Magnetic resonance probing of ground state in the mixed valence correlated topological insulator SmB6

Introducing of topological insulator concept for fluctuating valence compound – samarium hexaboride – has recently initiated a new round of studies aimed to clarify the nature of the ground state in this extraordinary system with strong electron correlations. Here we discuss the data of magnetic resonance in the pristine single crystals of SmB6 measured in 60 GHz cavity experiments at temperatures 1.8–300 K. The microwave study as well as the DC resistivity and Hall effect measurements performed for the different states of SmB6 [110] surface prove definitely the existence of the layer with metallic conductivity increasing under lowering temperature below 5 K. Four lines with the g-factors g ≈ 2 are found to contribute to the ESR-like absorption spectrum that may be attributed to intrinsic paramagnetic centers on the sample’s surface, which are robust with respect to the surface treatment. The temperature dependence of integrated intensity I(T) for main paramagnetic signal is found to demonstrate anomalous critical behavior I(T) ~ (T* − T)ν with characteristic temperature T* = 5.34 ± 0.05 K and exponent ν = 0.38 ± 0.03 indicating possible magnetic transition at the SmB6 [110] surface. Additional resonant magnetoabsorption line, which may be associated with either donor-like defects or cyclotron resonance mode corresponding to the mass mc ~ 1.2m0, is reported.

In order to compute the relative parts P s and P b for the power absorbed at the surface and in the bulk respectively, it is necessary to separate bulk and surface conductivities contributing to DC resistivity temperature dependence (T). To estimate, we assume that bulk conductivity in the plateau region follows the law  b (T )~exp(E a /k B T ) denoted by red dashed line in the panel b. In this case the total conductivity from the surface layer  s (T )=const (thus we neglect metallic temperature dependence of the surface layer shown in Figure 1 of the main text and possible deviations from the exponential law for the bulk states). The  s value (black dashed line in panel b) is found by fitting of experimental data by equation (T)=1/[1/ s +1/ b (T )]. This procedure gives P s (T ), P b (T ) and P s /P b (T ) temperature dependences shown in the panel b. It is visible that P s (T ) and P b (T ) are equal at T~6 K and lowering temperature results in the strong enhancement of the surface absorption with respect to the bulk. Namely, for T<4 K, almost all microwave power is absorbed inside the surface layer (see Supplementary Note 1 for calculation details).  Supplementary Table 1.
Parameters of the model calculation for T=2 K (see Supplementary Notes 3,4 and 5).
Supplementary Note 1: Model of the microwave power absorption in SmB 6 thin plate.
In the calculation, we assume that surface layer depth, a, is small with respect to the sample thickness, d. Indeed, in the topological Kondo insulator model, the parameter a is about size of the unit cell. As long as d >>a, it is possible to consider SmB 6 sample as -layer with the conductivity  s at the surface covering volume with the conductivity  b . For experimental layout used in cavity experiments ( Figure 1 of the main text) it is sufficient to consider geometry shown in Supplementary Figure 1, panel a (i.e. the problem becomes essentially one-dimensional). In the case of semi-infinite sample (d) the ratio of the microwave power absorbed in the front -layer, P s , to the microwave power absorbed in the sample bulk, P b , may be derived from a straightforward calculation, which yields P s /P b = s a/ b  b , where  b is the skin depth in the sample volume. The bulk conductivity  b (T )=1/ b (T ) can be directly obtained from experimental data, whereas determination of  s from  s is not so simple (see Supplementary  Figure 1, panel b). In the parallel resistor model used for modeling of the (T ) temperature dependence, the  s must be enhanced with respect to 1/ s by the factor l eff /a, where macroscopic length l eff is comparable with the sample sizes and depends of the measurement method. For the standard four-probe schema applied for the sample having the shape of a rectangular parallelepiped, the effective length is the ratio of the sample cross-section square and crosssection perimeter. For the Van der Paw method l eff is about L/4, where L is the distance between contacts at the sample surface. In any case, the final result acquires the form P s /P b = b l eff / s  b , and the ratio P s /P b does not depend on the thickness of the surface layer if the parameters  b and  s , following from experimental resistivity data,  b and  s , are introduced and a<<l eff ,  b . The finite sample thickness reduces the magnitude of the power absorbed in the sample bulk by ) . At the same time, in the considered case, the surface at the end of the sample will contribute to surface absorption as well (see Supplementary Figure 1, panel a), which will enhance P s by ) . This gives For an estimate we have used expression , which, in the plateau region can be re-written in the form . The temperature T 0 denotes the crossing point of asymptotics  s =const and  b~e xp(E a /k B T ) (See Supplementary Figure 1, panel a). Thus Equation (1s) depends on two dimensionless parameters l eff / 0 and d/ 0 and resistivity activation energy E a . Our experiments for the surface S1 correspond to the case E a 5 meV, l eff / 0 =2.18 and d/ 0 =1.25, which gives temperature dependences P s (T ), P b (T ) and P s /P b (T ) shown in Supplementary  Figure 1, panel a.

Supplementary Note 2: Low temperature magnetization field dependence.
The estimate of the paramagnetic centers concentration in the pristine SmB 6 with the S1 surface state may be obtained by the field dependence of magnetization M(B) analysis. The M(B) curve in SmB 6 is almost linear (Supplementary Fig. 2). Hereafter the magnetization is presented in the units of Bohr magneton per formulae unit. A small nonlinear contribution to M(B), which becomes more pronounced at low temperatures ( Supplementary Fig. 2), may be clearly detected in the data. For elucidating this part of M(B), the derivative M/B may be considered ( Supplementary Fig. 3). The data in Supplementary Figs. 2,3 indicate that magnetization of SmB 6 may be expressed by a superposition of two terms where function ) , ( T  B    represents the non-linear part and satisfies conditions . We assume that the main linear contribution corresponding presumably to the Van Vleck type response of the mixed valence SmB 6 matrix is complemented by the small saturating term ) , , which may be associated with the localized magnetic moments (LMM) responsible for the ESR signal observed below characteristic temperature T * =5.34 K.

Supplementary Note 3: Calculations schema.
In the strongly correlated electron systems, the ) , ( T B  function is not known exactly. Moreover, the microscopic nature of paramagnetic centers in the mixed valence material SmB 6 with strong charge and related spin fluctuations discovered in the present work by the ESR technique is also unknown. Therefore, it is reasonable to choose for ) , ( T B  some model form, which will allow estimating saturating magnetization M sat for the LMM part M of total magnetization in Equation (2s). In the present work we shall restrict ourselves with finding of this parameter only at the lowest available temperature, for which the magnitude of non-linear part of M(B) is maximal, and leave the analysis of the temperature evolution of the M(B) field dependences for future investigations. Thus we modeled ) , ( T B  at fixed temperature by Brillouin function B J (x) with the argument x=(T)B. For fixed quantum number J this function contains three free parameters (, a and M sat ), which may be found from the best fit of experimental M/B curve with the help of the derivative of Equation (2s).
LMM are "constructed" from the Sm 3+ magnetic ions, it is possible to expect essentially inhomogeneous distribution in space. Taking into account the experimental value of Sm valence v at T<5 K in the sample bulk v~2.52 [Supplementary reference 1] and the enhancement of this value at the sample surface up to v~2.66 as calculated in accordance with Supplementary reference 2 (see Supplementary Fig. 4) it is possible to expect that average number ~0.06% of Sm 3+ ions may contribute to ESR, whereas at the topologically protected surface of SmB 6 this number may be enhanced by ~30%.