## Spin Berry's Phase

When a quantum mechanical system performs an adiabatic cyclic path in the space of the parameters that affect its state (such as, for example, the magnetic field) its wave function may acquire an additional phase rather than go back to its original value. This quantity, called the Berry's phase, is associated with the topological properties of the parameter space and has been observed in materials such as graphene and bismuth. **Murakawa et al.** (p. 1490) observe a Berry's phase equal to π in the material BiTeI in which the phenomenon is predicted to be a consequence of a very strong coupling of spin and orbital degrees of freedom realized through the so-called Rashba effect.

## Abstract

The motion of electrons in a solid has a profound effect on its topological properties and may result in a nonzero Berry’s phase, a geometric quantum phase encoded in the system’s electronic wave function. Despite its ubiquity, there are few experimental observations of Berry’s phase of bulk states. Here, we report detection of a nontrivial π Berry’s phase in the bulk Rashba semiconductor BiTeI via analysis of the Shubnikov–de Haas (SdH) effect. The extremely large Rashba splitting in this material enables the separation of SdH oscillations, stemming from the spin-split inner and outer Fermi surfaces. For both Fermi surfaces, we observe a systematic π-phase shift in SdH oscillations, consistent with the theoretically predicted nontrivial π Berry’s phase in Rashba systems.

Quantum mechanical systems undergoing adiabatic evolution on a closed path in parameter space acquire a geometrical phase known as Berry’s phase, ϕ_{B} (*1*, *2*). A number of emergent phenomena—including the anomalous (*3*) and quantum (*4*) Hall effects, charge pumping (*5*), and topological insulating and superconducting phases (*6*)—are driven by a nontrivial (that is, nonzero) ϕ_{B}. A nontrivial ϕ_{B} can, for example, be realized for charge carriers that have *k*-space cyclotron orbits enclosing a Dirac point (*7*–*10*). In general, any closed cyclotron orbit is quantized under an external magnetic field *B*, according to the Lifshitz-Onsager quantization rule
(1)Here *A _{n}* is the extremal cross-sectional area of the Fermi surface (FS) related to the Landau level (LL)

*n*; γ is defined as and can take values from 0 to 1, depending on the value of ϕ

_{B}(

*7*);

*ħ*is Planck’s constant

*h*divided by 2π; and

*e*is the elementary charge. The quantity γ can be experimentally accessed by analyzing the LL fan diagram of Shubnikov–de Haas (SdH) oscillations. A nontrivial Berry’s phase has been observed in pseudo-spin Dirac systems such as graphene (

*10*), as well as elemental bismuth (

*11*), bulk SrMnBi

_{2}(

*12*), and, potentially, graphite (

*13*–

*15*), although in that case the experimental situation remains unresolved. Clear detection in physical-spin Dirac systems such as topological insulators has been complicated by large Zeeman energy effects and bulk conduction (

*16*–

*19*).

Systems described by the Rashba Hamiltonian also possess a Dirac point and provide an alternative path to realizing a nontrivial ϕ_{B}. The Dirac point in this class of noncentrosymmetric systems results from the crossing between energy bands spin-split by the Rashba spin-obit Hamiltonian , where λ is the Rashba parameter, **e** is the unit vector along which the system breaks inversion symmetry (*20*), **s** is the spin, and **p** is the momentum vector. Because of this interaction, the energy bands are linearly dispersed around **p** = 0, and the resulting FS consists of two pockets, an inner FS (IFS) and an outer FS (OFS). For each FS, **s** is locked to be normal to **p**, thereby forming a helical spin texture around the Dirac point at **p** = 0. It has been predicted that in systems where both Rashba and Dresselhaus spin-orbit interactions are present, both FSs obtain a Berry’s phase expressed as , with β corresponding to the Dresselhaus parameter (*21*–*23*). Accordingly, in the case of pure Rashba spin splitting (RSS) (β = 0), both inner and outer FSs are expected to obtain a nontrivial π Berry’s phase. Consequences of this Rashba coupling and phase have been studied in two dimensions in semiconductor heterostructures using weak antilocalization (*24*), ensemble averaging in interferometers (*25*), commensurability oscillations (*26*), and SdH oscillations (*27*, *28*). Given the relatively small RSS in these systems, the IFS and OFS have similar areas, and the SdH oscillations show beating patterns that obscure the underlying oscillation index structure. In addition, when the spin is aligned to the external magnetic field by a larger Zeeman energy, RSS disappears, and the Berry’s phase takes on the trivial value.

Recently, a large RSS in a bulk material has been found in the polar semiconductor BiTeI (*29*–*32*). This material is composed of alternating Bi, Te, and I atoms stacked along the hexagonal *c* axis and is typically electron-doped by native defects, as found in many chalcogenide semiconductors. Because of the absence of inversion symmetry and the strong polarity of the system, accompanied by the strong spin-orbit interaction of Bi, an extremely large RSS occurs around the hexagonal face center of the Brillouin zone, referred to as the *A* point (*30*). Both angle-resolved photoemission spectroscopy and optical spectroscopy (*29*, *31*–*34*) reveal that the RSS in BiTeI approaches 400 meV, with a Dirac point located ~110 meV above the conduction band minimum, in good agreement with relativistic first-principles calculations (*30*, *35*).

Figure 1 shows the calculated band dispersion around the *A* point (Fig. 1A) and a typical FS for a Fermi level *E*_{F} located slightly above the Dirac point (Fig. 1B), corresponding to the samples measured here. The large RSS causes the ratio of the extremal cross-sectional areas of the IFS and OFS to be extremely large, diverging as *E*_{F} approaches the Dirac point. In terms of SdH experiments for samples with finite electron mobility, this conveniently decouples the two sets of oscillations by confining the IFS and OFS to the low- and high-field regime, respectively. Moreover, the observed giant RSS in BiTeI can dominate the Zeeman effect, even in high magnetic fields. Therefore, BiTeI is an ideal system for investigating the Berry’s phase originating from the Rashba spin-split band.

Figure 2 shows the in-plane magnetoresistivity (ρ_{xx}) of a BiTeI sample (sample A) up to 14 T applied along the [001] *c* axis at 1.8 K. In this sample, the low-field Hall mobility is ~300 cm^{2}/V·s at 1.8 K. The Hall density (~4.0 × 10^{19} cm^{−3}) would place *E*_{F} slightly above the Dirac point. As can be seen, the SdH oscillations stemming from the IFS (<4 T) and OFS (>10 T) are well separated from each other; thus, they can be separately analyzed. Focusing first on the IFS, Fig. 3A shows the SdH oscillations at various temperatures. Because the IFS extremal cross-sectional area (*A*_{IFS}) is so small, 3.4 T is sufficient to reach the quantum limit, where all electron states in the IFS are condensed into the lowest LL. This can be seen by taking the negative second derivative of resistivity (–*d*^{2}ρ_{xx}/*dB*^{2}) (Fig. 3B). Here, the lowest-index maximum (corresponding to the peak in resistivity) appears at 3.4 T (1/*B* = 0.294 T^{–1}), and the oscillation disappears above 5 T. The period of the oscillation [∆(1/*B*) = 0.294 T^{–1}] corresponds to *A*_{IFS} = 3.1 × 10^{−4} Å^{−}^{2}.

The oscillatory component ∆ρ_{xx} is plotted in Fig. 3C, after subtraction of the nonoscillating background deduced by fitting a fourth-order polynomial, based on the resistivity values at the node positions of the SdH oscillations (Fig. 3A). From the temperature dependence of the oscillation amplitude at 3.4 T (Fig. 3D), the electron effective mass *m** for the IFS () is determined to be (0.023 ± 0.001)*m*_{0} (where *m*_{0} is the free electron mass), following the Lifshitz-Kosevich formula for a three-dimensional (3D) system (*36*–*38*)
(2)Here, ρ_{0} is the nonoscillatory component of the resistivity at *B* = 0, *T*_{D} is the Dingle temperature, *k*_{B} Boltzmann’s constant, and the cyclotron frequency ω_{c} = *eB*/*m**. Considering *A*_{IFS} = 3.1 × 10^{–4} Å^{–}^{2}, first-principles calculations indicate *E*_{F} = 151 meV above the conduction band minimum. At this *E*_{F}, the calculated is 0.021*m*_{0}, in agreement with the observed value. We can thus confidently assign this set of SdH oscillations to the IFS.

In the expression for ρ_{xx} in 3D systems (*36*–*40*)(3)1/*B*_{F} is the SdH frequency, and δ is a phase shift determined by the dimensionality, taking the value δ = 0 (or δ = ±1/8) for the 2D (or 3D) case (*41*–*43*). In this formula, values of |γ – δ| = |1/2 – ϕ_{B}/2π – δ| between 0 and 1/8 indicate a nontrivial π Berry’s phase, with the precise value determined by the degree of two-dimensionality via δ. To obtain the Berry’s phase, the relation between 1/*B* and Landau index number *n* is plotted in Fig. 3E. Here, peak and valley positions of the oscillation are determined using –*d*^{2}ρ_{xx}/*dB*^{2} (Fig. 3B). We assign integer indices to the ρ_{xx} peak positions in 1/*B* and half integer indices to the ρ_{xx} valley positions. The interpolation line of *n* versus 1/*B* in sample A has an intercept between –1/8 and 0, indicating a nontrivial Berry’s phase for the IFS. A similar trend is reproduced for other samples B, C, and D with *E*_{F} ~ 244, 194, and 154 meV, respectively (Fig. 3E). In a pure Rashba system (γ = 0), these results indicate that δ deviates from the 3D limit |δ| = 1/8, likely because of the quasi-2D nature of the system. In sample B, the SdH oscillation originating from the IFS can be seen up to 39 T with the respective ∆(1/*B*) = 0.0259 T^{–1}, corresponding to a much larger *A*_{IFS} = 3.65 × 10^{–3} Å^{–2}. The strict linearity of this index plot up to the quantum limit is a consequence of negligible Zeeman splitting and the fact that the OFS dominates the variation of the chemical potential at this magnetic field. This allows us to avoid index shifting near the quantum limit, as observed, for example, in graphite (*14*), in much the same manner as has been observed for specific field orientations of bismuth (*11*).

Next, we turn to SdH oscillations originating from the OFS. These oscillations are clearly observed in the higher–magnetic field region. Figure 4A shows ρ_{xx} of sample A up to 56 T at various temperatures. In this case, clear SdH oscillations are observed above 10 T at 1.5 K and can be discerned even at 100 K above 40 T. The oscillatory component is deduced by subtracting a fourth-order polynomial, as discussed previously, and is plotted as a function of 1/*B* in Fig. 4B. The period of the oscillation [∆(1/*B*) = 0.00288 T^{–1}] corresponds to an OFS extremal cross-sectional area of *A*_{OFS} = 3.4 × 10^{–2} Å^{–}^{2}, which is very consistent with the calculated *A*_{OFS} = 3.43 × 10^{–2} Å^{–}^{2} at *E*_{F} = 151 meV. This provides an important self-consistency check, given that this *E*_{F} was derived from the IFS SdH results. From the temperature dependence of the peak amplitude at 53.6 T, is determined to be (0.183 ± 0.003)*m*_{0} (Fig. 4C), also in good agreement with the calculated value *m** = 0.182*m*_{0}. To obtain the Berry’s phase for OFS, the fan diagram is plotted in Fig. 4D. Because a longer extrapolation is required in the case of the OFS, we measured five samples with varying *E*_{F} for the determination of the intercept value. Here the integer index *n*, corresponding to the ρ_{xx} maximum, is assigned such that a linear extrapolation of the index plots yields an intercept closest to zero index (*n* = 0). As can be seen in Fig. 4D, the index plots uniformly exhibit a linear dependence on *n* with the lowest integer index *n* = 6, This linearity again suggests that the Zeeman effect is negligible and that the observed oscillations are far from the OFS quantum limit. Similar to the case of the IFS, here again the intercept |γ – δ| is in the range 0 to 1/8. Thus, we observe a systematic π-phase shift in SdH oscillations, consistent with the theoretical prediction that a pure Rashba system will exhibit a nontrivial π Berry’s phase for both the IFS and OFS.

Observation of this effect is made possible by the extremely large Rashba energy in BiTeI. It is interesting to note that, unlike the other candidate spin Berry’s phase systems such as semiconductor heterostructures and surface state of topological insulators, BiTeI is a 3D electronic system (albeit, with a quasi-2D electronic structure). Accordingly, this system may provide the opportunity to explore the dependence of Berry’s phase on the trajectory-dependent spin evolution. Moreover, the prediction that pressure can drive BiTeI through a quantum phase transition to a topological phase provides a test to compare Berry’s phase stemming from different physical origins (*35*).

## Supplementary Materials

www.sciencemag.org/content/342/6165/1490/suppl/DC1

Materials and Methods

Figs. S1 to S3

Reference (*44*)

## References and Notes

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**Acknowledgments:**We thank J. G. Checkelsky for fruitful discussions. This work was supported by the Funding Program for World-Leading Innovative Research and Development on Science and Technology (FIRST Program), Japan, and the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under contract DE-AC02-76SF00515 (C.B. and H.Y.H). This research was partly supported by Grants-in-Aid for Scientific Research (B) (no. 23340096) and (S) (no. 24224009) from the Ministry of Education, Culture, Sports, Science and Technology, Japan.