Specific heat can be measured by several kinds of methods, while since most of them are undertaken in vacuum, generally speaking, it is not easy. Generally, the specific heat can be measured through adiabatic method or non-adiabatic method. Usually the adiabatic method is more challenging since the precise measurement of heat amount and temperature variation in high vacuum is quite difficult. We thus use the relaxation method to detect specific heat. For a small specimen, the relaxation method is more advantageous than the adiabatic method.

The relaxation method is illustrated in Fig. 3. The central part of the measurement technique is a chip of sapphire crystal with high purity. It is desired that this crystal has an almost zero or very weak magnetic field dependent specific heat. This crystal is hung in vacuum through several wires with low thermal conductive. On this crystal there is a Cernox thermometer and a small heater made through the micro mechanical engineering technique. They are connected to the supporting frame by the wires just mentioned above. This assembly is called as a measuring platform. The measured sample is attached to the chip by thermal conductive gel. If we send a heating power P to the platform, the temperature of the platform will satisfy the following equation:

Fig. 3.

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	Fig. 3. The measuring platform of specific heat through the relaxation method.

where C_total is the total specific heat of the sample and the chip, T = T_chip is the temperature of the sample and the platform, and K_w is the thermal conductance of the linking wires.

The T_cryostat is the temperature of the supporting frame, which is normally the temperature of the base. One can see that the sample temperature is time dependent, by measuring the time dependence we can determine the total specific heat of the platform and the sample. Usually we send a rectangular-wave shaped heating power to the sample, and measure the temperature variation of the chip. Supposing the heating period starts from the moment t = 0, the temperature change can be written as

Here the relaxation constant τ = C_total/K_w, ΔT₀ is the equilibrium temperature that finally reaches by the system at a long time scale, and we know that K_w = ΔP/ΔT₀. Through the measured data of T (t), one can fit to the above equation and get the relaxation constant τ, then we can determine the total specific heat.

In a condensed matter, there are many kinds of elementary excitations which can contribute to the specific heat in different ways. The specific heat can be written in a general way as

Here C_el is the electronic contribution, C_ph represents the lattice vibration part, C_mag and C_hyp give the contributions of the paramagnetic centers and the nuclear moments. According to the solid state physics, the lattice vibration, or called as the phonon contribution should contribute the dominant part. The Debye model suggests that the phonon contribution in low temperature region is

where N is the Avogadro constant, and Θ_D is the Debye temperature. Usually it is believed that the specific heat of phonon contribution is temperature independent. However, in some system with strong spin–lattice coupling, such as in the iron based superconductors, the situation may change.

In superconducting state, the electron distribution at different momentum will change a lot, so that the internal energy will also change. We will present the contribution to specific heat in the superconducting state in coming sections.

As we expressed above, the specific heat detects the low energy quasiparticle excitations. In a superconductor, the quasiparticles satisfy the Bogoliubov dispersion

Here ħ²k²/2m is the kinetic energy counting from the Fermi level. We can calculate the quasiparticle density states (DOS) in the superconducting state. It is found that the DOS is fully gapped within the gap with singularities at the two energy gaps ±Δ for the s-wave superconductor. In the low temperature limit, through the BCS theory, we can derive the temperature dependence of specific heat in the s-wave superconductors as

where β = 1/k_BT, Δ (T) is the temperature dependent superconducting gap, and Δ₀ is the gap at zero temperature. One can see that in the low temperature limit, the specific heat coefficient C_es/T will be zero and it grow up gradually with increasing temperature, following a thermal activation manner.

For a d-wave superconductor, the sign of the superconducting gap changes crossing the Fermi surface and a typical gap function is Δ (K) = Δ₀ cos2θ, which is shown in Fig. 6. In cuprate superconductors, many experiments have shown that the gap has a d-wave pairing symmetry.^[3] Although there is a pseudogap effect^[4] which will be addressed later, experimentally, it is still shown that the condensate is constructed by Cooper pairs and the low energy physics can be described by the BCS theory. One can see that the gap has the maximum values at the four k-points (±π, 0) and (0, ±π), which is called as the anti-nodal direction. Along the (π, π) direction, the gap is zero, which is called as the nodal direction. The specific heat actually detects the small gap region, so that the quasiparticle dispersion in a d-wave superconductor can be written as^[5,6]

Fig. 6.

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	Fig. 6. The d-wave superconducting gap (Δ/Δ0) in the momentum space. Here the sign ‘+’ and ‘−’ indicates the opposite phase of the gap.

Here ħ is the Planck constant, v_F is the Fermi velocity which reflects the dispersion in the direction (k_⊥) perpendicular to the Fermi surface, while v_Δ is the gap slope around the nodal point, which reads as

and reflects the dispersion parallel to the Fermi surface (k_‖).^[7] This dispersion naturally leads to a linear relation between the DOS and energy near zero energy, which reads as

The DOS for the s-wave and d-wave superconductors is shown in Fig. 7. In the clean limit, the low temperature specific heat of an s-wave superconductor will be zero. While for a d-wave superconductor, the specific heat is described by a quadratic temperature dependence^[7,8]

Fig. 7.

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	Fig. 7. The density of states for (a) s-wave and (b) d-wave superconductors.

Here A is a constant which concerns the detailed atomic structure of the material, and k_B is the Boltzmann constant. This quadratic temperature dependence for a d-wave superconductor is in sharp contrast with that of a conventional superconductor which shows an exponential temperature dependence in the low temperature region. If temperature is increased, some quasiparticles will be generated immediately. For an s-wave superconductor, this excitation process is very slow versus temperature. For a d-wave superconductor, the quasiparticle DOS will be^[7]

Here m is the mass of the charge carrier, α_FL is the correction factor of the Fermi liquid, in the order of unity, and d is the distance between the double CuO planes in one unit cell of the cuprate superconductor. By measuring the low temperature penetration depth or the superfluid density, one can also determine the gap structure. For a d-wave superconductor, Δλ (T) will increase with T linearly, while for an s-wave superconductor, the penetration depth will increase with T in an exponential way.

For a type-II superconductor, when the external magnetic field is larger than the lower critical value H_c1, the magnetic flux will penetrate into the superconductor to form vortices. For a vortex, it contains a core region with the size of about the coherence length ξ. Solving the Ginzburg–Landau equation around a vortex would give the distribution of the supercurrent density j(r) and magnetic induction B(r), and the order parameter |Ψ (r)|. It is known that j(r) and B(r) will change in a scale of the penetration depth λ_L, while the order parameter will change in the scale of the coherence length ξ. For the vortex system, we have two contributions from the low energy quasiparticles.

The first contribution comes from the finite density of states within the vortex cores. By solving the Bogoliubov–de Gennes equations, researchers find that the gap value actually changes spatially with the cylindrical space of 2ξ.^[9] Detailed calculation shows that there are some quantized levels^[9,10] of these states with the energies of , and the energy discretion is about . For example, the lowest level is . Here E_F is the Fermi energy, and Δ₀ is the superconducting gap. In conventional superconductors, E_F ≫ Δ₀, the discretion between the energy levels is very small and very difficult to be observed. At the meantime, the measuring temperature is usually much larger than the energy discretion between different levels. This leads to a continuum spectrum of DOS near zero energy.^[11] This is just the origin of the usually observed zero energy peak within the vortex cores.^[12] For a single vortex, this contributes almost a constant to the total density of states. In Fig. 8, we show a series of STM spectra^[12] of a vortex core in 2H-NbSe₂. One can see that there is a high density of state peak at zero energy contributed by this vortex.

Fig. 8.

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	Fig. 8. Scanning tunneling spectrum with a vortex core in 2H-NbSe2. Figure from Ref. [12]. APS Copyright.

By increasing the magnetic field, the total contribution to specific heat of the vortex core part is just proportional to the density of vortices. We thus expect a linear relation of this part

The second contribution would come from the outside core of the vortices, mainly in the region of λ > r > ξ. In this part, the superconducting current is flowing but the order parameter has already established. The low energy quasiparticle excitations would be the same as in Meissner state if there would be no supercurrent. So the supercurrent will add a momentum to each Cooper pair. That is to say, the kinetic energy of the condensate will be shifted by a small amount due to this effect, and thus the contribution of DOS will also change. This is called as the Doppler effect. This extended density of states was first mentioned theoretically by Volovik.^[13] This little shift energy is δ E (k) = v_s · ħK with v_s the velocity of the superfluid. A calculation in the region of r > ξ shows that the superfluid velocity can be approximated as |v_s| = ħ/2mr with m the electron mass and r the distance to the center of the vortex core. For a single vortex, the contribution is about

Input the relation |υ_s| = ħ/2mr to the above equation, we have N_single ∝ R, combining with (when H_c2 ≫ H ≫ H_c1), we have . Since the vortex density is proportional to H, thus we have

This is the well-known Volovik relation^[13] for a d-wave superconductor. Here a is a vortex-structure factor which is close to 1, the pre-factor A ∝ 1/v_Δ ∝ 1/Δ₀ with Δ₀ the maximum d-wave gap.^[7,8,14]

Strictly speaking, the Volovik effect mentioned above, namely, works only for . Under other circumstances, for example, when , the electronic specific heat C_el (T,H) will contain a T ² term which is independent of the magnetic field and an H linear term.^[15] To cover a wider situation, Simon and Lee^[16] pointed out that in much wider magnetic field H and temperature region, the specific heat may have a scaling law

where F (x) is a scaling function. Up to now, there is no precise expression of F (x). However some empirical laws have been found,

Here η is close to unity.^[17] When y ≪ 1 (namely, in low temperature and high magnetic field), f (y) → 1, the above scaling law recovers the Volovik relation . When y ≫ 1 (in the high field region), we have f (y) → 1/y, the above scaling relation corresponds to the case of a linear H term.

In the above section, we have discussed about the low energy quasiparticle excitations in the clean limit, which in practical is not the case. In a real material, we cannot avoid the existence of defects, disorders, and impurities. These will have strong influence on the DOS excitation in the materials, especially for superconductors with nodes. In the dirty limit, namely, the mean free path is comparable to the coherence length, the small gap amplitude near nodes will be strongly suppressed and lead to the excitations of quasiparticles, leading to the clear deviation from the ideal clean limit.^[18] Taking this effect into account, Kübert et al. calculated the specific heat at T = 0, the magnetic field induced specific heat C_el (T,H) = γ (H)T and find that it is clearly deviated from the Volovik relation and satisfies the H logH law^[18]

where γ₀ is the de-pairing factor which depends on the scattering potential.

However, the above relation has not been confirmed by experiment. In most cases, the Volovik relation seems to be obeyed very well.^[7,8,19,20] The further theoretical advances show that the strong correlation effect may play an important role here.^[21] In Fig. 9, we show the calculated DOS versus energy by Garg et al.^[21] with different scattering density with the strong correlation effect (a) considered and (b) not considered. One can see that, with the strong correlation effect considered, the low energy quasiparticle density of states does not enhance with the impurity density too much. It seems that the gap near the nodes is somehow protected by the strong correlation effect. This explains why in the cuprate, with a lot of dopants, the predicted Volovik relation for a clean d-wave superconductor can still be observed.

Fig. 9.

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	Fig. 9. Density of states versus energy with the strong correlation effect considered (a) and not considered (b) under different impurity densities. The figure is copied from Ref. [21]. Copyright Nature Publishing Company.

Although the cuprate superconductivity has been discovered for 30 years, about the superconductivity mechanism, no consensus has been reached. However, in the process of research, some important progresses or new phenomena have been observed. Among them are (1) recognition of d-wave pairing symmetry,^[3] (2) pseudogap state above T_c,^[4] and (3) Fermi arc state^[22–25] or pocket state. These features are quite different from the conventional superconductor. Let us have a look one by one.

5.1.1. The d-wave pairing symmetry in cuprates

It becomes more and more clear that, the order parameter in cuprates is described by the d-wave gap function

Along the Fermi surface, supposing to be an isotropic circle like, namely, , we can see that the d-wave gap takes the maximum along (±π, 0) and (0, ±π). These directions are called as the anti-nodal direction. The diagonal direction, namely, the (π, π) direction and its equivalent symmetric directions are called as the nodal direction. Since the gap function will be zero at the crossing point of this direction and the Fermi surface. In Fig. 10, we show the Fermi surface, the gap function is shown in Fig. 6 in a schematically way. This d-wave has been well proved by angle resolved photoemission spectroscopy (ARPES),^[26,27] phase sensitive experiment,^[3] thermal conductivity,^[28] and specific heat.^[7,8,19,20]

Fig. 10.

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	Fig. 10. Schematic plot of Fermi surface in cuprate. The Fermi surface is highlighted by the solid red line around the BZ center. In underdoped region, this Fermi surface is truncated, showing only the segment or pocket near the nodal point.

5.1.3. Fermi arc

As we addressed above, in cuprate superconductors, the gapping process occurs near (π, 0) at temperatures far above T_c, leading to the formation of pseudogap. Later, ARPES data find that the emergent Fermi surface by increasing temperature is constructed by just segments, that is, the Fermi surface is incomplete and not continuous.^[16,32] The question relies whether the normal state has a truncated Fermi surface. Refined ARPES measurements with variable temperatures show that the Fermi surface shows Fermi arcs near the nodal point. Upon increasing temperature, the Fermi arcs extend to longer length and finally may connect into a complete closed Fermi surface.^[23–25] Researchers also find that, by increasing the doping level, the Fermi arc length is getting longer and longer at temperatures just above T_c. This truncated normal state Fermi surface will become complete at the doping level of around 0.19–0.20, indicating a quantum critical point.^[32] If the normal state shows the Fermi arc like Fermi surface, it is interesting to know how the superconducting gap is established on these Fermi arcs. We have done systematic measurements of specific heat and find that the superconducting condensation process may be understood in the following picture. Figure 11 shows the cartoon picture.

Fig. 11.

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	Fig. 11. Superconducting gap established on the Fermi arcs near the nodal point (π/2, π/2). The solid line represents the standard d-wave gap function Δ = Δ0 cos2θ, while the dashed and dotted lines show the cases for a real superconducting gap established on the Fermi arc near the nodal point. The superconducting gaps are determined by the value at the terminals of the Fermi arc.

In this Fermi arc picture, the central pairing strength is determined by the gap slope, namely, v_Δ. In an ideal d-wave superconductor, the maximum gap Δ₀ is proportional to the superconducting transition temperature. While in cuprate superconductors, the anti-nodal area is gapped away, therefore it is difficult to associate the maximum gap Δ₀ with T_c. Wen et al.^[7,8] made the detailed measurement on the low energy quasiparticle excitations in wide doping regime and found that the gap slope will increase towards underdoping level. The superconducting transition temperature T_c is however roughly proportional to the gap energy at the terminals. We will discuss this in the coming sections.

In the past 20 years, the d-wave superconducting gap symmetry has been well proved by many experimental tools. This includes the ARPES^[26,27] and the phase sensitive experiment.^[3] However, all these experiments are done by using surface tools which are very sensitive to the surface state, therefore bulk evidence is strongly needed. Specific heat can detect the bulk properties of the order parameter, and it contribute important information about the electronic states near the Fermi energy, therefore it is quite useful.

As we have addressed previously, for a d-wave superconductor, when the magnetic field is zero, the low energy quasi-particle excitations can be linear with energy, therefore the electronic specific heat will exhibit a power law behavior C_el (T) = αT ². When a magnetic field is applied, the Doppler shift effect of the superconducting condensate outside the vortex core will lead to extra quasiparticle excitations, leading to a linear term . In the experiment, the quadratic term αT² at zero field and the linear term at finite magnetic fields have been observed in YBCO by Moler et al.^[20] In determining the quadratic term αT ², there have been some controversy, because this term behaves as a combination of a linear term γ₀T and a phonon term β T ³. In fitting to the experimental data, it was found that the value of α may be negative, which is not reasonable. Theoretical estimate tells that α ∼ γ_n/T_c, which is a very small value in the low temperature region. Many groups have done careful experiments on low temperature specific heat.^{[7,8,24,33–40]} It seems quite difficult to detect or determine this term αT ² precisely.

In experiments of YBCO and La_2−xSr_xCuO₄ (LSCO), it was found that the Doppler shift induced enhancement of specific heat is more easy to be determined since it is purely a magnetic field induced contribution. For YBCO, several groups derived the term, and the pre-factor A they obtained was also consistent with each other. Later the group in Geneva determined the term in a slightly overdoped YBCO, and calculated several important quantities, such as v_Δ, H_c2, and ξ, which were close to the values determined from other experiments.^[33] For LSCO, the situation is similar, several groups investigated the specific heat under magnetic field and derived the term .^[7,8,39] The group in Tokyo investigated low temperature specific heat in LSCO single crystals under different magnetic fields.^[39] They concluded that for overdoped sample p = 0.19, the results obeyed the d-wave predictions, namely, the relationship, while in optimally doped sample p = 0.16 and underdoped sample p = 0.10, they claimed that it was in the dirty limit, that is, γ (H) ∝ H logH. Actually this group used the formula [C (T,H ‖ c) −C (T,H ‖ ab)]/T to derive the γ (H) term, they used the C(T,H‖ab) as the background contribution including the phonon and electronic contributions at zero field, which is certainly imprecise since the term C(T,H‖ab) may also have the contributions of low energy quasiparticle by vortices when the magnetic field is parallel to the ab-plane.

For a d-wave superconductor, we have shown that there is a scaling law , which has been well proved in some YBCO and LSCO samples.^{[7,8,20,32–40]} This consistency with the Volovik relation acts as a proof of the d-wave pairing symmetry.

We have also done a systematic investigation on LSCO single crystal samples. The advantage of this system is that the doping level can be easily tuned through the substitution of La by Sr. Comparing with the YBCO system, it is found that the LSCO system has rather weak Schotky anomaly,^[7,8] this significantly lowers down the uncertainty of the data analysis. In Fig. 12 we present the magnetic field enhanced specific heat versus magnetic field.

Fig. 12.

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	Fig. 12. Magnetic field induced enhancement of specific heat coefficient.[7,8,14] The experimental data are the values derived through the extrapolation to lower temperatures and normalized by the value at 12 T. The solid line is a theoretical fitting to the Volovik relation .

One can see that the data cover a very wide region, from very under doped to very overdoped region.^[7,8] The vertical coordinate shows the normalized value of Δγ = [C (H) −C (0)]/T at 12 T. The solid line is a theoretical fitting to the data by the Volovik relation, but clearly the A is doping level dependent. Actually this A value relies on the gap slope near the nodal point. Next we derive the gap slope near the nodes. From the scaling, we can get the doping dependence of the pre-factor A. For a 2D superconductor, assuming that the c-axis lattice constant is l_c, the number of conducting layers is n in one unit cell, we have^[41]

For LSCO, l_c = 13.28 Å, V_mol = 58 cm³/mol (the volume per mol), α_p is a dimensionless constant taking 0.5 (0.465) for a square (triangle) vortex lattice, n = 2 (the number of Cu–O planes in one unit cell), and Φ₀ is the flux quanta. The doping dependence of A and the calculated gap slope v_Δ are shown in Fig. 13. Shown together is the normal state specific heat coefficient γ_n determined from Hall effect measurements.^[42]

Fig. 13.

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	Fig. 13. (a) The pre-factor A of the Doppler shift effect and (b) the gap slope vΔ derived by using Eq. (36). Shown in (b) by circle symbols are the normal state specific heat coefficient γn.

From Fig. 13 we can see that the gap slope v_Δ increases monotonically towards underdoping. This is in sharp contrast with the decreasing of the superconducting temperature in the underdoped side. Actually the doping dependence of the gap slope follows the similar way of the pseudogap or the pseudogap temperature.^[4] To illustrate this more clearly, we show in Fig. 13 the doping dependence of v_Δ and T^*. The pseudogap temperature T^* here is obtained from Ref. [4], which is determined from the resistivity or spin susceptibility measurement. It is thus quite clear that the doping dependences of v_Δ and T^* are quite similar. According to the d-wave formula Δ (K) = Δ₀ cos2θ, we can relate the gap slope v_Δ with the maximum gap Δ₀ at the anti-nodal point as

Here k_F is the Fermi vector at the nodal point (π/2, π/2). From the ARPES data, we get k_F ≅ 0.7 Å⁻¹ which is almost doping independent. Therefore we can calculate the maximum gap at the momentum point (π, 0) based on the nodal gap slope v_Δ and show them in Fig. 14.

Fig. 14.

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	Fig. 14. Doping dependence of the maximum gap at (π, 0) (filled points) based on the nodal gap slope vΔ and the pseudogap temperature T* (open symbols). The pseudogap temperature T* was taken from Ref. [4], which was determined from the resistivity and spin susceptibility measurements.

The doping dependence of the maximum gap derived from the data of v_Δ is shown in Fig. 15. We can see that in the doping regime p ⩽ 0.19, there is a simple linear correlation between Δ₀ (denoted here as Δ_q) and T^*, namely, Δ_q ∼ 0.46k_BT^*. This is in sharp contrast with the really measured superconducting transition temperature T_c.^[7] This may suggest that the maximum gap has a close relation with the pseudogap. Through analyzing the thermal conductivity in the underdoped region, the group in Canada^[28] has derived the similar conclusion. One cannot use the maximum gap Δ₀ to derive the superconducting transition temperature. When p ⩾ 0.19, the derived Δ₀ has a similar doping dependence with T_c and surprisingly the d-wave relation Δ₀ ≅ 2.14k_BT_c is well satisfied. This is also consistent with other experiments.^[43,44]

Fig. 15.

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Fig. 15. (a) Doping dependence of the maximum gap Δ0 in the d-wave gap function derived from the nodal gap slope vΔ. The blue dashed line shows the doping dependence of the maximum gap of the d-wave gap. (b) The ratio between the obtained maximum gap and Tc. In the overdoped region, it gradually approaches the d-wave ratio Δ0 ≅ 2.14kBTc.[19] Below p = 0.19, the derived Δ0 clearly deviates from the theoretical prediction and goes up towards underdoping.

ARPES measurements show that the detected Fermi surface is not complete, but rather segment like.^[32] The Fermi surface appearing first is the locus of the nodal point (π/2, π/2) and it extends to anti-nodal direction. Therefore it is called as the Fermi arc state. The superconducting gap detected by specific heat gives the gap slope v_Δ which in usual case reflects the pairing strength. Our experiment shows that the gap slope increases towards underdoping, which may indicate that the pairing strength gets enhanced towards underdoping. However, we know that the superconducting transition temperature drops down towards underdoping. The opposite doping dependence of v_Δ and T_c tells that the superconducting transition temperature in the underdoped region may not just depend on the pairing strength, but also some other effect. We believe that it should have close relation with the Fermi arc state in the underdoped region. In the underdoped region, just above T_c the Fermi surfaces are constructed by the segments of Fermi arcs near the nodal point, while in the overdoped region, at the doping level of about p = 0.19, the Fermi surface suddenly becomes a complete one.^[42] Based on the specific heat data, we believe that the ground state when superconductivity is completely suppressed by the magnetic field is the Fermi arc state,^[41] in this case the charge carrier density is proportional to the hole number, namely, n_s ∝ p, while in the overdoped region the Fermi surface becomes complete, the charge carriers become electron like and the charge carrier number becomes n_s ∝ 1 + p.^[42] Concerning the underdoped electronic state, there are two different views: one picture assumes that the ground state will be a nodal metal state, which means that a nodal metal will be recovered when the superconductivity is suppressed by the magnetic field, the other view assumes that the recovered normal state will be a Fermi arc state. The two cases are shown below in view of specific heat. In the picture of Fermi arc, a finite DOS will appear when the superconductivity is killed completely.

In order to check whether it is a Fermi arc ground state or a nodal metal state, we have measured the specific heat for a very underdoped sample La_2−xSr_xCuO₄ (x = 0.06). After deducting the phonon contribution we present the magnetic field induced electronic specific heat in the zero temperature limit. One can see that with the increase of magnetic field, more and more density of states are recovered and the electronic specific coefficient increases. We further find that this enhanced part is roughly described by the Volovik relation for a d-wave superconductor, that is, . Therefore our data strongly supports the Fermi arc picture and the superconducting gap is built up upon the Fermi arc near nodes.

Fig. 16.

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Fig. 16. A schematic plot for the electronic specific heat when superconductivity is killed completely. (a) Fermi arc picture, a finite electronic specific heat coefficient will be detected in zero temperature limit. (b) Nodal metal picture, there is no finite density of states at the zero temperature limit. T* here is the pseudogap temperature which can be detected from differential specific heat and resistivity.[45]

Fig. 17.

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	Fig. 17. The specific heat at different magnetic fields subtracted by that at zero field for a typical underdoped LSCO single crystal p = 0.069.[7] The figure is copied from Ref. [7], APS copyright.

We have argued that the ground state of the underdoped cuprates is the Fermi arc metal. Now we show how the superconducting condensation is established based on the Fermi arc picture. From the above discussion we know that the slope of the superconducting gap v_Δ near the nodal point has been measured through our specific heat measurement. This nodal gap slope v_Δ reflects actually the pairing strength. The similar doping dependence of T^* and v_Δ hinges on the intimate relation between superconductivity and pseudogap. Then the question is why the superconducting transition temperature drops down in the underdoped region. Here based on the phase coherence picture we propose the superconducting gapping on the Fermi arcs to understand the superconducting transition.

In the conventional BCS superconductor, the pairing strength or the gap is related to the DOS at Fermi energy (N_F) through Δ = 2ħω_D exp(−1/N_Fλ), with ω_D the Debye frequency and λ the electron–phonon coupling strength. Clearly, this formula cannot be used to describe superconductivity in the underdoped region. We have determined the gap slope v_Δ near the nodal point. In the Fermi arc state, the effective DOS at the Fermi energy is significantly suppressed by the pseudogap effect. However, we have the energy at the terminals of the Fermi arc, as highlighted by the green solid lines in Fig. 18. Now we can calculate this energy and compare it with the superconducting transition temperature T_c.

Fig. 18.

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Fig. 18. The schematic picture for the superconducting condensation based on the Fermi arc picture. We now extend the momentum angle from 0 to π/2, corresponding to the momentum from (π, 0) to (0, π). The two terminals are corresponding to the anti-nodal points with the maximum gap. While at the angle of 45° it is at the nodal point with zero gap. For an underdoped sample, the length of the Fermi arc is finite and is highlighted by the horizontal dashed line. With more doping, this Fermi arc length becomes longer and longer. Below Tc, a superconducting gap will be formed on these Fermi arcs. This figure is adopted from Refs. [7,8]. The dashed blue line represents the pseudogap. The green solid line shows the energy scales on the terminals of the Fermi arcs.

In a normal metal, the electronic specific heat coefficient is . In the underdoped region, the effective DOS will be determined by the length of the Fermi arcs. Supposing the Fermi arc length is k_arc and we assume that the DOS on the Fermi arc is uniform, then we can imagine that N (E_F) ∝ k_arc. Detailed calculation^[41] gives . Therefore the normal state specific coefficient can be written as

With the increase of doping level p, the Fermi arc length k_arc increases linearly, which enhances γ_n (0). When it comes to the overdoped region, the Fermi surface becomes complete, the effective γ_n (0) will not increase with the doping level. In the inset of Fig. 19, we show the doping dependence of the effective specific heat coefficient γ_n (0) of the LSCO system dtermined from the specific hea measurements, and also converted from the integrated photoemission spectroscopy measurements and NMR measurements.^[41] In the doping region p ⩽ 0.2, it is clear that γ_n (0) increases linearly with p. In the region p > 0.2, the γ_n (0) tends to be stable versus doping.

Fig. 19.

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Fig. 19. Doping dependence of the measured superconducting transition temperature Tc (solid symbols) of the LSCO system and the calculated Tc based on the Fermi arc model (open symbols).[6,39] The solid line is the empiric relation[47] , where . The inset presents the normal state specific coefficient in the zero temperature limit γn (0) adopted from Ref. [46]. The solid line in the inset shows a fit to the data with γn (0) = ζ (p − pc)η with ζ = 182.6, pc = 0.03, and η = 1.54.

Now we try to calculate the energy at the terminals of the Fermi arc. We assume an energy scale Δ_sc which governs the superconducting transition temperature T_c. We can imagine that this energy scale is proportional to both the gap slope v_Δ and the Fermi arc length k_arc, thus we have

For a more precise calculation, we have . Using Eq. (38) we further have

From the above equation, we immediately understand why the superconducting transition temperature T_c drops down in the underdoped region, this is because the Fermi arc length becomes shorter towards underdoping, this is reflected from the decrease of γ_n (0). Although the gap slope v_Δ or the maximum gap Δ₀ increases towards underdoping, the superconducting energy scale or transition temperature is determined by the multiplication of the gap slope and the Fermi arc length. The calculated T_c is presented in Fig. 19 by the open symbols, the really measured T_c is shown by the filled symbols. In the calculation of the superconducting energy scale, we have fitted the doping dependence of γ_n(0) with a relation γ_n (0) = ζ (p − p_c)^η, the fitting yields ζ = 182.6, p_c = 0.03, and η = 1.54. In the underdoped side, both sets of data consist with each other very well. In the overdoped region, the calculated data are deviating from the experimental values, this is understandable since the fitted relation of γ_n(0) cannot be used anymore. In the heavily overdoped region, there might be electronic phase separation or the strong pair breaking by the impurity effect.

The Ba_0.6 K_0.2Fe₂As₂ system has been found to contain multiband by ARPES^[56] and STM^[57] in very early date. In Fig. 20, we show the raw data of specific heat in two optimally doped Ba_0.6K_0.2Fe₂As₂ samples. One can see several important facts. (1) The data in low temperature region exhibit an exponential behavior. When the phonon contribution is removed, this becomes more clear. (2) The specific heat anomaly at T_c is very sharp showing a jump of about 100 mJ/mol·K², this value is about 3–5 times higher than the bare value of band structure calculation. Therefore we can imagine that the electron–boson coupling is very strong in this system. In Fig. 21, we show the temperature dependence of the electronic specific heat and the theoretical fitting using an s-wave model. Now the normal state specific heat has been removed. One can see that the low temperature part exhibits a very flat feature indicating a full gap behavior. However, in the intermediate temperature, there is a hump which may be induced by a second band effect. Hardy et al. also did a measurement on the same system, they found that a two model fitting^[50] can give rise to a very good description of the data.

Fig. 20.

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	Fig. 20. (a) and (b) Raw data of temperature dependent specific heat coefficient C/T on two Ba0.6K0.4Fe2As2 samples at zero (dark square) and 9 T (red circle) magnetic fields. The insets show the specific heat anomaly near the transition point, showing a quite strong specific heat jump. The figure is copied from Ref. [51], APS Copyright.

Fig. 21.

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	Fig. 21. The electronic specific heat after removing the phonon and the normal state electronic contributions in Ba0.6K0.4Fe2As2. The figure is copied from Ref. [51], APS Copyright.

In order to check whether it is fully gapped, we present in Fig. 22 the magnetic field dependence of the electronic specific heat. One can see that in the zero temperature limit, the field induced specific heat shows a roughly linear field dependence, which strongly supports the fully gapped feature. Since Ba_0.6K_0.2Fe₂As₂ is a hole doped system, now the hole pocket takes the most weight of specific heat, therefore we can also conclude that the gap on the hole pocket shows a very good isotropic behavior in Ba_0.6K_0.4Fe₂As₂.

Fig. 22.

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	Fig. 22. Magnetic field dependence of specific heat of Ba0.6K0.4Fe2As2 in low temperature region. The figure is copied from Ref. [51], APS Copyright.

In the electron doped 122 system, the situation seems quite different from that in hole doped Ba_0.6K_0.4Fe₂As₂. First of all, a strong angle dependence has been found. In Fig. 23, we show the temperature dependence of the electronic specific heat in BaFe_2−xT_xAs₂ (T = Co and Ni). Now we use the very overdoped non-superconducting sample as the background to subtract the phonon contributions in the normal state. First of all, the isotropic s-wave fitting cannot work very well, with either single or double components. However, a combination of an s-wave and an extended s-wave can give a quite good fitting. To our surprise is that the single d-wave fitting can also give quite nice consistency with the data. This may suggest that a d-wave gap is also possible. However, the magnetic field dependence of specific heat shows a rough linear behavior again, which is against the d-wave fitting. Therefore we conclude that the gap may be highly anisotropy. But it remains unclear where the deep gap minimum locates. One possibility is that the gap minimum still locates at the hole pocket with a small segment of Fermi surface of gapless or gap minimum, due to the multi-orbital feature. The other possibility would be that the gap minimum locates at the electron Fermi pockets, since the samples now are electron doped and the electron pockets near the corner of the Brillion zone become quite large and the gap is highly anisotropic in these electron doped systems. In Fig. 24, we show the specific heat coefficient under magnetic field, one can see that they exhibit roughly a linear behavior. Combining this linear magnetic field dependence feature with the global fitting, we believe that the gap on the electron Fermi surface should be highly anisotropic.

Fig. 23.

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	Fig. 23. Temperature dependence of electronic specific heat of optimally doped (a) BaFe1.84Co0.16As2 and (b) BaFe1.9Ni0.1As2. One can see that it is quite different from the case of Ba0.6K0.4Fe2As2. The low temperature data show some kind of positive curvature, but it is certainly not flattened as an s-wave. The figure is copied from Ref. [52], APS Copyright.

Fig. 24.

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	Fig. 24. Magnetic field dependence of enhancement of electronic specific heat in BaFe1.84Co0.16As2 and BaFe1.9Ni0.1As2. A rough linear behavior suggests fully gapped feature. The figure is copied from Ref. [52], APS Copyright.

In order to study the superconducting gap structures of the FeSe bulk system, we have measured the specific heat of FeSe single crystals.^[55] The temperature dependence of specific heat at zero field for one sample is presented in Fig. 25(a). One can see that the specific heat anomaly is quite clear with a value of about 10 mJ/mol·K². The normal state specific heat can be fitted with a polynomial function and thus it can be removed. The electronic contribution of specific heat is shown in Fig. 25(b), one can see that the specific heat jump is quite sharp without a very small tail in the normal state. This kind of tail has been observed in many cuprate superconductors which was interpreted as the strong superconducting fluctuation. Besides that, we see a small specific heat anomaly or jumping-up at about 1.08 K, which we argue that it might be the long sought antiferromagnetic order. In Fig. 26, we present the two-band model fitting to the data with many different models: (a) single s-wave gap, Δ (θ) = Δ₀, (b) single d-wave Δ (θ) = Δ₀ cos2θ, (c) single extended s-wave gap Δ (θ) = Δ₀(1 + α cos2θ), (d) a combination of an s-wave and an extended s-wave. The best fitting as judged from both the fitting quality in wide temperature region and the entropy conservation of the low temperature anomaly is the two-band model fitting with a highly anisotropic gap Δ (θ) = Δ₀(1 + α cos2θ) with α ≈ 0.9–1, see Figs. 23(b) and 23(c). This indicates that the gap minimum is quite small, in the scale of about 0.15–0.3 mV. This conclusion is quite consistent with the STM measurements.^[56] The gaps on the hole derivative α-pocket and on the electron derivative ε-pocket both are highly anisotropic.

Fig. 25.

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	Fig. 25. (a) Specific heat of FeSe bulk sample. The red solid line represents the polynomial fit to the normal state. There is a small specific heat anomaly at about 1.08 K, which may be induced by the possible antiferromagnetic ordering. (b) The electronic specific heat after the normal state contribution is removed. The figure is copied from Ref. [55], APS Copyright.

Fig. 26.

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Fig. 26. Fitting to the electronic specific heat with many different models (see text). The best fitting, judged from both the fitting quality in wide temperature region and the entropy conservation of the low temperature anomaly, is the two-band model fitting with a highly anisotropic gap Δ (θ) = Δ0(1 + α cos2θ) with α ≈ 0.9–1, see panles (b) and (c). The figure is copied from Ref. [55], APS Copyright.

We also measured the magnetic field dependence of specific heat in FeSe bulk samples, the data are presented in Fig. 27. It is found that a magnetic field of about 14 T has already turned the sample completely into the normal state. However, as shown in Fig. 27(b), the magnetic field dependence of specific heat in the zero temperature limit can be fitted to a function with (p : q ≈ 2 : 5). This formula contains a linear part and a quadratic part, which is deviated from a linear behavior for an isotropic gap. However, it is also different from the Volovik relation, which clearly suggests that the d-wave pairing gap is not appropriate. A reasonable picture would be that the gap is highly anisotropic. This is consistent with the fitting to the temperature dependence of the electronic specific heat in Fig. 23 and is consistent with the STM data.^[58] In iron based superconductors, the gap structure is mainly nodeless but highly anisotropic. This is in general consistent with the spin fluctuation mediated pairing manner, leading to the gap structure of s^±.^[59,60]

Fig. 27.

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Fig. 27. Magnetic field induced change of specific heat coefficient. (a) The raw data (symbols) and the fitting curves for each magnetic field. The fitting range varies depending on the region of the specific heat anomaly which should not be involved in deriving the low temperature intercept. (b) The magnetic field induced specific heat coefficient Δγ = [C(h) −C(0)]/T |T→0. The red solid line gives a combination of two terms, i.e., with (p : q ≈ 2 : 5).