Plasmon effect on the Coulomb pseudopotential $\mu^*$ in the McMillan equation

TL;DR

This study investigates how the Coulomb pseudopotential $$ varies with carrier density in heavily doped semiconductors, revealing a decrease influenced by plasmon effects that could impact superconductivity.

Contribution

It demonstrates the n-dependence of  in the McMillan equation using first-principles calculations and experiments, highlighting plasmon effects in low carrier systems.

Findings

 decreases with increasing carrier density

 can become negative at high doping levels

Plasmon effects significantly influence superconductivity in semiconductors

Abstract

We examine the Coulomb pseudopotential $\mu^*$ in the McMillan equation applying to the superconductivity of heavily doped semiconductors. Systematic calculation using the first-principles calculation suggests that $\mu^*$ should be considered as a variable quantity depending on carrier density $n$ in semiconductors, although it is usually considered as a constant about 0.1. To clarify $n-$dependence of $\mu^*$, we solve the McMillan equation inversely for $\mu^*$ by combining the result of the first-principles calculation and that of experiments. It indicates that $\mu^*$ decreases with $n$ and becomes negative under $n \sim 5 \times 10^{-21}[{\rm cm^{-3}}]$. This reduction is explained by the effect of plasmon which may play an important role in the superconductivity of low carrier systems such as heavily doped semiconductors.

Equations

$$T_{\mathrm{c}}\simeq \frac{{\omega }_{\log }}{1.2}\exp (-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.62\lambda )}),$$

$${\mu }^{\ast }=\frac{\mu }{1+\mu \ln (E_{\mathrm{F}}/{\omega }_{\mathrm{D}})},$$

$$\mu =\frac{q_{\mathrm{TF}}^2}{8p_{\mathrm{F}}^2}\ln (1+\frac{4p_{\mathrm{F}}^2}{q_{\mathrm{TF}}^2}).$$

$${\mu }^{\ast }\simeq \frac{1.04(1+\lambda )}{(1+0.62\lambda )\ln (1.2T_{\mathrm{c}}/{\omega }_{\log })}+\frac{\lambda }{1+0.62\lambda }.$$

$$\varphi (x)=-{\int }_{-1}^{\infty }\frac{d{x}^{\prime }}{2x^{\prime }}\tanh (E_{\mathrm{F}}{x}^{\prime }/2T_{\mathrm{c}})\varphi (x^{\prime })K(x,x^{\prime }),$$

$$K(x,x^{\prime })=\frac{\sqrt{(1+x^{\prime })}}{4g_{\nu }}$$

$$\times {\int }_{-1}^{1}dt\frac{{\omega }_{\mathrm{p}}+(q_{\mathrm{TF}}^2/q^2)(1+q^2/q_{\mathrm{TF}}^2{)}^{1/2}(|x|+|x^{\prime }|)}{(1+q^2/q_{\mathrm{TF}}^2{)}^{1/2}\{{\omega }_{\mathrm{p}}(1+q^2/q_{\mathrm{TF}}^2{)}^{1/2}+(|x|+|x^{\prime }|)\}},$$

$$T_{\mathrm{c}}=1.134E_{\mathrm{F}}\exp (\frac{1}{{\mu }_{\mathrm{pl}}}),$$

$$\frac{1}{{\mu }_{\mathrm{pl}}}=\frac{1}{{\mu }_0}+{\int }_{-1}^{\infty }\frac{dx}{2|x|}[\varphi (x)K(0,x)/{\mu }_0-\theta (1-|x|)].$$

$$\varphi (x)=K(x,0)/{\mu }_0$$

$$-{\int }_{-1}^{\infty }\frac{d{x}^{\prime }}{2|x^{\prime }|}\varphi (x^{\prime })[K(x,x^{\prime })-K(x,0)K(0,x^{\prime })/{\mu }_0)].$$

$$T_{\mathrm{c}}=\frac{{\omega }_{\mathrm{fi}}}{1.2}\exp (\frac{1.04}{{\mu }_{\mathrm{pl}}^{\ast }}),$$

$${\mu }_{\mathrm{pl}}^{\ast }=\frac{1.04{\mu }_{\mathrm{pl}}}{1+{\mu }_{\mathrm{pl}}\log (1.3608E_{\mathrm{F}}/{\omega }_{\mathrm{fi}})}.$$

Full text

Since the superconductivity in heavily Born-doped diamond was observedEkimov , much effort has been done to find the superconductivity of not only diamond but semiconductors such as Silicon and Germanium.Takano ; Klein ; Kawano ; Okazaki ; Marcenat ; Grockowiak ; Herrmannsdorfer ; Herrmannsdorfer2 ; Koonce ; Kriener If the BCS theoryBCS can be applied to these systems, the transition temperature of superconductivity, $T_{\rm c}$, may be roughly described as $T_{\rm c}\sim\omega_{\rm ph}\exp(-\frac{1}{VN(E_{\rm F})})$, where $\omega_{\rm ph}$ is a characteristic phonon frequency of the system, $V$ is an effective attraction between electrons, and $N(E_{\rm F})$ is the density of state at the Fermi energy. Because higher $\omega_{\rm ph}$ may be expected to lead higher $T_{\rm c}$, the superconductivity of diamond has attracted much interest due to its high phonon frequency up to about 2000K as the Debye frequency $\omega_{\rm D}$. Indeed, experimentsTakano ; Klein ; Kawano ; Okazaki show that $T_{\rm c}$ reaches about 10K or more at carrier density, $n\sim\times 10^{22}{\rm cm^{-3}}$, which is corresponding to boron concentration $\rho\sim 8\%$. Recent progress in implant technology of dopant also enable to produce the heavily doped Si and Ge with carrier concentration up to several percent and to observe its superconductivity of $T_{\rm c}\sim$ 0.7K in boron-doped SiMarcenat ; Grockowiak and 1K in gallium-doped Ge.Herrmannsdorfer ; Herrmannsdorfer2 .

In theoretical point of view, many works using the first-principles calculation address the superconductivity of semiconductors by assuming a traditional phonon mechanism.Boeri ; Xiang ; Ma ; Moussa ; Subedi ; Sano In these theories, so-called the McMillan equationMcMillan ; Allen is used to estimate $T_{\rm c}$ as a more sophisticated formula than the BCS theory,

[TABLE]

where $\omega_{\rm log}$ is a logarithmic average frequency, $\lambda$ is the electron-phonon coupling constant, and $\mu^{*}$ is the Coulomb pseudopotential. Although the equation is produced as a semi-empirical formulation to fit the numerical solution of the Elashberug equation, estimated $T_{c}$ is well consistent with many experiments of usual metals.

Since it is not easy to determine $\mu^{*}$ within the first-principles calculation, its value is usually treated as a constant about 0.1, based on the work of Morel and Anderson.Morel They drive $\mu^{*}$ within Thoms-Fermi approximation

[TABLE]

where $E_{\rm F}$ is the Fermi energy and $\mu$ is the screened Coulomb potential

[TABLE]

Here, $q_{\rm TF}$ is the Thomas-Fermi wave number and $p_{\rm F}$ is the Fermi momentum. It is interpreted that $\mu$ is renormalized to $\mu^{*}$ by the difference of the energy scale between phonon and Coulomb potential. In usual metals, a typical value of $\mu$ is 0.2 and $\ln(E_{\rm F}/\omega_{\rm D})\sim 5$, and then $\mu^{*}$ is roughly given by $\sim 0.1$. When we adapt the above $\mu^{*}$ into Eq.1, we can estimate a reasonable value of $T_{\rm c}$ for many realistic materials. Because the first-principles calculation gives us accurate values of $\lambda$ and $\omega_{\rm log}$ without ambiguity. It even succeeds in prediction of new phonon mediated high-temperature superconductors such as SH3Li and LaH10Liu at high pressure. Therefor, the above formulation might seem to give us a useful recipe to estimate $T_{\rm c}$ of not only usual metal but also heavily doped semiconductors.

However, comparison of theory and experiment for semiconductors seems to be not well, although many theoretical works analyze $T_{\rm c}$ using the McMillan equation. Especially, $n$-dependence of $T_{\rm c}$ can not been explained at all.Bustarret If we wish to explain $n$-dependence of $T_{\rm c}$ in the framework of the McMillan formulation, we should assume $\mu^{*}$ not a constant but a variable depending on $n$. In the case of diamond superconductor, Klein et al. have given $\mu^{*}$ as a function of $n$ concretely. They substitute experimental result of $T_{\rm c}$ as a function of $n$ into the McMillan equation and solve it inversely for $\mu^{*}$. It shows that $\mu^{*}$ decreases with $n$ and the value seems to be even negative under $\sim 10^{21}{\rm cm^{-3}}$. In their work, however, the cause of this reduction is not discussed beyond phenomenological analysis.

This result is very curious, because $\mu^{*}$ represents the effect of repulsion as the screened Coulomb potential and should be positive. Furthermore, Eqs.2 and 3 indicate that $\mu^{*}$ monotonically increases with the decrease of $n$.mu-n-depend Therefore, it is hard to understand the behavior of $\mu^{*}$, and the result might seem suspicious. In this letter, we reexamine this problem and clarify the puzzling behavior of $\mu^{*}$ based on the plasmon mechanism which has been discussed on the superconductivity of the uniform dilute electron gas.Takada1 ; Takada2 ; Rietschel It would give a reasonable interpretation for $n$-dependence of $\mu^{*}$ and recover the usefulness of the McMillan formulation applying to the superconductivity of semiconductors.

At first, we clarify the discrepancy between results of experiments and that of McMillan equation with a fixed $\mu^{*}$. To do it, we calculate $n$-dependence of $\lambda$ and $T_{\rm c}$ for several semiconductors, and compare $T_{\rm c}$ systematically. Calculations are performed using the ’Quantum ESPRESSO’, which is a useful computer code of the pseudopotential method.QE We adopt the rigid-band approximation(RBA)Subedi , which is the method to introduce fictitious carrier in the pure system by assuming the rigid-band. It changes only the Fermi energy of the system according to carrier density. Although it is a simple approximation, it allows us to obtain systematic results for any semiconductors at any carrier density.mesh

In Fig.1a, we show the result of $\lambda$ for diamond(empty circles), Si(empty squares), germanium(empty rhombus), strontium titanate(empty triangle), and silicon carbide(empty down-pointing triangle). Here we also show the results of the super-cell approximation(SCA)Xiang and the virtual crystal approximation(VCA)Ma , which are represented by solid circles and cross, respectively. Further, doted line indicates a fitting for the result of diamond, which is given by $0.11\times(n/10^{21})^{0.5}$. In the case of diamond, $\lambda$ of the SCAXiang is larger than that of the RBA as shown in the figure. Since the SCA produces virtual periodicity of dopant in the system, the value of $N(E_{\rm F})$ and $\lambda$ may be overestimated.Shirakawa ; Yanase

On the other hand, the result of the VCAMa is almost consistent with that of the RBA. Indeed, both methods should be equivalent to each other in the limit $n\to 0$. In the RBA, the effect of the impurity potential as a dopant is completely neglected except the effect of carrier doping. Then, the value of $\lambda$ may be underestimated, especially in the diamond system.Yanase However, if we attention to $n$-dependence of $\lambda$, we find that these methods give similar results. Further, our systematic calculation indicates that $n$-dependence is approximately described by a power function with respect to $n$, and the value of its power is near a half for all semiconductors within our calculation. For example, the powers of Si and Ge are given by 0.56 and 0.52, respectively, which are close to that of diamond. Because semiconductors have similar band structures near $E_{\rm F}$ in the case of hole doping, these results may be plausible.power

In Fig.1(b), we compare experimental $T_{\rm c}$ with theoretical one. Symbols indicate $T_{\rm c}$ obtained by the experiments for diamond(crossEkimov , double circlesKlein and solid circlesKawano ), Si(empty squaresGrockowiak ), Ge(empty rhombusHerrmannsdorfer2 ), strontium titanate(empty triangleKoonce ), and silicon carbide(empty down-pointing triangleKriener ). Solid lines represent the theoretical result calculated by our analysis for diamond and doted lines are that of Si with $\mu^{*}=-0.05,0,$ and 0.1. Here, to calculate $T_{\rm c}$ as a function of $n$, we use the power function discussed in the above. We also use $\omega_{\rm log}$ as 1600K for diamond and 430K for Si in Eq.1, which are typical values and $n$-dependences of these are neglected for simplicity. As shown in Fig.1(b), it indicates that $n$-dependences of $T_{\rm c}$ obtained by theory are far from that of experiments, even if we use negative $\mu^{*}$.Bustarret As long as a constant $\mu^{*}$ is adopted to Eq.1, it is difficult to explain $n$-dependences of $T_{\rm c}$. This result suggests that we need to change the traditional interpretation of $\mu^{*}$.

To focus on the behavior of $\mu^{*}$, we solve the McMillan equation inversely for $\mu^{*}$,

[TABLE]

Substituting experimental $T_{c}$ and $\lambda$ of the RBA to the above equation, we estimate $\mu^{*}$ of several semiconductors.wlog In Fig.2, we show $\mu^{*}$ as a function of $n$ with the result of Klein et al.Klein . They used $\lambda$ obtained by the SCA and the VCA to calculate $\mu^{*}$. In the case of diamond, $\lambda$ of the former is larger than that of the latter. Therefore the line of $\mu^{*}$ by the SCA(open circles) is on the upper side than that of the VCA(crosses) and the RBA(solid circles) in the figure. Because, experimental $T_{\rm c}$ of the ref. [3] is smaller than that of ref. [4], it causes the difference between the results represented by double circles and that of solid circles. Figure 2 clearly indicates that $\mu^{*}$ is not a constant but a parameter depending on $n$. It decreases with $n$, and becomes negative in the case of $n\stackrel{{\scriptstyle<}}{{{}_{\sim}}}3\times 10^{21}{\rm cm^{-3}}$. Our result also suggests that the behavior of $\mu^{*}$ is not limited in diamond, but common in all semiconductors.

In the above, we show $n$-dependence of $\mu^{*}$ obtained by the phenomenological method using the experimental $T_{\rm c}$. To explain this behavior theoretically, we obtain $\mu^{*}$ based on the plasmon mechanism and compare it with the above result. Then, we consider the gap equation of the uniform electron gas system based on the work of TakadaTakada1 ,

[TABLE]

where $\phi(x)$ is a renormalized gap function defined as $\phi(0)=1$. Here, the kernel $K(x,x^{\prime})$ is given by

[TABLE]

where $g_{\rm\nu}$ is valley degeneracy of the band, the frequency of plasmon $\omega_{\rm p}$ is given as $\sqrt{4\pi{\rm e}^{2}n/m^{*}}$, $q^{2}$ is defined by $2m^{*}E_{\rm F}\{2+x+x^{\prime}-2t\sqrt{(1+x)(1+x^{\prime})}\}$, and $q_{\rm TF}^{2}=4g_{\rm\nu}{\rm e}^{2}m^{*}p_{\rm F}/\pi$, with the effective mass $m^{*}$. Here, the dielectric constant is set to be unity for simplicity.

Using $K(x,x^{\prime})$ and $\phi(x)$, $T_{\rm c}$ is given by

[TABLE]

where

[TABLE]

Here, $\mu_{0}=K(0,0)$ and $\phi(x)$ is determined by

[TABLE]

We noted that $\mu_{0}$ is corresponding to $\mu$ of Eq.3 with valley degeneracy $g_{\rm\nu}$.Takada1 . Thus, $\mu_{\rm pl}$ can be regarded as an effective Coulomb potential including the effect of plasmon.

Of cause, the above analysis is driven to the uniform electron system and there is no energy scale of phonon. So, we introduce a fictitious phonon frequency $\omega_{\rm fi}$ to Eq.7 as

[TABLE]

where

[TABLE]

When we set $\omega_{\rm fi}$ to $\omega_{\rm log}$, it is completely corresponding to the McMillan equation in the case of $\lambda\to 0$. By replacing $\mu^{*}$ with $\mu_{\rm pl}^{*}$, we can expect that Eq.1 gives more reasonable $T_{\rm c}$ including the effect of plasmon in the case of $\lambda<<1$. To confirm above consideration, we solve the above gap equation numerically and calculate $\mu_{\rm pl}$ as a function of $n$. In Fig.2, we show $\mu_{\rm pl}$ and $\mu_{\rm pl}^{*}$, where $\omega_{\rm fi}$ is set as 1600K. Here, the valley degeneracy and the effective mass of carrier are set as 3 and 0.5, respectively, to adapt our analysis to a typical semiconductor.e-mass In the present systems, the value of $E_{\rm F}/\omega_{\rm fi}$ is not so largeadiabatic and then, the difference between $\mu_{\rm pl}$ and $\mu_{\rm pl}^{*}$ is small.

As shown in Fig.2, the $n$-dependence of $\mu_{\rm pl}^{*}$ is well consistent with that of $\mu^{*}$ calculated by the experimental $T_{\rm c}$. It suggests that the McMillan equation replacing $\mu^{*}$ to $\mu_{\rm pl}^{*}$ is valid to analyze the superconductivity of not only for diamond but also for other semiconductors. In Fig.2, the dash dotted line represents $\mu_{0}/(1+\mu_{0}\ln(E_{\rm F}/\omega_{\rm D}))$, which is corresponding to the traditional $\mu^{*}$ without the effect of plasmon. It suggests that the effect of plasmon reduces $\mu^{*}$ to about 0.2 at $n\simeq 1.0\times 10^{21}[{\rm cm^{-3}}]$, and its contribution is fairly large as well as phonon. For $n\stackrel{{\scriptstyle>}}{{{}_{\sim}}}0.2\times 10^{21}[{\rm cm^{-3}}]$, $\lambda$ is smaller than 0.05 as shown in Fig.1. In this region, the absolute value of $\mu^{*}$ becomes larger than $\lambda$ and the mechanism of plasmon may be superior to that of phonon to rise $T_{\rm c}$. In fact, $T_{\rm c}$ of strontium titanate increases with the decrease of $n$, as shown in Fig.1, although $\lambda$ decreases with $n$. It indicates that the effect of plasmon dominates the superconductivity in this region.Takada2 After all, the overall behavior of $n$-dependence of $\mu^{*}$ shows that the mechanism of the superconductivity is characterized by the interplay between plasmon and phonon.Hanke

In summary, we examine the Coulomb pseudopotential $\mu^{*}$ on the McMillan formulation in the case of applying to the heavily doped semiconductors. Systematic calculation of the electron-phonon coupling constant $\lambda$ by the first-principles calculation indicates that $\mu^{*}$ should be not a constant, but a variable depending on carrier density $n$. The $n-$dependence of $\mu^{*}$ indicates that it decreases with $n$ and it becomes negative under $n\sim 2\times 10^{-21}[{\rm cm^{-3}}]$. We find that the reduction of $\mu^{*}$ is interpreted by the effect of plasmon which may play an important role in the superconductivity of low carrier systems. The effect is fairly large and seems to be beyond the effect of phonon for $n\stackrel{{\scriptstyle<}}{{{}_{\sim}}}1\times 10^{-21}[{\rm cm^{-3}}]$. Although our analysis is limited in the case of $\lambda<<1$, it expands the validity of the McMillan equation into the superconductivity of semiconductors and explains the $n$-dependence of $T_{\rm c}$ very well in the case of $n\stackrel{{\scriptstyle<}}{{{}_{\sim}}}5\times 10^{-21}[{\rm cm^{-3}}]$.

More sophisticated analysis including the plasmon effect has been already studied on the superconductivity of lithium.Akashi However, the formulation is rather complicated and it may be hard task to apply to the superconductivity of semiconductors systematically. Although the McMillan equation is more phenomenological method, it is widely accepted and useful to estimate $T_{\rm c}$ of many materials. We think that the reinterpretation of $\mu^{*}$ discussed in our analysis develops practical usefulness of the McMillan equation much further.

ACKNOWLEDGMENTS

This work was supported by JPSJ KAKENHI Grant Number 15K05168.