Number Parity effects in the normal state of $SrTiO_{3}$

Number parity effects in superconductors were expected as soon as the Bardeen-Cooper-Schrieffer (BCS) microscopic model was developed (Bardeen et al., 1957). Indeed, the BCS ground state corresponds to a coherent superposition of pair states in which the number of particles has even parity and the total number $N$ is not fixed. Under these circumstances, the charge displacement operator $exp(i\phi)$ (canonically conjugate with the number operator $\hat{N}$) has a fixed expectation value, which leads to the common notion that a macroscopic BCS superconductor has a complex order parameter $\Delta$ with a rigid phase $\phi$. As soon as the BCS state is projected Schrieffer (1999) onto fixed N, it becomes clear that one has to differentiate between two cases: (a) if the total number $N=2n$ is even, all particles can participate in pair states and the ground state resembles the usual grand canonical BCS ground state; (b) if $N=2n+1$ is odd, the ground state will inevitably contain not only pairs, but also an unpaired electron (more precisely: the ground state will contain a Bogoliubov quasiparticle).

Intuitively, one would expect that the $N$ vs $N+1$ (even/odd) difference in a superconductor or any kind of paired fermionic state must be experimentally observable only if N is relatively small. Indeed, inspired by the success of the BCS theory, Bohr, Mottelson and Pines Bohr et al. (1958) were the first of many who studied pairing and even-odd effects in nuclear matter, with particle numbers around $N\sim 10^{2}$. It was therefore, even more surprising, when Mooij et al. Geerligs et al. (1990), Tinkham and coworkers Tuominen et al. (1992), as well as Devoret and his colleagues Lafarge et al. (1993a, b) showed experimentally measurable difference between Coulomb blockaded mesoscopic superconducting islands that contain a billion, and a billion plus one electrons. As it turns out, the magnitude of $N$ was less important. Instead, the quality of the Coulomb blockade turned out to be crucial: the superconducting islands had to be isolated from their environment with ultrasmall tunnel junctions and highly resistive electromagnetic environment, in order to ensure that $N$ is a fixed, good quantum number.

The pioneering experiments on number parity effects in conventional superconductors were performed on single-electron (SET) devices consisting of lithographically patterned aluminum islands Grabert and Devoret (1992). Even-odd effects emerged below a crossover temperature $T^{*}$ that was always much lower than the superconducting transition temperature: $T^{*}\ll T_{c}$. Rather than being directly correlated with $T_{c}$, $T^{*}$ is set by the experimentally measurable even-odd free energy difference $\delta{\cal F}_{\rm e/o}\sim\Delta_{0}-k_{B}T\log N_{\rm eff}$. Here $\Delta_{0}$ is the low temperature energy gap, and $N_{\rm eff}$ is the effective number of states Tuominen et al. (1992); Lafarge et al. (1993a, b); Jankó et al. (1994) available for the unpaired electron to explore in the odd number parity state of the superconducting island. Within this parity projected framework Tuominen et al. (1992); Jankó et al. (1994) $T^{*}$ corresponds to the temperature at which $\delta{\cal F}_{\rm e/o}$ becomes negligibly small: $T^{*}\sim\Delta_{0}/(k_{B}\log N_{\rm eff})$. For typical device parameters in these early experiments, the crossover temperature was measured to be around $T^{*}\sim 10^{2}{\rm mK}$ for aluminum island with $T_{c}\sim 1{\rm K}$. Consequently, the effective number of states was typically around $N_{\rm eff}\sim 10^{4}$.

The experiments by Levy and his coworkers Cheng et al. (2015) on SET devices constructed on STO/LAO provided experimental evidence for a spectacular departure from the conventional number parity effects described above. Levy and his colleagues detected $T^{*}\sim 900{\rm mK}$, much higher than the superconducting transition temperature $T_{c}\sim 300{\rm mK}$ measured for these devices. Even-odd effects remained detectable well into the ”normal” phase of the superconductor, and persisted in magnetic fields $B^{*}\sim 1\sim 4{\rm T}$, much higher than the upper critical field of the device. Furthermore, the extracted $N_{\rm eff}$ is also drastically different: $N_{\rm eff}\sim 2-3$.

A possible and relatively straightforward interpretation of novel experimental developments suggest that preformed pairs Geshkenbein et al. (1997); Seo et al. (2019); Tagliavini et al. (2016) persist into the normal state of STO-LAO well above the superconducting transition temperature. Consequently, fundamental changes must be made to the theoretical description of the number parity effects in this novel preformed pair phase. This paper is devoted to the presentation of a phenomenological theoretical framework aimed at providing a description of number parity effects in the normal phase of STO/LAO devices. Given the fact that the details of the microscopic mechanism behind the superconducting and preformed pair state of STO/LAO are not yet established, we use a phenomenological fermion-boson model Alexandrov (1996) that allows us to describe a normal phase where both pairs and unpaired particles are present. Furthermore, the model allows pairs to decay into unpaired particles, and particles to form pairs. This theoretical picture provides in a natural way a finite pair lifetime Cheng et al. (2015) in the preformed pair state. We find, after performing the number parity projection developed earlier by Ambegaokar, Smith and one of us Jankó et al. (1994), that the finite pair lifetime has drastic effect on the magnitude of $N_{\rm eff}$. In fact, as we will show in detail below, the theoretical framework we develop in this paper can reproduce not only $T^{*}\gg T_{c}$, but also $N_{\rm eff}\sim\mathcal{O}(1)$.

Generally speaking, the materials for making the single-electron devices can be separated into three categories (see Fig. 1): gapless materials, BCS and unconventional superconductors. Their density of states are shown, respectively, in panels (a), (b) and (c) of Fig. 1. While the single electron transistors with ultrasmall islands and discrete energy spectrum have also been investigated extensively Smith and Ambegaokar (1996); von Delft et al. (1996), we will not discuss this regime here. According to our calculations, the density of states at the Fermi level should be vanishingly small in order to obtain a finite even/odd free energy difference. As a result, the single electron transistors made from BCS superconductors and unconventional superconductors (as shown in panels (b) and (c)) are expected to show even-odd effects, and a superconducting gap above $T_{c}$ is necessary to cause $T^{*}\gg T_{c}$. The effective excitation number for the unpaired electrons in the odd parity states is highly dependent on the density of states at $E=\Delta$, since the smallest excitation energy is assumed to be $\Delta$ Tuominen et al. (1992); Lafarge et al. (1993a, b). In BCS superconductors (see panel (b)), the density of states at $E=\Delta$ is known to have a van Hove singularity, and this results in a large $N_{\rm eff}\sim 10^{4}$. In unconventional superconductors (see panel (c)), the van Hove singularity is broadened by the presence of low energy quasiparticles, which results in a small $N_{\rm eff}\sim\mathcal{O}(1)$.

Several possible microscopic superconducting mechanisms of the electron system at the STO/LAO interface have been proposed recently by different groups. Ruhman and Lee Ruhman and Lee (2016) suggested on the plasmon-induced superconducting mechanism. A nonperturbative approach within the plasmon model is being developed by Edelman and Littlewood Edelman and Littlewood . Kedem et al. Edge et al. (2015); Kedem (2018) related the mechanism to the ferroelectric mode. Arce-Gamboa and Guzmán-Verri Arce-Gamboa and Guzmán-Verri (2018) discovered the influence of strain on the ferroelectric mode and obtained the dependence of superconducting transition temperature on cation doping. On the contrary, Wölfle and Balastsky Wölfle and Balatsky (2018) proposed the transverse optical phonons may be the glue for electron pairing. While these theories can explain the origin of superconducting gap selfconsistently, some important experimental facts in remain unexplained. First of all, the pairing gap should persist above $T_{c}$. This is very important for the even-odd effects above $T_{c}$ in the single electron transistorsRichter (2013). Next, the van Hove singularity in the density of states should be broadened out. This will lead to a small $N_{\rm eff}$ consistent with the experiments of Levy et al Cheng et al. (2015). The detailed discussion of these results will be presented in the sections below.

This paper is organized as follows. In Section II, two important physical quantities, the even/odd free energy difference and the effective excitation number for the unpaired electron in the odd parity state, are related to the density of states within the phenomenological Dynes formula. In Section III, the boson-fermion model is introduced, and the analytic form of its electron Green’s function is provided. The density of states and the physical quantities of the even-odd effect predicted by the model are calculated in Section IV. Finally, we present our conclusions in Section V.

As mentioned in the introduction, there is no consensus yet on the microscopic theory of superconductivity in STO. In order to reproduce a Dynes-like density of states, we turn to the phenomenological boson-fermion model. For a single band model, electrons are assumed to have a Bogoliubov quasiparticle dispersion $E_{\mathbf{k}}=\sqrt{\epsilon^{2}_{\mathbf{k}}+\Delta^{2}}$, where $\epsilon_{\mathbf{k}}=\frac{\hbar^{2}k^{2}}{2m^{*}}-\mu_{F}$. The Hamiltonian of the electrons can be written as:

The superconducting gap of the 2D electron system at the interface of STO/LAO, $\Delta$ vanishes at $T_{s}\sim 300{\rm mK}$, and it turns into superconducting state at $T_{c}\sim 190{\rm mK}$ Richter (2013). This suggests between $T_{c}$ and $T_{s}$, the superconducting phase is destroyed, but the superconducting gap is preserved. This regime corresponds to the preformed pair state. The present model is devoted to studying the preformed pair state and superconducting state. Notice that the coherence length of pairs is $\sim{\rm 70-100nm}$ in (001)-STO/LAO and ${\rm 40-75nm}$ for (011)-STO/LAO Herranz et al. (2015), which is very small compared to the coherence length in conventional superconductors. In order to give an approximate description of preformed pairs, we introduce a bosonic field $\hat{b}_{\mathbf{q}}$ with elementary charge unit $-2e$. For a small momentum $\mathbf{q}$, the dispersion of the pairs is approximated as $\xi_{\mathbf{q}}=\xi_{0}+\hbar v|q|-\mu_{b}$ Schrieffer (1999), and the Hamiltonian for the bare bosonic field is:

The total Hamiltonian is

The total particle number is defined as $\hat{N}=\sum_{\mathbf{k},\sigma}{\hat{c}^{\dagger}_{\mathbf{k},\sigma}\hat{c}_{\mathbf{k},\sigma}}+2\sum_{\mathbf{q}}{\hat{b}^{\dagger}_{\mathbf{q}}\hat{b}_{\mathbf{q}}}$, and it can be proven that $[\hat{N},\hat{H}]=0$. The first order approximation of the self-energy is

With Eqs. 17,18, the one-particle Green’s function can be written as:

where $\epsilon_{\mathbf{k}}=\sqrt{\Delta^{2}+(\frac{\hbar^{2}(k^{2}-k_{F}^{2})}{2m^{*}})^{2}}\approx\sqrt{\Delta^{2}+(\hbar v^{*}_{F}(k-k_{F}))^{2}}$, $v^{*}_{F}\equiv\frac{\hbar k_{F}}{m^{*}}$, $\Sigma^{\prime}=\Sigma^{\prime}_{\mathbf{k}}(\omega)|_{k=k_{F}}$, $\Gamma=\Gamma_{\mathbf{k}}(\omega)|_{k=k_{F}}$. Numerical calculations show that $\Sigma^{\prime}\ll\Delta$ at low temperature and consequently $\Sigma^{\prime}$is negligible.

The density of states is

where $A(\mathbf{k},\omega)=-\frac{1}{\pi}{\rm Im}(G(\mathbf{k},\omega))$ and $L\approx 500{\rm nm}$ is the length of the small island in the middle of the single electron transistor. Notice that only $\epsilon_{\mathbf{k}}$ is dependent on the momentum $\mathbf{k}$ in the spectral weight function $A(\mathbf{k},\omega)$. This allows us to deduce an exact result in the mathematical expression of $D(\omega)$ with the residue theorem (see Appendix B).

As shown in Fig. 3, the decay and formation of the electron pairs produces many gap states and broadens the van Hove singularity in the density of states. The effects can reduce the even/odd free energy difference $\delta{\cal F}_{\rm e/o}$, and the effective excitation number for the unpaired electron in the odd parity state $N_{\rm eff}$, and this can be measured in experiments. In addition, zero superconducting gap makes a finite spectral function at Fermi level, and in that case, the density of states, $D(\omega)$, is finite at the Fermi level. This can destroy the even-odd effects. As shown in Fig. 3, the results on the even-odd effects calculated by the boson-fermion model is very similar to those we obtained from Dynes’ model density of states.

In the present paper, we argue that the even-odd effects seen in the normal state of STO originate from superconducting preformed pairing. This in turn imposes severe constraints on the density of states and consequently any microscopic model aimed at explaining the superconducting and normal state of STO/LAO. First, the density of states at Fermi level should be zero below and above $T_{c}$. A superconducting gap is required for the electron system to demonstrate even-odd effects above $T_{c}$. Next, the gap states are necessary to reduce the even/odd free energy difference and weaken the even-odd effects. Finally, the van Hove singularity needs to be broadened out in order to obtain a small $N_{\rm eff}$. These constraints for the density of states are not immediately satisfied by most current microscopic theories. On the other hand, the theories are successful in explaining the microscopic origin of the superconducting gap $\Delta$ and the dispersion of Bogoliubov quasiparticles.

The broadening of the van Hove singularity in the density of states may be a fingerprint of the lifetime effects of electron pairs. The signals in the single electron transistor experiments are very sensitive in detecting the lifetime effects, as well as the existence of the superconducting gap, the gap states and the broadening of the van Hove singularity. Moreover, compared to the experimental condition of scanning tunneling spectroscopy, that of the single electron devices can be relatively more easily satisfied in some strongly interacting electron systems. Furthermore, the decay and formation of electron pairs may widely exist in many different types of superconductors, including BCS superconductors. The application of the single electron transistor devices to study novel superconductors is therefore very promising.

The microscopic origin of phenomenological interaction potential is still unknown. Phonon-electron interactions, electron-electron interactions, etc. may participate in the actual microscopic mechanismYang and Liu . Further theoretical and experimental investigations are needed to elucidate the detailed microscopic model of the superconducting and normal state of SrTiO₃.