the approximation c c (E) = A c (E − E c ) ac and Γ(x) is the gamma function. To obtain the detailed expression for a c or a v , let us consider the following derivation. We used the expression for S and c based on relaxation time approximation [53]: where v(E), τ (E), f (E) , and g(E) are band carrier velocity, carrier relaxation (scattering) time, the Fermi-Dirac distribution function, and the energy density per unit volume, respectively. By comparing Equations (29) and (34), we conclude that . Furthermore, we use the following approximations, The relaxation time is assumed to satisfy τ (E) = τ 0 E ′ [54], with τ 0 a constant. Other authors proposed different forms such as τ (E) = with a 1 and a 2 being constants and r a real parameter between 0 and -3 [55] and g(E) = By substituting all those expressions into Equation (34), we obtain We have obtained the dependence of the chemical potential on size as shown in Equation (17). We also assume Ec and Ev change similarly with size so that one can write . This is acceptable since when the material dimension changes, the location of energy states changes, including the boundary of conduction and the valence band. Based on this assumption, we can then write and the Seebeck coefficient and the conductivity in Equations (33) and (34) transform into In Equations (40) and (41), we have introduced new parameters: Using Equations (40)- (42), we obtain the final expression for the thermoelectric power factor and then Equation (43) is the general expression for size and dimensionality dependence of the thermoelectric power factor. Since A c ∝ 1/D 3−d (Equation (36)), we obtain the thermoelectric power factor that changes with size Now let us search for the optimum size giving rise to the maximum thermoelectric power factor. We search for the size giving the optimum power factor. The optimum thermoelectric power factor satisfies dP F c /dD = 0 .
By substituting Ac from Equation (36) into Equation (43), the criterion for the optimum power factor is Equation (44) can be solved if we know the expression for χ(D). Based on Equation (17), the final expression for χ(D) depends on ϕ(D) . For the moment, let us consider a special case of very small size such that 1/D σ >> ϕ(D) . In this case, we can approximate χ(D) ≈ C ′ /D σ so that Equation (44) can be approximated as For the moment, let us also write y = η c C ′ /D σ , and Equation (45) becomes The solution for Equation (46) is straightforward, i.e.
It is clear that the size of the material that produces the optimum power factor strongly depends on η c , σ , a c , and dimensionality. D opt increases linearly with η 1/σ c and has a complex dependence on σ . The optimum sizes for 0d, 1d, and 2d structures are and respectively. It is clear that if dimensionality increases, the denominators in Equations (48)-(50) decrease, resulting in increases in D opt . Therefore, for a specific material, the thickness of the thin film that produces the optimum power factor is larger than the diameter of nanowires producing the optimum power factor. Now let us estimate the parameter a c . This parameter determines the optimum power factor. From the parameter a c , we can then predict the power factor relaxation time, r , based on Equation (37). For d = 3 , we have Since the size of 3d material is defined as infinity, the optimum size of 3d material is infinity, too. This leads to the conclusion that a c = 1 . Using this value, the optimum sizes for 0d, 1d, and 2d are and respectively. Interestingly, we obtain that the optimum size changes are nearly quantized when we move from 0d to 2d. Let us estimate the optimum sizes as expressed in Equations (52)-(54). If we look back at Equations (24) or (25), we can conclude that C is approximated as C ′ ≈ 2ℏ 2 π 2 /m ef f E g (∞) . For example, for CdS, .18m e and m * k = 0.53m e [56], so m ef f = 0.134m e and C ′ ≈ 4.64 × 10 −18 m e . For CdSe, the reported value is E v (∞) − µ(∞) ≈ 0.7eV eV so that η c ≈ 28 at room temperature. Using Equations (52)-(54) we obtain the estimated values D ef f (0d) ≈ 10.7 nm, D ef f (1d) ≈ 11.4 nm, and D ef f (0d) ≈ 12.9 nm, respectively.
As explained above, the power factor σ might be different for different materials [34][35][36][37][38][39]. Therefore, in predicting the optimum sizes that generate the optimum thermoelectric power factor we must first understand the value of σ . As an illustration, in Figure 2 we plot how the optimum diameter changes with σ for 0d, 1d, and 2d materials. In calculations, we use C' ≈ 4.64x0 −18 m σ and η c ≈ 28 , i.e. the parameters belong to CdS materials. It is clearly seen that for all dimensions, the optimum diameters increase quickly with the increase of σ , and as the dimension decreases, the optimum diameter is larger at a fixed σ . Furthermore, using Equation (37) we have the estimation for r as r = a c − d/2 = 1 − d/2 . Therefore, for 1d, 2d, and 3d materials we obtain r = 0.5, r = 0, and r = -0.5, respectively. Those values are precisely the same as used by Hung et al. [51]. The difference is that their values were estimated, while in our work those values occur automatically. The size dependence of the power factor for small sizes can be approximated as where Ω(T, d) is a factor that depends on T and d , but is independent of the size. For bulk materials, where d = 3 and D → ∞ we have Hence, the ratio of the optimal power factor to that of the same material in the bulk states becomes To make the right-hand side of Equation (57)    For all dimensions, the power factors relative to the corresponding values at the bulk states initially increase with size and then decrease as the size gets larger. At very small sizes, the power factor approaches zero. We also identified the presence of the optimal size where the power factors are maximum. The reduction of power factor at very small sizes can be explained in terms of the reduction of electrical conductivity (increase in resistivity) of material at very small sizes. There are several reports on the metal-to-semiconductor transition as the size is reduced [58,59]. For example, the change of electrical conductivity of metal nanoparticles on size changes similarly to Figure 3 [60]. In Figure 3, we can also observe that at a very small size, the higher power factors belong to materials with lower dimensions. Meanwhile, at larger sizes, the reverse condition occurs, where larger dimension materials have a higher power factor. The thermoelectric power factor of bulk Si is 0.004 W/m K 2 [61]. The power factor of Si NW with a cross-section of 6 nm × 6 nm was reported as 0.0068 W/m K 2 [62]. This size corresponds to D/Λ ≈ 0.56 .
If we compare Equation (58) with Equation (9) in [51], we will notice that where v is the carrier mobility. It is then clear that ξ 3−d ∝ 1/d. If we look at Figure 3, at D/Λ ≈ 0.56 , the relative power factor for WN (1d) material is around 0.5 in scale of ξ 3−d . Therefore, the true power factor must be 3 × 0.5 = 1.5 -fold of the corresponding value at bulk state. Based on the data reported by Curtin and Boewrs [62], the enhancement of the power factor relative to that in the bulk state is 0.0068/0.004 = 1.7, very close to our prediction (circle symbol in Figure 3). In addition, Boukai et al. [63] reported that the power factor on SiNW with dimension of 20 nm at 200 K is 0.007 W/m K 2 . Since Λ ≈ T −1/2 , at this temperature, the de Broglie wavelength becomes 13.2 nm. Therefore, in the report of Boukai et al. [63], we have D/Λ ≈ 1.5 .
By looking at Figure 3, at this value the relative power factor is also around 0.5 in scale of ξ 3−d , or the true enhancement of the power factor relative to that in the bulk state is around 1.5-fold. Again, this prediction is consistent with the experimental data of 0.007/0.004 = 1.75-fold (triangle symbol). Hochbaum observed a power factor of SiNW with diameter 50 nm at 300 K of as much as 0.00175 W/m K 2 [64]. The enhancement relative to the corresponding value at bulk state was 0.00175/0.004 = 0.4375. When we put it in Figure 3, this value must be divided by 3 to result in 0.146. The de Broglie wavelength of silicon at 300 K is around 12 nm so for this nanowire we have d/Λ ≈ 4.2 . This result is indicated by a square symbol in Figure 3. At small sizes, the power factor increases with the square of temperature, while at large sizes, the power factor decreases with temperature. At a certain size, there is an optimum temperature where the power factor is maximum. The optimum temperature can be derived from Equation (47). In this equation, the temperature occurs in η c , given by Equation (42). Therefore, for a certain size, the optimum temperature satisfies the following equation: Therefore, for a material of a certain size, one can control the operating temperature to achieve the optimum power factor. Figure 3. Variation of the thermoelectric power factors for 0d, 1d, and 2d materials when the ratio D/Λ changes. In calculations, we use the parameters belonging to CdS: C' ≈ 4.64 × 10 −18 m σ and ηc 8. In addition, we use σ = 2 and Λ = 10nm . Circle symbol is the experimental power factor of SiNW with cross-section of 6 nm × 6 nm at 300 K [62], triangle symbol is for the experimental power factor of SiNW with cross-section of 20 nm at 200 K [63], and square symbol is for the experimental power factor of SiNW with cross-section of 50 nm at 300 K [64].
The optimum power factor occurs at different sizes when the de Broglie wavelength changes. For example, Figure 4 shows the plot of the power factor when Λ = 5 nm. We can observe that materials with higher dimensions have larger power factors at arbitrary sizes than the lower-dimension materials. This figure is different from Figure 3, where the crossover occurs at D/Λ ≈ 1 . The locations of optimum power factors are also different from Figure 3, although the optimum sizes for lower dimensions are still smaller than those of higher dimensions. Finally, we can state that the proposed model has been able to explain the dependence of the thermoelectric power factor on nanostructure size. We showed that the power factor initially increases when reducing the particle size; however, it then approaches zero when further reducing the material size. At very small sizes, the material can change the behavior dramatically. For example, the transition from metal to semiconductor or even to insulator can occur, dramatically reducing the electrical conductivity to imply the loss of carrier transport ability. We have also tested the model with the experimental data belonging to SiNW both for sizes larger and smaller than the de Broglie wavelength. Both predictions were fairly consistent with the experimental results.

Conclusion
We have obtained a general formula for the size dependence of the chemical potential from a very basic formulation, i.e. using the configuration equation as the foundation for deriving the Fermi-Dirac distribution function. The general formula can explain all reports regarding the size dependence of the chemical potential that have been obtained experimentally or theoretically. The chemical potential generally increases with decreasing material dimensions. These properties are very important, since there are many advantageous properties of materials that are controlled by the chemical potential. We have also derived the general equation to describe the size-dependent thermoelectric power factor, which is more general that the equation introduced elsewhere.