[1] Dmitrii V. Semenok

[3]Emil A. Yuzbashyan

[1]Center for High Pressure Science & Technology Advanced Research (HPSTAR),
Bldg. 8E, ZPark, 10 Xibeiwang East Rd, Beijing, Haidian 100193, China

2]Physics Department, Columbia University, 538 West 120th Street, New York, NY
10027, USA

[3]Department of Physics and Astronomy, Center for Materials Theory, Rutgers
University, Piscataway, NJ 08854, USA

# Fundamental limits on the electron-phonon coupling and superconducting
$T_{c}$

<EMAIL_ADDRESS>Boris L. Altshuler<EMAIL_ADDRESS>*  [  *

###### Abstract

We establish fundamental upper bounds on the electron-phonon interaction
strength and superconducting transition temperature $T_{c}$ in metals stemming
from an intrinsic instability of the equilibrium between electrons and the
crystal lattice with respect to strong interaction. Our theory explains why
the largest observed electron-phonon couplings $\lambda\leq 4$ as well as the
mechanism of metastable superconductivity with higher $T_{c}$. We conclude
based on our theory and analysis of experimental data that room-temperature
superconductivity is feasible but only in hydrogen compounds.

##  1 Introduction

The interaction of electrons with lattice vibrations (phonons) is at the heart
of the theory of metals. It determines superconducting $T_{c}$, electrical and
thermal conductivities, and many other physical properties[1, 2]. Existing
theoretical frameworks, such as the Bardeen-Cooper-Schiffer (BCS)[3],
Migdal[4], and Eliashberg[5] theories, provide no limits on the electron-
phonon interaction and, as a result, on $T_{c}$. In this paper, we show that
there are in fact intrinsic upper bounds on both these quantities dictated by
the stability of the metal with respect to the electron-lattice interaction.
We compare our results with experimental data and argue that room-temperature
superconductivity is entirely realistic but only in hydrogen-rich compounds.

Figure 1: The evolution of superconducting materials in 20th and 21st
centuries as exemplified by metals and intermetallic alloys (grey squares),
which are well described by the BCS theory, unconventional superconductors
(orange circles) and hydrides (blue rhombi). The inset shows the distribution
of superconducting transition temperatures for the hydrides of various metals.
Shown in grey are the metals whose polyhydrides have not been investigated for
superconductivity.

The question, “What is the maximum possible superconducting $T_{c}$?" has
remained unanswered since the discovery of superconductivity by Onnes in
1911[6], despite remarkable progress in this area[7, 8, 9, 10, 11, 12]. At the
same time, the $T_{c}$ of real materials never exceeded 133 K at atmospheric
pressure and 160 K at elevated pressures ($\sim$30 GPa) in more than a hundred
years (1911-2011) of experimental experience. It is believed that metallic
hydrogen is a superconductor with one of the highest critical temperatures[13,
14]. This is because $T_{c}$ is proportional to lattice vibration frequencies,
which are highest in this material since hydrogen is the lightest element.
Unfortunately, producing metallic hydrogen requires pressures in excess of 450
GPa[15, 16], at the limit of reach of current experimental techniques for
transport measurements. There is, however, an ingenious solution – alloying
hydrogen with other elements[17]. This provides an effective chemical pressure
thus reducing the external pressure necessary to produce a stable metal.

And indeed, compressed polyhydrides became the leaders in the quest for the
highest $T_{c}$ since the discovery of record superconductivity in 2014 and
2018 in the cubic hydrides of sulfur (H3S, max $T_{c}=200$ K) and lanthanum
(LaH10, max $T_{c}=250$ K). This “hydride revolution” gave rise to justified
hopes of room-temperature superconductivity in the near future, see Fig. 1.
Moreover, several incorrect reports of $T_{c}$ exceeding room temperature in
the ternary systems C-S-H[18, 19], Y-Pd-H[19], and Lu-N-H[20] emerged but were
quickly refuted. At the present moment, the maximum well reproduced critical
temperature is $T_{c}=250$ K in LaH10, (La,Y)H10[21, 22, 23], (La, Sc)H10[24],
and other ternary compounds containing lanthanum. At the same time, the
question whether room-temperature superconductivity is attainable remains
open.

Here we will approach this problem from a different perspective. Our recent
studies within the Migdal-Eliashberg (ME) theory revealed that the metallic
and superconducting states are unstable with respect to strong electron-phonon
interaction[25]. A traditional measure of the strength of this interaction is
the electron-phonon coupling constant $\lambda$ defined as the sum of the
couplings $\lambda_{k}$ to the individual lattice vibration modes. This
characterization of the interaction strength with a single number is not
unique, and we introduce below another metric $\xi$ – the stability parameter
of the metal. It corresponds to the contribution of the electron-phonon
interaction to the electronic specific heat and is also a sum of $\lambda_{k}$
but with unequal weights.

Indeed, we will see that the dynamical stability of a metal requires
$\xi<\xi_{*}=1$. The metallic state ceases to be the global minimum of the
free energy at a smaller value $\xi_{c}<\xi_{*}$ via a first order phase
transition and is no longer even a local minimum when $\xi>\xi_{*}$. For a
fixed phonon spectrum, $\lambda$ and $\xi$ are directly proportional to each
other, so that this condition translates into
$\lambda<\lambda_{c}<\lambda_{*}$. The value of $\lambda_{*}$ is smallest,
$\lambda_{*}=3.69$, for dispersionless (Einstein) phonons. When $\lambda$
exceeds $\lambda_{c}$ ($\xi$ exceeds $\xi_{c}<1$), the crystal structure of
the metal becomes unstable and is either destroyed or undergoes a
reconstruction lowering the value of $\lambda$ below $\lambda_{c}$. This
result provides an effective tool to investigate the maximum transition
temperature of electron-phonon superconductors thanks to an upper bound on
$T_{c}$ within the ME theory in terms of $\lambda$ and the average square
phonon frequency[8, 26].

##  2 Analysis of experimental data

Presently, superconductors with highest critical temperatures are the
polyhydrides. They also have some of the highest values of the electron-phonon
interaction constant $\lambda\sim 2-3$ and of the stability parameter $\xi\sim
0.2-0.5$ (Table 1). Interestingly, the values of $\lambda$ for hydrides lie
approximately within the same range as for soft superconducting metals Pb and
Bi and their alloys, even though the critical temperatures are tens of times
larger. This begs the question: Can the electron-phonon interaction be
arbitrarily large or is it inherently bounded from above?

Table 1: Various superconducting metals, compounds, and compressed
polyhydrides and the values of $T_{c},$ electron-phonon coupling constant
$\lambda$, stability parameter $\xi$, and average logarithmic
($\omega_{\mathrm{log}}$) and maximum ($\omega_{\mathrm{max}}$) phonon
frequencies for them.

Compound |  $\bm{T}_{c}$, K | $\bm{\lambda}$ | $\bm{\xi}$ |  $\bm{\omega}_{\mathrm{\textbf{log}}},$ K |  $\bm{\omega}_{\bm{\max}},$ K
---|---|---|---|---|---
Experimental Materials
Ga (amorph.)[34] | 8.6 | 2.25 | 0.16 | 62 | 302
Pb (amorph.)[34] | 7.2 | 1.91 | 0.29 | 35 | 155
Bi (amorph.)[34] | 6.1 | 1.84 – 2.46 | 0.19 | 42 | 163
Hg[34] | 4.2 | 1.0 – 1.6 | 0.14 | 86 | 165
Nb[34] | 9.2 | 0.82 – 1.05 | 0.2 | 163 | 325
Nb3Sn[35] | 17.9 | 1.6 – 1.8 | 0.27 | 142 | 422
MgB2[36] | 40 | 0.87 | 0.16 | 680 | 1624
PbBi (amorph.)[37] | 7.0 | 3.0 | 0.26 | 33.3 | 150
PbBi3 (amorph.)[37] | 6.8 | 2.78 | 0.23 | 33.4 | 154
H3S (157 GPa)[29] | 190 | 1.84 | 0.32 | 1078 | 2704
LaH10 (214 GPa)[21, 22] | 245 | 2.06 | 0.43 | 1340 | 3145
YH6 (165 GPa)[38] | 224 | 1.71 | 0.39 | 1333 | 2450
ThH9 (150 GPa)[39] | 146 | 1.73 | 0.32 | 957 | 3362
ThH10 (170 GPa)[39, 40] | 161 | 1.65 | 0.31 | 1116 – 1470 | 3122
YH9 (205 GPa)[27, 41] | 235 | 2.66 | 0.36 | 916 – 1054 | 3167
(La,Y)H10 (180 GPa)[24] | 253 | 3.87 | 0.47 | 868 | 3119
(La,Ce)H9 (123 GPa)[32, 42] | 190 | 2.27 | 0.33 | 582 – 915 | 2894
Theoretical Calculations
CaH6 (172 GPa)[47] | 215 | 2.69 | 0.58* | 950 | 2877
LaH16 (250 GPa)[44] | 141 | 1.89 | 0.31 | 1511 | 3481
ScH12 (200 GPa)[45] | 325 | 2.85 | 0.47 | 1189 | 2959
Li2MgH16 (250 GPa)[46] | 473 | 3.30 | 0.49 | 1111 | 3783
MgH6[47] | 263 | 3.29 | 0.58 | 1408 | 3719
Hydrogen (I41/amd, 500 GPa)[48] | 374 | 2.85 | 0.45 | 1616 | 4089

  * *

Theoretical calculations for $P=150$ GPa without accounting for the
anharmonicity.

Figure 2: Distribution of superconducting materials in the
$\lambda-\omega_{\mathrm{log}}$ plane. (a) Open red circles and filled blue
rhombi show theoretically predicted and experimentally synthesized hydrides,
respectively. Filled grey squares correspond to simple metals, alloys, and
MgB2. Dashed grey lines are defined by the equation $T_{c}^{\max}\equiv
0.182\omega_{\mathrm{log}}\sqrt{\lambda}=\mathrm{const}$. (b) Comparison of
experimentally observed and theoretically calculated critical temperatures
$T_{c}$ for various superconductors with
$T_{c}^{\max}=0.182\omega_{\mathrm{log}}\sqrt{\lambda}$. Observe that for all
experimentally realized materials $T_{c}<T_{c}^{\max}$.

Hydride superconductors or superhydrides (Figs. 2a and 3) can be synthesized
only at high pressure $P$, about $100-200$ GPa, in special diamond anvil
cells. Subsequent lowering of the pressure leads to a domelike dependence of
$T_{c}$ on $P$ characteristic of hydrides[21, 22, 27, 28, 29]. Increasing $P$
does not lead to the destruction of the polyhydrides but is accompanied by an
increase of phonon frequencies, decrease of the electron-phonon interaction
strength, and, most importantly, decrease of $T_{c}$. An appropriate measure
of the magnitude of phonon frequencies at these $\lambda$[26] is the average
logarithmic frequency $\omega_{\mathrm{log}}$ defined as
$\ln\omega_{\mathrm{log}}=\frac{1}{\lambda}\sum_{k}\lambda_{k}\ln\omega_{k}$.

The situation seen when decreasing the pressure is almost the opposite –
phonons soften, while $\lambda$ and $\xi$ grow (Fig. 3b). It may seem that
decreasing the pressure further and thus increasing $\lambda$, we can make
$T_{c}$ arbitrarily large, since $T_{c}\propto\sqrt{\lambda}$ at large
$\lambda$ according to the ME theory[26, 30]. In reality, this turns out to be
impossible, since the growth of the electron-phonon interaction in hydrides
inevitably leads to the distortion of the crystal structure, lowering of its
symmetry, diffusion and partial loss of hydrogen, and, as a result, to an
abrupt lowering of the superconducting $T_{c}$ (Fig. 3a). Good examples of
this process are the decompression of D3S[29, 31], LaH10[21], YH9[27], and
CeH9[28] as well as recent studies of ternary lanthanum-cerium superhydrides
(La,Ce)H9[32, 33].

Figure 3: Instability and decomposition of polyhydrides with decreasing
pressure. (a) Experimentally observed dependence of the critical temperature
$T_{c}$ on pressure for certain polyhydrides and deuterium sulfide (D3S). As
the pressure is lowered below its critical value $P_{c}$, $T_{c}$ abruptly
decreases. This is accompanied by a distortion of the crystal structure and
partial loss of hydrogen. (b) Dependence of the stability parameter $\xi$ on
pressure for several experimentally synthesized hydrides. The $\xi=0.5$ line
marks the supposed first order phase transition, i.e., upon crossing this
line, the destruction of the structure is merely a function of time and the
height of the kinetic barrier. Filled symbols correspond to experimental data,
empty ones are theoretical calculations. The point $\xi=0.96$ (LaH10) is an
extrapolation obtained by multiplying the Eliashberg function
$\alpha^{2}F(\omega)$ for LaH10 at 129 GPa by a factor of 1.5. (c)
Decomposition of LaH10 upon lowering the pressure below $P_{c}=138$ GPa
accompanied by a distortion of its cubic structure, partial loss of hydrogen
(H2) and formation of lower hydrides, such as LaH9.

##  3 Fundamental limit on the electron-phonon interaction strength in metals

When does a metal stop being a metal as the pull between conduction electrons
and phonons increases? In a new study[25], we discovered that the specific
heat of conduction electrons ($C_{\mathrm{el}}$) turns negative when the
electron-phonon coupling constant exceeds a certain threshold $\lambda_{*}$.
The precise value of $\lambda_{*}$ depends on the phonon spectrum and, in
particular, $\lambda_{*}=3.69$ for Einstein phonons. Stability analysis of the
kinetic equations (Supplemental Material) shows that when $C_{\mathrm{el}}<0$,
the metal is unstable with respect to an infinitesimal difference between the
electron and phonon temperatures, $T_{\mathrm{el}}$ and $T_{\mathrm{ph}}$,
respectively. Specifically, if initially $T_{\mathrm{el}}>T_{\mathrm{ph}}$,
$T_{\mathrm{el}}$ will grow exponentially and the system will never
equilibrate. This indicates that metals with $\lambda>\lambda_{*}$ cannot
exist in nature even in a metastable state.

This result goes beyond effective electron-phonon models, such as Holstein or
Frölich Hamiltonians. By not renormalizing phonons and instead taking the
Eliashberg function from ab initio calculations or experiment[26, 49], we
avoid artificial phonon softening at bare electron-phonon coupling
$\lambda_{0}\approx 0.5$[4, 5, 50], which is a result of double counting the
static electronic contribution (overscreening) inherit in such models[51, 52,
53]. Moreover, this spurious lattice instability does not provide a sharp
upper limit on the physical electron-phonon coupling $\lambda$, because
$\lambda\to\infty$ as $\lambda_{0}\to 0.5$. In contrast, our result does
furnish such a limit for arbitrary phonon spectrum without any phonon
softening. In particular, our upper bound cannot be overcome by invoking many
phonon modes with comparable couplings[10] as this does not affect the ME
expression for $C_{\mathrm{el}}$.

As noted above, the stability requirement $C_{\mathrm{el}}\geq 0$ is
equivalent to a certain upper limit on the electron-phonon interaction. To see
this, consider a well-known expression for the electronic specific heat at
temperature $T$[54, 55, 56] (in units $\hbar=k_{B}=1$),

$C_{\mathrm{el}}=\frac{2}{3}\pi^{2}\nu_{0}T\left[1-\int_{0}^{\infty}g\left(\frac{\omega}{2\pi
T}\right)\frac{2\alpha^{2}F(\omega)}{\omega}d\omega\right]\\!,$ (1)

where $\nu_{0}$ is the density of states at the Fermi energy,
$g(x)=6x+12x^{2}\mathrm{Im}\psi^{\prime}(\i
x)+6x^{3}\mathrm{Re}\psi^{\prime\prime}(\i x),$ and $\psi(x)$ is the digamma
function. The combination $\alpha^{2}F(\omega)$ is the Eliashberg function
defined as

$\alpha^{2}F(\omega)=\sum_{k}\frac{\lambda_{k}\omega_{k}}{2}\delta(\omega-\omega_{k}),$
(2)

where $\omega_{k}$ are the frequencies of lattice vibration modes.

With the help of Eq. (1), the stability condition $C_{\mathrm{el}}>0$ for all
$T>T_{c}$ takes the form

$\xi\equiv\max_{T}\left\\{\int_{0}^{\infty}g\left(\frac{\omega}{2\pi
T}\right)\frac{2\alpha^{2}F(\omega)}{\omega}d\omega\right\\}<1.$ (3)

The maximum is with respect to the temperature $T$. The stability parameter
$\xi$ is another measure of the electron-phonon interaction strength. Just as
$\lambda$, it grows linearly with the overall scale of the Eliashberg function
but is proportional to a weighted average of $\lambda_{k}$,
$\xi=\sum_{k=1}^{N}\lambda_{k}G_{k}$, where
$G_{k}=g\left(\frac{\omega_{k}}{2\pi T_{\max}}\right)$.

Figure 4: Values of the stability parameter $\xi$ for various hydrides as a
function of (a) average logarithmic frequency $\omega_{\mathrm{log}}$, (b)
electron-phonon coupling constant $\lambda$, and (c) superconducting $T_{c}$.
Filled and empty symbols correspond to experiment and theoretical
calculations, respectively. Dashed violet lines indicate the trend apparent
among the best hydride superconductors.

The phase transition from the metal (superconductor below $T_{c}$) to a new
state as we increase the electron-phonon coupling is of the first order[25].
$C_{\mathrm{el}}=0$ marks the point where the metal becomes unstable with
respect to small deviations from the thermal equilibrium, i.e., ceases to be a
local minimum of the free energy. In first order phase transitions, such a
local (absolute) instability is preceded by a metastable region
$\xi_{c}<\xi<\xi_{*}=1$. The metal is no longer the global minimum of the free
energy past the phase-transition point $\xi_{c}$. Therefore, $\xi_{*}=1$
provides a fundamental upper bound on $\xi$ in metals and superconductors.
This implies $\xi_{c}<1$, but the exact value of $\xi_{c}$ is nonuniversal and
can depend on, for example, the lattice and electronic band structures, number
of carries per lattice site etc.

We analyzed data for numerous metals and superhydrides for which the
Eliashberg functions are known and found that all of them conform to our upper
bound $\xi<1$. Hydrides loose stability at $\xi\leq 0.5$, while $\xi$ for
other metals in Table 1 and Fig. 4 is significantly lower, suggesting that
$\xi_{c}=0.5$ for hydrides and $\xi_{c}<0.5$ for conventional metals. We
attribute $\xi=0.58$ in CaH6 and MgH6 to the fact that these are not actually
existing hydrides at these pressures. Moreover, if our conjecture that
$\xi_{c}\leq 0.5$ holds, these two materials can only exist in a metastable
state.

The condition $\xi<\xi_{*}=1$ implies also an upper bound
$\lambda<\lambda_{*}$ on the standard interaction parameter $\lambda$ defined
as

$\lambda=\int_{0}^{\infty}d\omega\frac{2\alpha^{2}F(\omega)}{\omega}=\sum_{k=1}^{N}\lambda_{k}.$
(4)

The value of $\lambda_{*}$ varies with the shape of the Eliashberg function
with a minimum $\lambda_{*}=3.69$ attained for Einstein phonons (see
Supplemental Material). For Debye phonons ($\alpha^{2}F\propto\omega^{2}$ for
$\omega\leq\omega_{D}$ and zero otherwise), $\lambda_{*}=4.72$. For the
Eliashberg function of the same shape as that for a material with given $\xi$
and $\lambda$ (i.e., differing from it only by an overall scale), we have
(Supplemental Material)

$\lambda_{*}=\frac{\lambda}{\xi},\quad\lambda_{c}=\frac{\lambda\xi_{c}}{\xi}=\lambda_{*}\xi_{c}.$
(5)

In particular, using Table 1, we obtain $\lambda_{*}=7.39$ and (assuming
$\xi_{c}=0.5$) $\lambda_{c}=3.69$ for YH9 at 205 GPa and $\lambda_{c}=2.36$
for Debye phonons. The values of $\lambda_{c}$ for all experimentally realized
hydrides are in the range from 2.19 for YH6 to 4.12 for (La,Y)H10. This
indicates that stability must be the reason why $\lambda$ in actually existing
metals is limited to $\lambda<4$.

We argued previously that in the new state the system lowers its energy by
either opening a gap or, at least, lowering the density of states at the Fermi
energy[25]. We also saw that the superconducting state is more resilient
against perturbations than the normal state as it already has a gap.
Therefore, we expect robust metastable superconductivity when $\lambda$ is
quenched from $\lambda<\lambda_{c}$ to $\lambda>\lambda_{c}$ while the system
is superconducting. This creates an opportunity of attaining metastable
superconductivity with substantially higher $T_{c}$ in polyhydrides at lower
pressures than usual, since $\lambda$ generally increases with decreasing $P$,
see Fig. 2. Moreover, metastable superconductivity of this type has likely
been observed in pressure-quenched FeSe[57], but a more thorough analysis of
these experiments is necessary to establish this with certainty.

##  4 Fundamental limit on $T_{c}$ in phonon mediated superconductors

In the formal limit $\lambda\to\infty$ of the ME theory, $T_{c}$
asymptotically approaches $0.18\sqrt{\lambda\langle\omega^{2}\rangle}$, while
at finite $\lambda$ it falls below this asymptote[8, 26, 58], i.e.,

$T_{c}<0.18\sqrt{\lambda\langle\omega^{2}\rangle}.$ (6)

This provides an upper bound on $T_{c}$ in terms of $\lambda$ and the average
square frequency
$\langle\omega^{2}\rangle=\frac{2}{\lambda}\int_{0}^{\infty}\alpha^{2}F(\omega)\omega
d\omega$.

The requirement $C_{\mathrm{el}}>0$ imposes a constraint $\lambda<\lambda_{*}$
on $\lambda$. Let $\omega_{\max}$ be the maximum available phonon frequency.
Spreading out the phonon spectral function to frequencies below
$\omega_{\max}$ increases $\lambda_{*}$, while decreasing
$\langle\omega^{2}\rangle$ and results in an overall decrease of $T_{c}$
(Supplemental Material). Therefore, the critical temperature for Einstein
phonons ($T_{c}^{E}$) with maximal $\lambda=\lambda_{*}=3.69$ and frequency
$\omega_{\max}$ provides a rigorous upper bound on $T_{c}$ for stable and
metastable superconducting metals,

$T_{c}<0.32\,\omega_{\max},$ (7)

where we used a standard algorithm[26, 59] to numerically compute
$T_{c}^{E}=0.3175\,\omega_{\max}$. We define a metal as a good conductor with
a Fermi energy $E_{F}$ much larger than typical phonon frequencies. This is
the only essential assumption that goes into the Eliashberg theory from which
we derived Eq. (7).

Recall that metals are stable for $\lambda<\lambda_{c}$ and metastable for
$\lambda_{c}<\lambda<\lambda_{*}$. Assuming $\xi_{c}=1/2$ (see above), Eq. (5)
obtains $\lambda_{c}=\lambda_{*}/2$. Similarly calculating
$T_{c}^{E}=0.1995\,\omega_{\max}$ for Einstein phonons with $\lambda=3.69/2$,
we determine an upper bound on $T_{c}$ for stable metals,

$T_{c}<0.20\,\omega_{\max}.$ (8)

Metals with $0.20\,\omega_{\max}<T_{c}<0.32\,\omega_{\max}$ can therefore only
exist in a metastable state.

Table 1 lists the values of $\omega_{\max}$ we extracted for various materials
from their Eliashberg functions. In the Debye model $\omega_{\max}$ is equal
to the Debye frequency $\omega_{D}$. However, since real solids do not conform
to the Debye model, $\omega_{D}$ is not uniquely defined. In particular, its
value depends on the quantity used to extract it (e.g., the phonon specific
heat vs resistivity). We therefore prefer to use the edge of the phonon
spectrum (Eliashberg function) as $\omega_{\max}$ whenever possible. Note that
both our upper bounds (7) and (8) are significantly more generous than the
heuristic bound of $0.1\omega_{D}$ proposed in [9]. However, observe that, for
example, (La,Ce)H9-10 at 123 GPa with $T_{c}=190$ K and $\omega_{D}=1107$ K,
extracted from the temperature dependence of the electrical resistance,
violates the latter bound, so it apparently does not hold for compressed
hydrides. Indeed, to fall below the $0.1\omega_{D}$ bound, (La,Ce)H9-10 must
have a Debye frequency of about 2000 K, which exceeds all calculated and
experimental values of $\omega_{D}$ ever obtained for synthesized
superhydrides (see Table S4 in Supplemental Material).

Our results suggest that room-temperature superconductivity is achievable in
metals with $\omega_{\mathrm{max}}>1500$ K and $\lambda\geq 2$, which up until
now has only been observed in hydrides at high pressure. Moreover, since we
have condensed the entire phonon weight to $\omega=\omega_{\max}$ to maximize
$T_{c}$, the maximum of the Eliashberg function cannot be far below
$\omega=1500$ K in a room-temperature superconductor.

We can obtain an absolute numerical upper bound on $T_{c}$ by observing that
the maximum phonon frequency cannot exceed the ionic plasma frequency and that
interactions will only renormalize $\omega_{\max}$ down[60]. The ionic plasma
frequency is inversely proportional to the square root of the ionic mass, and
we therefore expect it to be the highest in the metallic hydrogen. According
to recent ab-initio calculations of the Eliashberg function for solid atomic
hydrogen at 500 GPa[48], $\omega_{\max}=3000\,\mathrm{cm}^{-1}=4320$ K for
harmonic phonons (as the ones in the Eliashberg theory), see also [11].
However, this Eliashberg function has a sharp maximum below $\omega_{0}=3000$
K dropping very quickly from the maximum to zero at $\omega_{\max}$. In
addition, theoretical estimates of the Debye frequency for the metallic
hydrogen are in the range 3000–3500 K[14, 61]. It is therefore safe to replace
$\omega_{\max}$ in Eqs. (7) and (8) with $\omega_{0}=3000$ K, and we obtain
the following absolute upper bounds on $T_{c}$ in stable and metastable
metals:

$T_{c}^{\mathrm{stable}}<\frac{600\,\mathrm{K}}{\sqrt{A}},\quad
T_{c}^{\mathrm{metastable}}<\frac{950\,\mathrm{K}}{\sqrt{A}},$ (9)

where $A$ is the atomic mass (in atomic units) of the lightest element in the
material.

It is important to note that $\omega_{\max}$ grows (roughly linearly) with
pressure, see, e.g., [11] and Fig. S1a in Supplemental Material. For the above
bounds, we used $\omega_{\max}$ at 500 GPa – the current upper limit for
transport measurements. However, there is a negative correlation between
$\omega_{\max}$ and other characteristic phonon frequencies and the electron-
phonon interaction strength (Figs. 2 and S1b), which we did not take into
account in our analysis, such that increasing the pressure beyond 500 GPa will
likely result in an overall decrease rather than increase of $T_{c}$.

##  5 Conclusion

We established intrinsic upper bounds on the electron-phonon interaction
strength and superconducting $T_{c}$ in metals. Materials where this
interaction exceeds a certain threshold cannot exist in nature in a metallic
state, similar to the states of the Van der Waals gas with negative
compressibility. Just as in the case of the Van der Waals gas, there is an
absolute instability (negative electronic specific heat) that signals a first
order phase transition without immediately telling us what the new phase is.
However, based on additional considerations[25] and experiment, we believe
that the tendency is towards a lattice reconstruction.

It is generally known that superconducting $T_{c}=\kappa\omega_{\max}$, where
$\omega_{\max}$ is the maximum or some other characteristic phonon frequency
and the coefficient $\kappa$ is a monotonically increasing function of the
strength of the electron-phonon interaction (overall height of the Eliashberg
function). As a result, a fundamental bound on $T_{c}$ is impossible without a
limit on the interaction strength. Prior work suggested that $\kappa$ might be
somehow limited by stability, but was unsuccessful in furnishing definitive
evidence of an instability as well as in determining its character and
criteria for it. Our work fills this crucial gap and provides the precise
stability limit.

It is clear from the bounds (9) that stable room-temperature superconductivity
can only be achieved in hydrogen $(A=1)$ and deuterium $(A=2)$ compounds.
Substitution of hydrogen with deuterium typically decreases $T_{c}$ by a
factor of roughly 1.4[21, 27, 28, 29, 38, 62] (isotope effect) consistent with
Eq. (9). Helium-3 and 4 are extremely unlikely candidates[63], and, in
addition, $T_{c}<300$ K already for $A=4$. Similarly, the $T_{c}$ for phonon
mediated superconductivity in cuprates (lightest element – 16O) is limited by
150 K, which is coincidentally about the record $T_{c}$ for these materials.

At the same time, there are no fundamental reasons why $T_{c}$ cannot exceed
the room temperature in hydrides, at least at a sufficiently high pressure.
All we need to engineer such superconductors, is $\lambda=2-3$ that primarily
comes from phonons with frequencies near or above 1500 K. Moreover, we saw
that the upper bound on $T_{c}$ increases 1.59 times for metastable metals and
that even further increase is possible if the pressure (electron-phonon
coupling) is quenched while the material is in the superconducting state as in
recent FeSe experiments[57].

Dmitri V. Semenok’s research was supported in part by the National Natural
Science Foundation of China (NSFC, grant No. 1231101238) and Beijing Natural
Science Foundation (grant No. IS23017).

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Supplemental Material

##  1 Stability analysis

To evaluate the stability of a metal with respect to electron-phonon
interactions, we use the standard kinetic equation for the electron
distribution function $f(E,t)$[1, 2],

$\left(1-\frac{\partial\Sigma}{\partial E}\right)\frac{\partial f}{\partial
t}+\frac{\partial\Sigma}{\partial t}\frac{\partial f}{\partial
E}=a_{\mathrm{ep}}I_{\mathrm{ep}}(E)+a_{\mathrm{ee}}I_{\mathrm{ee}}(E),$ (S1)

where we added nonnegative constant coefficients $a_{\mathrm{eph}}$ and
$a_{\mathrm{ee}}$ for convenience. The electron self-energy is

$\Sigma=\fint
dE^{\prime}\int_{0}^{\infty}\\!\\!\\!d\omega\,\alpha^{2}F(\omega)\frac{f(E^{\prime}+\omega)-f(E^{\prime}-\omega)}{E-E^{\prime}},$
(S2)

and the electron-phonon collision integral is

$I_{\mathrm{ep}}(E)=-2\pi\int_{0}^{\infty}\\!\\!\\!d\omega\,\alpha^{2}F(\omega)\left\\{N_{0}(T_{\mathrm{ph}})[2f-f_{+}-f_{-}]+f(f_{+}-f_{-})+f-f_{+}\right\\},$
(S3)

where $f\equiv f(E,t)$, $f_{\pm}\equiv f(E\pm\omega,t)$, and

$N_{0}(T_{\mathrm{ph}})=\frac{1}{e^{\omega/T_{\mathrm{ph}}}-1}$ (S4)

is the equilibrium Bose (phonon) distribution at temperature
$T_{\mathrm{ph}}$. The kinetic equation (S1) assumes spatially uniform initial
conditions and, as usual, that phonons remain in thermal equilibrium, see
below and, e.g., Ref. [2], which also provides an explicit expression for the
electron-electron collision integral $I_{\mathrm{ee}}.$

We will prove that the metal is unstable when $C_{\mathrm{el}}<0$ by
contradiction. Let

$f(E,t=0)=f_{0}(T)\equiv\frac{1}{e^{E/T}+1}$ (S5)

be the equilibrium Fermi distribution with temperature $T$ that is slightly
higher than the phonon temperature $T_{\mathrm{ph}}$. Experimentally, initial
conditions of this type are created by heating the electrons with an
ultrashort laser pulse[3, 4, 5]. Note, however, that we do not assume the two-
temperature model, but merely chose an initial condition of the form (S5). The
true fermion distribution $f(E,t)$ will generally become nonthermal in the
course of the actual time evolution of the system.

Suppose the system is stable with respect to the electron-phonon interaction
($a_{\mathrm{ep}}=1$, $a_{\mathrm{ee}}=0$). Since the phonon specific heat is
much larger than that of the electrons, $C_{\mathrm{ph}}/|C_{\mathrm{el}}|\sim
E_{F}/\omega_{D}\gg 1$, the change in the phonon temperature and, more
generally, the deviation of the phonon distribution from the equilibrium Bose
distribution $N_{0}(T_{\mathrm{ph}})$ is negligible. Phonons serve as a
thermal bath for the electrons, and electrons equilibrate at temperature
$T_{\mathrm{ph}}$ up to corrections of order $\omega_{D}/E_{F}$, which are
beyond the accuracy of the Eliashberg theory anyway.

Linearizing the kinetic equation and $f(E,t)$ around the equilibrium at
$T=T_{\mathrm{ph}}$ with the help of the usual substitution[6],

$f(E,t)=f_{0}+\delta f\equiv
f_{0}(T_{\mathrm{ph}})+\frac{f_{0}(T_{\mathrm{ph}})\left[1-f_{0}(T_{\mathrm{ph}})\right]}{T_{\mathrm{ph}}}\varphi(E,t),$
(S6)

we obtain after some algebra

$\int\\!\\!dE^{\prime}A(E,E^{\prime})\dot{\varphi}(E^{\prime},t)=-\int\\!\\!dE^{\prime}\left[a_{\mathrm{ep}}\lambda
M_{\mathrm{ep}}(E,E^{\prime})+a_{\mathrm{ee}}M_{\mathrm{ee}}(E,E^{\prime})\right]\varphi(E^{\prime},t),$
(S7)

where $A(E,E^{\prime})$, $M_{\mathrm{ep}}(E,E^{\prime})$, and
$M_{\mathrm{ee}}(E,E^{\prime})$ are real symmetric integration kernels
(matrices) and $\dot{\varphi}\equiv\partial\varphi/\partial t$. Letting
$\varphi(E,t)=e^{-\gamma t}\psi(E)$, we reduce the problem to a generalized
eigenvalue equation of the form $\gamma A\psi=M\psi$. Matrices
$M_{\mathrm{ep}}$ and $M_{\mathrm{ee}}$ are positively defined, because they
are independent of $\lambda$ and the system is obviously stable ($\gamma>0$)
in the weak coupling limit $\lambda\to 0$. The assumption that the system is
stable with respect to the electron-phonon interaction (i.e., for $M=\lambda
M_{\mathrm{ep}}$) implies that $A$ is positively defined as well. Since a
linear combination of two positively defined real symmetric matrices with
positive coefficients is similarly positively defined, it follows that it must
also stable for $M=a_{\mathrm{ep}}\lambda
M_{\mathrm{ep}}+a_{\mathrm{ee}}M_{\mathrm{ep}}$ for any $a_{\mathrm{ep}}>0$
and $a_{\mathrm{ee}}>0$.

Therefore, if the metal is stable with respect to the electron-phonon
interaction, Eq. (S1) must also be linearly stable for any choice of
$a_{\mathrm{ep}}>0$ and $a_{\mathrm{ee}}>0$. Let $a_{\mathrm{ee}}\gg
a_{\mathrm{ep}}>0$. Then, electron-electron collisions dominate, and the
electron distribution thermalizes essentially instantaneously with temperature
$T(t)$, i.e., $f(E,t)=f_{0}(T(t))$, corresponding to the instantaneous energy
density $\epsilon(t)$ of the electronic subsystem. However, since electron-
electron collisions conserve $\epsilon(t)$ (this follows formally from
$\int\\!dEEI_{\mathrm{ee}}(E)=0$), the latter is able to relax only through
electron-phonon collisions. Multiplying Eq. (S1) by $E$, integrating over $E$,
and using $f(E,t)=f_{0}(T(t))$, we obtain

$C_{\mathrm{el}}\frac{dT}{dt}=2\pi
a_{\mathrm{ep}}\int_{0}^{\infty}\\!\\!\\!d\omega\alpha^{2}F(\omega)\left[N_{0}(T_{\mathrm{ph}})-N_{0}(T)\right],$
(S8)

where

$C_{\mathrm{el}}=\int\\!\\!dEE\left\\{\left(1-\frac{\partial\Sigma_{0}}{\partial
E}\right)\frac{\partial f_{0}}{\partial T}+\frac{\partial\Sigma_{0}}{\partial
T}\frac{\partial f_{0}}{\partial E}\right\\}$ (S9)

is the electronic specific heat. Here $\Sigma_{0}$ and $f_{0}$ are the
equilibrium electron self-energy and distribution function at temperature $T$.
Eq. (S9) for $C_{\mathrm{el}}$ was derived by Prange and Kadanoff in 1964[1]
and later shown to be equivalent to Eq. (1) [7, 8].

The right hand side of Eq. (S8) is negative when $T>T_{\mathrm{ph}}$
indicating that the heat flows from hotter electrons to the colder phonon
bath. Linearizing this equation in $(T-T_{\mathrm{ph}})$, we see that $T$
grows exponentially when $C_{\mathrm{el}}<0$. Note that Eq. (S8) does not
describe the actual dynamics of the system but is a consequence of the
assumption that it is stable with respect to the electron-phonon interaction.
Thus, $C_{\mathrm{el}}<0$ is a sufficient condition of the instability.

##  2 Derivation of upper bounds on the electron-phonon interaction constant
$\lambda$ and $T_{c}$

It is helpful to disentangle the notions of the shape, height (overall scale),
and support (spread) of the Eliashberg function $\alpha^{2}F(\omega)$ from
each other. To this end, we introduce a normalized shape function

$P(\omega)=\frac{1}{\lambda}\frac{2\alpha^{2}F(\omega)}{\omega}.$ (S10)

By the definition of $\lambda$ in Eq. (4) and since $\alpha^{2}F(\omega)\geq
0$,

$\int_{0}^{\infty}\\!\\!\\!P(\omega)d\omega=1,\quad P(\omega)\geq 0,$ (S11)

so we can think of $P(\omega)$ as the distribution function of phonon
frequencies. It determines the shape of the Eliashberg function, while
$\lambda$ controls its overall height. Note that we can specify $\lambda$ and
$P(\omega)$ independently of each other.

Now it is straightforward to determine the upper bound $\lambda_{*}$ on the
electron-phonon coupling $\lambda$ from the stability condition (3),

$\lambda_{*}=\frac{1}{\max_{T}\left\\{\int_{0}^{\infty}g\left(\frac{\omega}{2\pi
T}\right)P(\omega)d\omega\right\\}}.$ (S12)

The function $g(x)$ reaches its single maximum $g_{\max}=0.2709$ at
$x_{\max}=0.3273$. The upper bound $\lambda_{*}$ is smallest,
$\lambda_{*}=1/g_{\max}=3.6915$, for Einstein phonons, i.e., for
$P(\omega)=\delta(\omega-\omega_{E})$, because in this case the denominator of
Eq. (S12) assumes its maximum possible value $g_{\max}$. Similarly, Eq. (S12)
provides $\lambda_{*}$ for other phonon spectra, e.g., for the Debye model
[$P(\omega)=2\omega/\omega_{D}^{2}$ for $\omega<\omega_{D}$ and zero
otherwise] or for the same spectrum [same $P(\omega)$] as that for any of the
materials in Table 1. Furthermore, the definition of the stability parameter
$\xi$ in Eq. (3) together with Eq. (S12) imply Eq. (5).

To bound $T_{c}$, we need to bound both $\lambda$ and the phonon spectrum.
Suppose the highest phonon frequency is $\omega_{\max}$ and let
$x=\omega/\omega_{\max}$. Then,

$\lambda\langle\omega^{2}\rangle=\omega_{\max}^{2}\frac{\int_{0}^{1}x^{2}P(x)dx}{\max_{\tau}\left\\{\int_{0}^{1}g\left(\frac{x}{\tau}\right)P(x)dx\right\\}}.$
(S13)

By analyzing the variational derivative of the right hand side of this
equation with respect to $P(x)$, we find that it is maximal for
$P(x)=\delta(x-1)$, i.e., for Einstein phonons with frequency
$\omega_{E}=\omega_{\max}$. Taking this into account, we immediately obtain
Eq. (7) from Eq. (6).

##  3 Extended Data

Table S1: Data for Fig. 1: evolution of superconducting materials in 20th – 21st centuries. Material | Year of discovery | Critical temperature, K
---|---|---
Low-temperature superconductors[9]
Hg | 1911 | 4.2
Pb | 1914 | 7.3
Nb | 1933 | 9.7
NbN | 1941 | 16
Nb3Sn | 1952 | 18.3
V3Si | 1953 | 17.1
Nb3Ge | 1973 | 23.2
NbTi | 1962 | 9.2
CaC6[10] | 2005 | 11.5
MgB2 | 2001 | 40
Unconventional superconductors[11]
KWO3 | 1967 | 6
LiTi2O4 | 1973 | 1.2
BaPbBiO3 | 1975 | 13
La2BaCuO4 | 1986 | 30
YBa2Cu3O7 | 1987 | 90
BaKBiO3 | 1988 | 20
BiSrCaCu2O6 | 1988 | 105
TlBa2Ca2Cu3O9 | 1989 | 110
HgBa2CaCu2O6 | 1993 | 120
GdFeAsO | 2008 | 53.5
SrFFeAs | 2009 | 72
| Superhydrides[9, 12] |
LaH10 | 2018 | 250
ThH9 | 2020 | 146
ThH10 | 2020 | 161
YH6 | 2021 | 224
YH9 | 2021 | 243
CeH9 | 2021 | 100
CeH10 | 2021 | 115
BaH12 | 2021 | 20
YH4 | 2022 | 82
(La,Y)H10 | 2021 | 253
Lu4H23 | 2023 | 70
SnH4 | 2022 | 74
PH3 | 2015 | 100
SbH4 | 2023 | 115
(La,Ce)H9-10 | 2023 | 190
NbH3 | 2024 | 42
Table S2: Parameters of the electron-phonon interaction in metals,
polyhydrides and, other compounds. We used this data to draw Fig. 2.

Compound | Pressure, GPa | EPC strength ($\bm{\lambda}$) |  $\bm{\omega}_{\mathrm{log}}$, K
---|---|---|---
Theoretical predictions
ScH9[13] | 300 | 1.94 | 1156
VH8[14, 15] | 200 | 1.13 | 876
CrH3[16] | 81 | 0.95 | 568
ZrH6[17, 18] | 295 | 1.7 | 914
ZrH10[19] | 250 | 1.77 | 1068
NbH4[20, 21] | 300 | 0.82 | 938
SrH6[22] | 100 | 1.65 | 1316
HfH10[19] | 250 | 2.77 | 677
TaH6[23] | 300 | 1.56 | 1151
TiH14[22, 24] | 200 | 0.81 | 1063
Hf3H13[22] | 100 | 1.1 | 497
RaH12[22] | 200 | 1.36 | 998
MgH6[25] | 300 | 3.29 | 1450
CaYH12[26] | 200 | 2.2 | 1230
MgCaH12[27] | 200 | 2.53 | 1400
YH10[28, 29] | 250 | 2.58 | 1282
ScH12[30] | 200 | 2.85 | 1189
Li2MgH16[31] | 250 | 3.3 | 1111
ScH4[13, 32] | 250 | 0.81 | 1892
Metals and alloys (experiment)
Nb[33] | 0 | 1.05 | 229
V[33] | 0 | 0.83 | 330
Sn[33] | 0 | 0.72 | 165
Ta[33] | 0 | 0.73 | 213
Hg[33] | 0 | 1.3 | 86
a-Ga[33] | 0 | 2.25 | 62
a-Pb*[33] | 0 | 1.91 | 35
a-Bi[33] | 0 | 2.46 | 42
Nb3Sn[33] | 0 | 1.7 | 142
MgB2**[34] | 0 | 0.87 | 680
a-PbBi[35] | 0 | 3.0 | 33.3
a-PbBi3[35] | 0 | 2.78 | 33.4
Hydrides (experiment)
(La,Y)H10[36] | 180 | 3.87 | 868
(La,Ce)H9-10[37] | 123 | 2.27 | 915
H3S[38] | 157 | 1.84 | 1080
LaH10[39] | 163 | 2.67 | 1118
YH9[12] | 200 | 2.75 | 885
YH6[40] | 170 | 2.24 | 1330
ThH10[41] | 170 | 1.91 | 1210
ThH9[41] | 150 | 1.73 | 960
YH4[40] | 155 | 1.1 | 1080
CeH9[42] | 110 | 1.46 | 650
CeH10[43] | 100 | 2.0 | 1000
SnH4[44] | 190 | 1.24 | 890
CaH6[45] | 170 | 2.69 | 950

  * *

Amorphous film of Pb

  * **

Given for comparison

Table S3: Values of the stability parameter $\xi,$ electron-phonon coupling
constant $\lambda$, and the average logarithmic frequency
$\omega_{\mathrm{log}}$ for several hydrides at different pressures $P$.

Compound |  $\bm{P}$, GPa | $\bm{\xi}$ | $\bm{\lambda}$ |  $\bm{\omega}_{\mathrm{log}},$ K
---|---|---|---|---
CeH9 | 100 | 0.175 |  |
120 | 0.165 |  |
150 | 0.152 |  |
200 | 0.131 |  |
LaH10 |  $\sim$50* | 0.96 | 5.32 |
129 | 0.64 | 3.62 | 887
163 | 0.53 | 2.67 | 1119
214 | 0.43 | 2.06 | 1340
264 | 0.37 | 1.73 | 1469
H3S | 135 | 0.42 |  |
157 | 0.39 |  |
ThH10 | 100 | 0.46 | 2.57 | 1048
200 | 0.32 | 1.58 | 1184
300 | 0.25 | 1.33 | 1150
CaH6 | 150** | 0.58 | 2.97 | 964
160 | 0.41 | 1.96 | 1204
170 | 0.40 | 1.89 | 1419
180 | 0.39 | 1.83 | 1249
190 | 0.39 | 1.84 | 1333

  * *

Extrapolation obtained by scaling the Eliashberg function at 129 GPa by a
factor of 1.5. Extrapolated $T_{c}=353$ K.

  * **

Without accounting for the anharmonicity

Figure S1: Statistical analysis of the results of DFT calculations of electron-phonon interaction parameters in metals, polyhydrides, and other compounds. (a) Correlation between the logarithmic average phonon frequency $\omega_{\mathrm{log}}$ in Kelvin and pressure $P$ in GPa. For about 95% of the compounds $\omega_{\mathrm{log}}$ lies below the line $500+5P$. (b) Correlation between $\omega_{\mathrm{log}}$ in Kelvin and the electron-phonon coupling parameter $\lambda$. For about 95% of the compounds $\omega_{\mathrm{log}}$ in Kelvin falls below the line $2500-500\lambda$. Red areas indicate unlikely combinations of $\omega_{\mathrm{log}}$, $P$ and $\lambda$. Data for the plot are from Refs. [12, 13, 22, 49, 50, 51, 52, 53]. Table S4: Debye temperatures ($T_{D}$) for some superconducting metals, intermetallic compounds and compressed polyhydrides. Compound |  $\bm{T}_{c}$, K | EPC parameter ($\bm{\lambda}$) |  $\bm{T}_{D}$, K
---|---|---|---
Hg[33] | 4.16 | 1.0 – 1.6 | 72
Nb[33] | 9.22 | 0.82 – 1.05 | 277
Nb3Sn[46] | 17.9 | 1.6 -1.8 | 270
ThH10 (170 GPa) | 161 | 1.65 | 1350
YH9 (205 GPa)[12, 47] | 235 | 2.66 | 1275
(La,Ce)H9 (123 GPa)[37, 48] | 190 | 2.27 | 1107

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