Introduction

Thermoelectric energy conversion efficiency is characterized by the dimensionless thermoelectric figure of merit zT. The optimization of zT is well known to be a challenging task1,2, because thermoelectric transport coefficients are significantly correlated with each other. In 1996, Mahan and Sofo suggested that zT can be written as a functional of the electronic transport distribution function Σ(E), and studied the problem of best thermoelectrics from the point of view of functional optimization1. They showed mathematically that for a specified lattice thermal conductivity, the maximal zT is achieved when the electronic thermal conductivity κe = 0, which means that the electronic transport distribution function should be a delta function. This is the so-called Mahan–Sofo limit, which was believed to give the best thermoelectrics.

Following the classical work of Mahan and Sofo, Kim et al. discussed the influence of dimension on thermoelectric performance using the Landauer–Büttiker formalism and pointed out that the upper limit performance of delta-shaped transmission function is up to about 50% improvement over a parabolic band in power factor3. In 2011, Zhou et al. acknowledged the mathematical rigor of the Mahan–Sofo theory and further pointed out that the existence of delta-shaped density of states cannot guarantee the appearance of delta-shaped transport distribution function4. In the same year, Fan et al. revealed that the Mahan–Sofo limit is the optimal result for a constant integral of Σ(E) over the energy E5. Moreover, they suggested that under the assumption of bounded Σ(E), boxcar-typed functions would become the optimal Σ(E) for maximizing zT, which is also confirmed by Maassen6. In the last two works, the Mahan–Sofo limit is simply excluded by the bounded requirement on Σ(E). The general validity of the Mahan–Sofo result, however, has never been doubted, as far as we know.

In this work, we show that there is a significant loophole in the original study of Mahan and Sofo. Then, we redefine the optimization problem of zT in the quantum limit and apply the calculus of variations to find the best transmission probability that maximizes zT. We show that the best transmission probability in the quantum limit is a boxcar function instead of a delta function, giving the best zT exceeding the well-known Mahan–Sofo limit. We also suggest that topological materials are candidates for achieving optimal thermoelectric performance.

Results and discussion

Landauer–Büttiker formalism

The Landauer–Büttiker formalism is applicable to thermoelectric transport ranging from ballistic to diffusive regimes3,7, which can reproduce the Boltzmann transport equation in the diffusive limit. Here, we write zT as a functional of transmission probability \({{{\mathcal{T}}}}\) in the language of the Landauer–Büttiker formalism in the quantum limit and search for the optimal zT. In the Landauer–Büttiker formalism, the electrical conductance G, Seebeck coefficient S, and thermal conductance contributed by electrons Ke can be written as8

$$G=\frac{{e}^{2}}{h}{I}_{0},$$
(1)
$$S=\frac{{k}_{{{{\rm{B}}}}}}{e}\frac{{I}_{1}}{{I}_{0}},$$
(2)
$${K}_{{{{\rm{e}}}}}=\frac{{k}_{{{{\rm{B}}}}}^{2}T}{h}({I}_{2}-{I}_{1}^{2}/{I}_{0}),$$
(3)

where e(<0) is the electron charge, h is the Planck constant, kB is the Boltzmann constant, T is the temperature, and In(n = 0, 1, 2) are dimensionless integrals defined as

$${I}_{n}=\int\nolimits_{-\infty }^{+\infty }{{{\rm{d}}}}x\,\frac{{x}^{n}{{{{\rm{e}}}}}^{x}}{{({{{{\rm{e}}}}}^{x}+1)}^{2}}M(x){{{\mathcal{T}}}}(x),$$
(4)

where x = (E − μ)/kBT with μ the chemical potential and E the energy. M(x) is the total density of modes for spin-up and spin-down electrons, counting the number of conducting channels at a given energy E8,9,

$$M(E)={{{\rm{number}}}}\,{{{\rm{of}}}}\,{{{\rm{modes}}}}\,{{{\rm{at}}}}\,{{{\rm{energy}}}}\,E.$$
(5)

The electronic transmission probability \({{{\mathcal{T}}}}(x)\) (\(0\le {{{\mathcal{T}}}}(x)\le 1\)) is equal to 1 in the ballistic limit and to λ(x)/L in the diffusive limit, where λ(x) is the mean free path and L is the transport length3. The product \(M(x){{{\mathcal{T}}}}(x)\) is the so-called electronic transmission function10. The quasi-classical formula \({{{\mathcal{T}}}}(x)=\lambda (x)/(\lambda (x)+L)\) applies to ballistic-diffusive transport when quantum interference effects can be neglected10,11.

The thermal conductance contributed by lattice vibrations (Kl) is12,13

$${K}_{l}=\frac{{k}_{{{{\rm{B}}}}}^{2}T}{h}\beta ,$$
(6)

and β is a dimensionless integral defined as

$$\beta =\int\nolimits_{0}^{+\infty }{{{\rm{d}}}}x\,\frac{{x}^{2}{{{{\rm{e}}}}}^{x}}{{({{{{\rm{e}}}}}^{x}-1)}^{2}}{M}_{p}(x){{{{\mathcal{T}}}}}_{p}(x),$$
(7)

where Mp and \({{{{\mathcal{T}}}}}_{p}\) are phonon density of modes and phonon transmission probability, respectively. For ballistic transport in the zero-temperature limit, β = (π2/3)Nac, where Nac is the number of acoustic phonons14.

Then, the thermoelectric figure of merit is written as

$$zT=\frac{G{S}^{2}T}{{K}_{{{{\rm{e}}}}}+{K}_{l}}=\frac{{I}_{1}^{2}/{I}_{0}}{{I}_{2}-{I}_{1}^{2}/{I}_{0}+\beta }.$$
(8)

The formulas of ref. 1 derived by the Boltzmann transport equation is reproduced here when choosing the transport distribution \({{\Sigma }}(E)\propto (1/h)M(E){{{\mathcal{T}}}}(E)\)3 and the parameter α ∝ β−1. Note that M(E) contains information on band structures, \({{{\mathcal{T}}}}(E)\) contains information on scattering, and Σ(E) is bounded by M(E).

Loophole of the Mahan–Sofo Limit

Now we revisit Mahan and Sofo’s optimization procedure in the language of the Landauer–Büttiker formalism. Based on Eq. (8), zT is expressed as1

$$\begin{array}{rc}zT&=\frac{\xi }{1-\xi +A},\end{array}$$
(9)

where \(\xi =\frac{{I}_{1}^{2}}{{I}_{0}{I}_{2}}\) and \(A=\frac{\beta }{{I}_{2}}\). For a given system, β describes the contribution of phonons and the electronic transport quantities M(x) and \({{{\mathcal{T}}}}(x)\) define In. By a simple statistic argument, one can get ξ ≤ 11. Then, Eq. (9) implies that zT can be enhanced by increasing ξ and decreasing A. Thus the upper bound of zT is

$$zT\le \frac{1}{A}=\frac{{K}_{0}}{{K}_{l}},$$
(10)

where \({K}_{0}={k}_{{{{\rm{B}}}}}^{2}T{I}_{2}/h\). Finally, Mahan and Sofo argued that to achieve the upper limit, ξ = 1 is required, which implies Ke = 0. The condition can be fulfilled only by choosing a distribution with zero variance. Therefore, the Dirac delta distribution was suggested as the optimal distribution function.

However, there is a significant loophole in the Mahan–Sofo argument: ξ and A are not independent arguments in Eq. (9). Both of them are functionals of \(M(x){{{\mathcal{T}}}}(x)\). If ξ is defined, the form of function \(M(x){{{\mathcal{T}}}}(x)\) would be constrained, and A would not be allowed to change freely. For instance, if we choose ξ = 1, \(M(x){{{\mathcal{T}}}}(x)\) should be a delta function. Then I2 is determined, so is A. Intuitively, when Ke ≪ Kl, zT will benefit more from maximizing GS2, rather than minimizing Ke, so it is unreasonable to require Ke = 0 for maximizing zT in this situation. Thus, it is possible to find a transmission function that gives zT exceeding the Mahan–Sofo limit.

We will give a counter-example to make this point more clearly. Compare zT of two types of transmission function \(M(x){{{\mathcal{T}}}}(x)\): (i) a delta function δ(x − b) and (ii) a step function Θ(x − b), where b = 2.4 is the best peak position given by Mahan and Sofo1.

This example is sufficient to prove that the Mahan and Sofos result is not general. For the first case, zT turns out to be

$$z{T}_{{{{\rm{delta}}}}}^{\max }=\frac{0.439}{\beta }.$$

For the second case,

$$z{T}_{{{{\rm{step}}}}}^{\max }=\frac{0.987}{0.087+\beta }.$$

The results of \(z{T}^{\max }\) are presented in Fig. 1. It shows that for β ≥ 0.07 (corresponding to \(z{T}^{\max }\le 6.5\)), the step-shaped transmission function gives zT larger than the delta-shaped one. So the Mahan–Sofo optimization procedure is not perfect, and optimizing zT is still an open question. A simple argument for the necessity of restriction for the optimization of zT can be found in Supplementary Note 1.

Fig. 1: Comparison of the maximum zT of a step-function transmission function with a delta-function one.
figure 1

a \(z{T}^{\max }\) as a function of β for the step-function transmission function Θ(x − b) and delta-function transmission function δ(x − b) with b = 2.4. The insets b and c illustrate the step and delta functions, respectively.

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Optimal z T under an arbitrary M(x)

In light of the loophole in the original derivation1 of optimal zT, we directly apply the calculus of variations to maximize zT under an arbitrary upper bound of the electronic density of modes M(x) and find that the optimal transport probability is a boxcar-shaped function as (details in Methods):

$${{{\mathcal{T}}}}(x)=\left\{\begin{array}{lll}1,&{{{\rm{if}}}}&{x}_{1}\le x\le {x}_{2};\\ 0,&{{{\rm{else}}}}.\end{array}\right.$$
(11)

The x1 and x2 can be solved numerically through the following equations:

$${x}_{1}+{x}_{2}=2\frac{{I}_{2}({x}_{1},{x}_{2})+\beta }{{I}_{1}({x}_{1},{x}_{2})},$$
(12)
$${x}_{1}{x}_{2}=\frac{{I}_{2}({x}_{1},{x}_{2})+\beta }{{I}_{0}({x}_{1},{x}_{2})}.$$
(13)

In the following, we investigate the optimal zT under a constant density of modes M0, boxcar-shaped M(x), and discuss the realization of materials.

Constant M 0

Firstly, we focus on systems with a constant M0. For a clear demonstration, we solve Eqs. (12) and (13) for M0/β varies from 10−3 to 40. Without loss of generality, we assume that x1 > 0.

The results of x1, x2, and optimal zT (\(z{T}^{\max }\)) with different M0/β are shown in Fig. 2a–c. As shown in the figure, zT increases sublinearly as M0/β increases, which agrees well with the common sense that a larger density of modes and lower thermal conductance lead to higher zT. The optimal position x1 and x2 increases and decreases with the increase of M0/β, respectively. As a result, the width of the optimal boxcar function decreases as M0/β increases. As M0/β → 0, which may occur in insulators with large thermal conductance and low electronic density of modes, the optimal boxcar function approaches a step function6. As M0/β increases, the optimal transport width keeps decreasing. These results actually give the upper limits of zT under a given M0/β. It clearly shows that the optimal zT requires not only the boxcar shape but also the optimal position of the boxcar function. Also, it demonstrates that the upper bound of zT largely depends on the lattice thermal conductance.

Fig. 2: Optimal transmission probability and zT.
figure 2

a Endpoints of optimal transport window x1, x2 versus M0/β, respectively. b Optimal \({{{\mathcal{T}}}}(x)\) with maximized zT. c Maximized zT versus M0/β.

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Note that the electronic thermal conductance K0 and Ke are nonzero when the optimal zT is achieved (see Supplementary Note 2).

Boxcar-shaped M(x)

Our optimization procedure can also be applied to arbitrary-shape M(x). In the following, we examine the case of a boxcar-shaped M(x). The optimal zT for a multi-step M(x) can be found in the Supplementary Note 3. A typical band structure of a one-dimensional monoatomic chain, which has two spin-degenerate parabolic bands, is15

$$\varepsilon (k)={E}_{0}-2\gamma \cos ka$$
(14)

with E0 the onsite energy, γ the nearest hopping intensity, and a the lattice constant (Fig. 3a). The band structure is derived from a one-dimensional s-band tight-binding model. As shown in Fig. 3b, its density of modes is in a boxcar shape, and the upper bound M(x) is 2 (including spin degeneracy).

Fig. 3: Variation of zT with the width of the transport window in a 1D chain model.
figure 3

a Schematic plot of a one-dimensional monoatomic chain model with lattice constant a, onsite energy E0, and hopping integral γ. b (left panel) The band structure of the one-dimensional spin-degenerate monoatomic chain and (right panel) its density of modes M(ε) with hopping integral γ = 0.1E0. c Variation of zT as a function of Δx ≡ x2 − x1, which is the width of the transport window, with M0/β = 5.65, \({x}_{1,2}=\bar{x}\mp {{\Delta }}x/2\), μ = 0, T = 300 K, and E0 = 95.65 meV such that \(\bar{x}=3.7\).

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Optimization of zT, in this case, is equivalent to the optimization of a system with constant upper bound M0 = 2. We can tune the width of M(x) by straining the lattice16,17 so as to change the hopping intensity γ and fit the width of the best transport window Δx.

To find out the optimal transport window, we plot the variation of zT as a function of Δx. We choose M0/β = 5.65 (corresponding to the vertical dashed line in Fig. 2a such that β = 0.354 and the room-temperature lattice thermal conductance is about 0.0306 nW/K. As shown in Fig. 3c, zT varies significantly as the transport window width Δx changes. The optimal zT reaches 4, and the corresponding transport window width is about 3.3kBT ~ 85 meV. These results agree well with Fig. 2. We recall that the width of the transport window for a delta function is 0. Because delta-function type transmission is believed to be best for higher zT, one may expect that zT would monotonically decrease with Δx. Our calculations again show that a delta-shaped transport distribution function indeed does not guarantee the best thermoelectrics.

Hick and Dresselhaus18 also proposed a 1D model, which has a parabolic band. They estimated transport coefficients under relaxation time approximation and optimized band structure by changing width and tuning Fermi level. However, their optimization relied on relaxation time approximation, which demands that electrons on different energy levels have equal relaxation time, while we do not take such kind of assumption on the scattering mechanism.

Materials realization

For a typical case with a spin-degenerate conduction channel M0 = 2, room-temperature thermal conductivity κl = 2Wm−1 K−1, length L = 20 μm, and area of the cross-section S = 50 nm × 50 nm, the lattice thermal conductance is Kl = 0.25 nW/K with β = 0.897 and M0/β is 0.69. According to Fig. 2a, for a typical material with M0/β = 0.69, zT is at most 0.75. This upper bound of zT agrees well with the thermoelectric performance of layered IV-V-VI semiconductors19.

However, to be competitive with conventional machines in efficiency, zT of a thermoelectric device should be larger than 41. Therefore, as illustrated in Fig. 2a–c, we should find a material with M0/β > 5.65, and boxcar-function \({{{\mathcal{T}}}}(x)\) with the interval between 2.1kBT and 5.4kBT above the Fermi level μ.

At room temperature T = 300 K, the optimal interval is about 54–140 meV, and the transport window is about 85 meV. Except for engineering the band edges to achieve the optimal interval, the lattice thermal conductance can be reduced to at least 0.031 nW/K such that M0/β > 5.65. Defects and disorders can be introduced for lowering the lattice thermal conductance, however, it is not so easy to maintain the boxcar-shaped transmission for the electrons.

In fact, the boxcar-shaped transmission function with large M0/β can be realized in 2D topological materials20. For example, there are topologically protected gapless 1D edge states within the bulk band gap of a 2D topological insulator (Fig. 4a). Within the band gap, electrons of the edge modes are immune from elastic scattering, which means \({{{\mathcal{T}}}}\approx 1\); while in the bulk-band regions, the edge-mode electrons are allowed to be scattered into bulk states, so \({{{\mathcal{T}}}}\) can be gradually reduced to nearly zero by introducing disorders within the bulk region and by increasing the transport length. Then, the shape of the transmission probability can approach a boxcar shape (Fig. 4a, b). Importantly, the introduced disorder scattering reduces thermal conductance contributed by phonons but has little influence on the transport of topologically protected edge modes. Further, the materials can be made very narrow/thin to reduce the lattice thermal conductance. As long as the edge states do not overlap, the transmission of edge states maintains the box-shape feature. These treatments lead to large M0/β and enhanced zT, as proposed by ref. 20.

Fig. 4: Scenarios for achieving the best zT in topological materials.
figure 4

a (left panel) A schematic band structure of 2D topological materials with topological edge states within the bulk band gap20, and (right panel) a boxcar-shaped transmission probability \({{{\mathcal{T}}}}(\varepsilon )\), which could be realized when considering the energy-dependent elastic scatterings. b Disorders and defects in 2D topological materials, which scatter bulk states and phonons but have minor influence on transport of edge states.

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Similar argument can also be seen in ref. 21, though phonon thermal current was neglected. Since our method treats phonon thermal conductance as a parameter, thus can be applied to general cases with non-negligible thermal conduction from phonons.

Since transmission probability in real materials usually deviates from perfect boxcar shape, we discuss the effect of random noise on transmission in Supplementary Note 4.

In summary, we revisited the classical Mahan–Sofo study of optimizing zT1, and redid the optimization under the framework of Landauer–Büttiker formulae by the calculus of variations. For a given arbitrary upper bound of the density of modes M(x), it is proved that the boxcar-function-type transmission probability \({{{\mathcal{T}}}}(\varepsilon )\) leads to a maximal zT. Such a boxcar-type transmission can be realized in the quantum ballistic limit or in topological material systems. We showed how the best \({{{\mathcal{T}}}}\) changes its form with the material properties M(x) and β. Materials with larger M or lower β have higher upper bounds of zT. To get zT > 4, the candidate materials must have M0/β > 5.65, which demands a reduction of thermal conductance without significantly suppressing the electronic transmission. Our findings give a guide for optimal zT in the quantum ballistic limit and demonstrate that topological materials are promising candidates for realizing optimal thermoelectric efficiencies.

Methods

Optimization of z T through calculus of variations

We add some variation \(\delta {{{\mathcal{T}}}}(x)\) to \({{{\mathcal{T}}}}(x)\), and get the variation of zT. First, we get the variation of In, i.e.,

$$\delta {I}_{n}=\int\nolimits_{-\infty }^{+\infty }{x}^{n}M(x)\delta {{{\mathcal{T}}}}(x)\frac{{{{{\rm{e}}}}}^{x}}{{\left({{{{\rm{e}}}}}^{x}+1\right)}^{2}}{{{\rm{d}}}}x.$$
(15)

We assume that \(\delta {{{\mathcal{T}}}}(x)\) is small enough so that δIn ≪ In. Thus the variation of zT up to the first order \(\delta {{{\mathcal{T}}}}(x)\) is:

$$\begin{array}{lll}\delta (zT)&\approx& \frac{2{I}_{1}\delta {I}_{1}}{{I}_{2}{I}_{0}-{I}_{1}^{2}+\beta {I}_{0}}-\frac{{I}_{1}^{2}}{{\left({I}_{2}{I}_{0}-{I}_{1}^{2}+\beta {I}_{0}\right)}^{2}}\\ &&\cdot \left({I}_{2}\delta {I}_{0}+{I}_{0}\delta {I}_{2}-2{I}_{1}\delta {I}_{1}+\beta \delta {I}_{0}\right)\\ &=& \frac{{I}_{1}^{2}{I}_{0}}{{\left({I}_{2}{I}_{0}-{I}_{1}^{2}+\beta {I}_{0}\right)}^{2}}{\displaystyle\int}\frac{{{{{\rm{e}}}}}^{x}}{{(1+{{{{\rm{e}}}}}^{x})}^{2}}g(x)M(x)\delta {{{\mathcal{T}}}}(x){{{\rm{d}}}}x,\end{array}$$
(16)

where

$$g(x)\equiv 2\frac{{I}_{2}+\beta }{{I}_{1}}x-\left({x}^{2}+\frac{{I}_{2}+\beta }{{I}_{0}}\right).$$
(17)

Notice that I0 ≥ 0, M(x) ≥ 0. For some x, if g(x) > 0, a positive \(\delta {{{\mathcal{T}}}}(x)\) will increase zT; if g(x) < 0, a negative \(\delta {{{\mathcal{T}}}}(x)\) will increase zT. Because \(0\le {{{\mathcal{T}}}}(x)\le 1\), if \({{{\mathcal{T}}}}(x)\) satisfies

$${{{\mathcal{T}}}}(x)=\left\{\begin{array}{ll}1, &{{{\rm{if}}}}\,g(x) \, \ge \, 0;\\ 0, &{{{\rm{if}}}}\,g(x) \, < \, 0,\end{array}\right.$$
(18)

zT will reach a maximum. As shown in Eq. (17), g(x) is a quadratic function of x. Let x1 < x2 be two roots of g(x) = 0, we know that for x1 < x < x2, g(x) > 0; for x < x1 or x > x2, g(x) < 0. So the best \({{{\mathcal{T}}}}(x)\) has the form of a boxcar function:

$${{{\mathcal{T}}}}(x)=\left\{\begin{array}{ll}1, &{{{\rm{if}}}}\,{x}_{1}\le x\le {x}_{2};\\ 0, &{{{\rm{else}}}}.\end{array}\right.$$
(19)

This result is consistent with that of Maassen’s6 under a constant upper bound of the transport distribution function in the diffusive transport regime. It is also revealed that the optimal transmission function under zero lattice thermal conductance is a boxcar function22.

Therefore, our work demonstrates that a boxcar-shaped transmission probability is the best one not only for a constant upper bound but also for an arbitrary upper bound M(x) with various lattice thermal conductance.

To solve x1, x2, we substitute Eq. (19) into (4) and define

$${I}_{n}({x}_{1},{x}_{2})\equiv \int\nolimits_{{x}_{1}}^{{x}_{2}}M(x){x}^{n}\frac{{{{{\rm{e}}}}}^{x}}{{\left({{{{\rm{e}}}}}^{x}+1\right)}^{2}}{{{\rm{d}}}}x.$$
(20)

In now become functions of x1 and x2. Substituting In(x1, x2) to g(x1) = g(x2) = 0, we get two simultaneous nonlinear equations. It is hard to solve these equations analytically, but we can solve them numerically. For convenience, we actually solve an equivalent form of g(x1) = g(x2) = 0 as:

$${x}_{1}+{x}_{2}=2\frac{{I}_{2}({x}_{1},{x}_{2})+\beta }{{I}_{1}({x}_{1},{x}_{2})},$$
(21)
$${x}_{1}{x}_{2}=\frac{{I}_{2}({x}_{1},{x}_{2})+\beta }{{I}_{0}({x}_{1},{x}_{2})}.$$
(22)

which are acquired by applying Vieta’s formulae. There may be multiple solutions, e.g., for an even function of M(x), if x1,2 with 0 < x1 < x2 is a solution, then −x2,1 is another. All these solutions give local maximums for zT, and the global maximum can be easily found by calculating zT for each solution and choosing the highest. Substituting Eq. (21), (22) into (8), we further get \({(zT)}_{{{{\rm{optimal}}}}}=4{x}_{1}{x}_{2}/{\left({x}_{2}-{x}_{1}\right)}^{2}\), which agrees well with the results in ref. 6. Although zT depends on thermal conductance, evaluating the optimal zT needs only the positions of optimal x1 and x2.

Optimal z T under a constant M 0

By assuming that M(ε) ≡ M0, we focus on optimal zT under a constant M0. Physically, it can be interpreted as considering a system with M0 conducting channels for electrons. Define

$${J}_{n}({x}_{1},{x}_{2})\equiv \int\nolimits_{{x}_{1}}^{{x}_{2}}{x}^{n}\frac{{{{{\rm{e}}}}}^{x}}{{\left({{{{\rm{e}}}}}^{x}+1\right)}^{2}}{{{\rm{d}}}}x,$$
(23)

and then Eq. (21), (22) can be altered by changing In to Jn and β to β/M0.

There are two dimensionless parameters β and M0, which are closely related to the numbers of the thermal and electronic conducting channels, respectively.

In the limit of β → ∞, lattice thermal conductance approaches infinity. Optimization of zT, in this case, is equivalent to maximizing the power factor GS26, and the optimal transmission is the step function \({{{\mathcal{T}}}}(x)={{\Theta }}(x-{x}_{\infty })\) with x∞ = 1.145. (Details in the Supplementary Note 5.)

In the limit of β → 0, lattice thermal conductance approaches 0, and the optimal transport window approaches 0 as Δx ∝ β1/3. When M(x) ≡ M0 is a constant, the limit x0 satisfies that \({x}_{0}\tanh ({x}_{0}/2)=3\), which leads to x0 = ± 3.24, and the optimal \(zT\approx 1.39{({M}_{0}/\beta )}^{2/3}\). (Details in the Supplementary Note 6.)