Semiconductors based on heavy p‑block elements (such as Bi, Sb, Te and Se) that crystallize in the rhombohedral tetradymite structure have long been of interest as the world’s best thermoelectrics (TEs) for room‑ temperature operation and have been continuously studied and developed for that application1. For simplicity, we clas‑sify them as ‘tetradymites’ after the mineral of nominal composition, Bi2Te2S. In the past decade, tetradymites have also emerged as important hosts for the topological surface state, a previously unexplored electronic state of matter.

Generally, semiconductors have surface states because the surface represents an interruption of the periodic potentials of the atoms in the bulk (these are Shockley and Tamm states2), which creates additional energy levels that are available to electrons at the surface. In materials with inverted electronic band energies (BOX 1), electrons in the surface states can be topologically protected from back‑scattering and can develop other unusual proper‑ties. Topological protection provides the surface state electrons with robustness to perturbations that do not break symmetry (time‑reversal symmetry in the case of tetradymites). In materials called topological insulators (TIs), electrical conduction in the bulk is expected to be suppressed, and the topological protection of the surface states strengthened. TIs are semiconductors in which the constituent atoms are heavy and of similar electro‑negativities; these materials have been reviewed exten‑sively from the perspective of materials science3–5 and have also been discussed from a chemical point of view6,7.

It has often been stated that materials with good TE properties are concomitantly materials with good TI properties8. Tetradymites are important materials in both of these areas of materials science (Bi2Se3 has been called ‘the hydrogen atom of TIs’).

Although topological physical concepts are novel, exciting and newsworthy, it is topological materials that have turned these concepts into reality and enabled the field to engage experimentalists as well as theoreticians. Two kinds of materials initially displayed topological properties3–5, the bismuth–antimony alloy semiconductor Bi1 − xSbx (x ≈ 5 to 20 at%) and thin‑film heterostructures of (Hg,Cd)Te. Both of these materials have limitations that have precluded their widespread experimental study. For bismuth–antimony alloys, the normal surface states that exist on the crystal surfaces in addition to the topological surface states create problems. For (Hg,Cd)Te, the desired effects are displayed only when the material is buried in thin‑film heterostructures, limiting the techniques that can be used to study it. The high vapour pressure of Hg and the sensitivity of the (Hg,Cd)Te properties to the Hg content complicate the control of these properties, such that (Hg,Cd)Te has been fabricated in only a small num‑ber of laboratories worldwide. By contrast, tetradymites are perfect materials to begin a large, new field of study — the crystals are relatively easy to prepare, the topological physics they display is elegant and new, and the energy dispersions of the surface states are relatively simple3–5,9,10.

Despite their theoretically ideal behaviour, the tetra‑dymite bulk materials that were initially studied, Bi2Se3

1Departments of Mechanical and Aerospace Engineering, Physics, and Materials Science and Engineering, The Ohio State University, Columbus, Ohio 43210, USA.2Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA.3Department of Physics and Materials Research Institute, the Pennsylvania State University, University Park, Pennsylvania 16802, USA.

Correspondence to J.P.H.  heremans.1@osu.edu

doi:10.1038/natrevmats.2017.49Published online 5 Sep 2017

Tetradymites as thermoelectrics and topological insulatorsJoseph P. Heremans1, Robert J. Cava2 and Nitin Samarth3

Abstract | Tetradymites are M2X3 compounds — in which M is a group V metal, usually Bi or Sb, and X is a group VI anion, Te, Se or S — that crystallize in a rhombohedral structure. Bi2Se3, Bi2Te3 and Sb2Te3 are archetypical tetradymites. Other mixtures of M and X elements produce common variants, such as Bi2Te2Se. Because tetradymites are based on heavy p‑block elements, strong spin‑orbit coupling greatly influences their electronic properties, both on the surface and in the bulk. Their surface electronic states are a cornerstone of frontier work on topological insulators. The bulk energy bands are characterized by small energy gaps, high group velocities, small effective masses and band inversion near the centre of the Brillouin zone. These properties are favourable for high‑efficiency thermoelectric materials but make it difficult to obtain an electrically insulating bulk, which is a requirement of topological insulators. This Review outlines recent progress made in bulk and thin‑film tetradymite materials for the optimization of their properties both as thermoelectrics and as topological insulators.

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and Bi2Te3, have carrier concentrations that are too high for the materials to be insulating in practice, which is explained later. Optimal TE carrier concentrations can be achieved easily1, but these materials still have not been systematically made with sufficiently low bulk carrier

concentrations to permit the study of their surface states without substantial interference from the bulk electronic states. There are several tetradymites for which this issue has been overcome, whereby a third or even fourth major elemental constituent is added. Both bulk crystal growth

Nature Reviews | Materials

Conventional bands

Atomic Levels

Inverted band extrema Inverted bands

Valence band Fermi surface of Bi

2Se

3

Conduction band Fermi surface of Bi

2Te

3

ZZΓ

E

ZZΓ

E

ZZΓ

E

ZZ

ΓΓ

FSp

in–o

rbit

in

tera

ctio

ns

Cry

stal

fiel

dsp

litti

ng

Che

mis

try

ZZ

Bi

Se

PBi–x,y,z PBi+

x +

iy,↑ PBi+

x +

iy,↓

PBi+x + iy,↓ PBi+

x + iy,↑

PSe–z,↑ PSe–

z,↓

PSe–x + iy,↑ PSe–

x + iy,↓

PSe–x + iy,↓PSe–

x + iy,↑

PSe–x,y,z

PSe–z

PBi+z

PSe–x,y

PSe+x,y,z

PBi+x,y,z

PBi+x,y

PBi+ PBi+z,

↓z,

↑

Box 1 | Effect of spin–orbit coupling on the band structures of heavy-atom tetradymites

The electronic band structure of a light-atom crystalline solid results from Coulombic interactions among valence electrons and charged nuclei and core electrons. Heavy-atom solids, such as tetradymites, display an additional coupling between the spin angular momentum of the electrons, S, and the localized magnetic fields induced by the orbital motion of the core electrons around the nuclei, which is characterized by the orbital angular momentum, L. In atomic physics, the L–S interaction energy splits discrete electron energy levels according to their spin. For atoms, the sum of the relativistic effects results in a correction of the atomic energy levels99, ΔEn,j,l =

mc2 4

2 4 j + ½–3αFSZ

nn⎞

⎠⎛⎝

⎞⎠

⎛⎝

where Z is the atomic number, c is the speed of light, αFS is the fine structure constant, j = l ± 1/2 is the total angular momentum quantum number (where l is the orbital quantum number), n is the principal quantum number, and m is the mass. For s and p electrons in the outer shells (n = 5 or 6) of heavy elements (Z >50), ΔEn,j,l lowers the energy levels significantly compared with the non-relativistic values, and also lowers the energy of states with j = l − 1/2 more than that of states with j = l + 1/2. When l is small enough and Z is large enough, relativistic corrections, including spin–orbit coupling (SOC), invert the relative positions of some of the levels. In MX (where M is a cation and X is an anion) compound semiconductors composed of light atoms, the cation s-like bands lie above the anion p-like bands; however, if M has a high enough Z, the relativistic corrections are large and negative enough to invert the bands, in analogy with their effect on atomic levels. The effect is pronounced at points of the Brillouin zone where the wave vector k (a congruent concept to the angular momentum L in atomic physics) is small: that is, near the Γ point or other high-symmetry points.

In tetradymites, SOC works as illustrated in the schematic diagram in the figure, which shows the evolution of the Bi and Se atomic px,y,z orbitals into Bi2Se3 bands in four stages10. Chemical bonding between Bi and Se splits their atomic p levels (the superscripts + and – denote the even or odd parity of the electronic states). Adding crystal-field splitting and SOC further modifies the levels. The lowest Se level reaches an energy lower than that of the highest Bi level, and thus, band inversion is realized.

The schematic dispersion relations for conventional MX or M2X3 compound semiconductors are shown in the upper part of the figure; the conduction band usually takes on the orbital character of the electron donor (M, green), whereas the valence band takes on that of the electron acceptor (X, purple). Tetradymites have two types of inverted bands. For reasons explained in the main text, a bandgap cannot exist at the Γ point of the Brillouin zone for tetradymites unless the states near the Γ point are hybridized with other bands. The effect of progressively increasing the SOC with respect to this hybridization is shown in the panels displaying inverted band extrema and inverted bands. For intermediate SOC values, the bands are inverted near the Γ point, but the band bending remains the same as for non-inverted bands with Γ-point extrema. When the SOC is stronger, the bands around the Γ point also invert their curvature, and the

extrema are located along the Γ–Z axis. Real tetradymites differ from this simplified explanation in that other band extrema can exist at other locations in the Brillouin zone, particularly along the Z–F direction, but they are not inverted. Moreover, the conduction and valence bands are not symmetric, and the ratio of the contributions from SOC and hybridization vary from band to band.

The growth of the valence band Fermi surface of Bi2Se3 with degenerate doping is shown in the bottom central part of the figure. This system is an example of a tetradymite with a band structure with inverted band extrema. The Fermi surface grows outward from the Γ point, starting as an ellipsoidal pocket (hole density, p = 5 × 1018 cm−3), then forming a rounded rectangular pocket (p = 1 × 1019 cm−3) and eventually branching out towards the next Brillouin zone (p = 8 × 1019 cm−3).

The growth of the conduction band Fermi surface of Bi2Te3 with degenerate doping, corresponding to a band structure with inverted bands, is also shown (lower right part of the figure). Extrema grow from outside in, appearing first along the Γ–Z direction and then in the Z–F direction (electron density, n = 5 × 1018 cm−3), and then growing into each other (n = 1 × 1019 cm−3) and towards the Γ point (n = 8 × 1019 cm−3). This is the behaviour of the conduction and valence bands of Sb2Te3 and of the conduction band of Bi2Te3; the valence band of Bi2Te3 does not reach a minimum along Γ–Z even at a doping level of p = 1020 cm−3.

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and thin‑film growth yield high‑quality tetradymites that permit detailed study of the characteristics of the topological surface‑state electrons.

In some cases, understanding the defect chemistry of tetradymites has been crucial to the development of high‑quality TI tetradymites. In other cases, improve‑ments arose more organically; solid solutions between two binary tetradymites with different dominant carrier types yielded carrier compensation and high bulk resistiv‑ity at intermediate compositions. Although in such cases it was not considered explicitly, the defect chemistry was the basis of the observed compensation behaviour. Many of the essential characteristics of topological surface‑state physics have been observed in such solid‑solution crys‑tals. Interestingly, defect chemistry is also often used to generate the optimum carrier concentrations in TEs. Tetradymites are generally indicated with whole number subscripts and are considered ‘line compounds’ (that is, intermediate phases appearing as lines in binary phase diagrams), although the defects that give rise to free car‑riers change the formulas so that the subscripts are not precisely integers. The line compound designation is chemical and as such does not imply that the materials are formally stoichiometric to high precision.

Although tetradymites are important materials in both TIs and TEs, the way materials are optimized for each application is quite different, primarily because of the target bulk carrier concentration ranges, which are <1016 cm−3 for TIs and 1019–1020 cm−3 for TEs. Furthermore, the TE field has embraced chemical com‑plexity in tetradymites (which often contain four or more elements) because complexity favours the vital character‑istic of decreased lattice thermal conductivity. By con‑trast, in TIs, the simplest of tetradymites — Bi2Se3 — is still the most studied, even though its bulk carrier con‑centrations are not sufficiently low to study the surface states without interference from the bulk states.

Initially, it was believed that topological protection implied that 3D TIs would have high‑mobility surface states. However, experimentally the situation is not that clear‑cut, even after extensive optimization of tetra dymite materials for this purpose. One reason for the lack of clarity is that time‑reversal symmetry only strictly pro‑tects 2D surface states against 180° back‑scattering. By contrast, breaking time‑reversal symmetry can lead to strongly topologically protected 1D edge states in quan‑tum anomalous Hall insulators. In this case, ballistic transport occurs over millimetre length scales at low temperatures, even in the presence of structural and mag‑netic disorder. To our knowledge, an equivalent effect has not been demonstrated in 2D states formed at the surface of bulk tetradymites, where surface‑state mobilities on the order of several thousand cm2 V−1 s−1 are observed.

In this Review, we start by defining tetradymites and showing their relation to other materials in the same chemical family. We then describe their band structures, their development as TIs and a handful of the experi‑ments that have been done to illustrate their topological physics3–5. The characteristics of the bulk materials and their defect chemistry are discussed, followed by those of thin‑film tetradymites. Finally, we present a short

description of TE tetradymites, building on REF. 1 but emphasizing links with TIs and the contrast between the requirements for materials optimization for the two purposes.

Materials formulations and structuresTetradymites have the formula M2X3, in which M is a heavy main‑group metal from group V (a pnictogen) and X is a chalcogen from group VI (FIG. 1). Thus, we consider M = As, Sb and Bi, and X = S, Se and Te as pos‑sible tetradymite‑formers. These elements have similar electronegativity, which in theory favours band inversion (BOX 1) but in practice is unfavourable for controlling the defect chemistry of a real material. This similarity in electronegativity produces anti‑site defects (that is, M atoms on the X sites or vice versa), which act as donors or acceptors, that are common in this family of materials. Anti‑site defects are the root problem in producing TI bulk insulators, but they are useful for achieving favour‑able carrier concentrations for TEs. The tetradymite structure shown in FIG. 1b has rhombohedral symmetry, space group R3–m, and three symmetry‑equivalent quin‑tuple layers of X1–M–X2–M–X1 stacked along the long c axis of the crystallographic unit cell5,7. X1 and X2 are two distinct types of X sites within the quintuple layers; X1 sites are located at the periphery of the layers, whereas X2 sites are in the middle. Each X1 atom bonds on one side to an M atom and on the other side to an X1 atom in another layer via van der Waals bonding. X2 atoms are bonded on both sides to M atoms; therefore, X2 bonds are more ionic in character.

For simple binary M2X3 compounds, the tetra dymite‑type structure competes with stibnite‑ and orpiment‑type structures (FIG. 1c,d), which have lower symmetry (for crystal structures, see the ICSD database). A schematic illustration that summarizes the occurrence of the orpiment, stibnite and tetradymite structures in these compounds is shown in FIG. 1a. The most stable binary tetradymites form from M = Sb and X = Te or M = Bi and X = Se, Te. For M = As, none of the most stable binary compounds has the tetradymite‑type structure.

The tetradymite family includes a wide range of solid solutions (TABLE 1) and distinctly ordered or partially ordered line compounds. Manipulating the elements present in these materials can help optimize the elec‑tronic properties of the solid solutions1,3,5,7; this has been accomplished for both TEs and TIs. In the 1970s11, Linus Pauling discovered that the nominal tetradymite mineral Bi2Te2S is actually Bi2Te1.8S1.2. The more electronegative S is primarily found in the more ionic X2 layer in the quintuple layer sandwich, and the resulting non‑stoichi‑ometry in the X1 layer is necessary to balance the relative layer sizes11. The most notable solid solutions are mix‑tures of Bi and Sb on the M site, and mixtures of Te, Se and S on the X sites. High‑efficiency TEs are frequently1 complex tetradymite solid solutions of the type (Bi1 − xSbx)2

(Te1 − ySey)3. A simpler, ordered phase in the normally random Bi2(Te1 − ySex)3 solid solution is Bi2Te2Se, which displays ordered or mostly ordered Se in X2 and Te in X1, an arrangement driven by electronegativity differences and of interest in both TIs5,7,12,13 and TEs14,15.

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Several compounds outside the V–VI chemical family also have the tetradymite crystal structure, including one of the polymorphs of In2Se3, stabilized by limited Bi sub‑stitutions16, as well as Hf2Te2P and Zr2Te2P (REFS 17,18). Zr2Te2P was proposed as a ‘topological metal’: that is, a host for topological surface states on a fully metallic bulk crystal surface.

Band structuresSpin–orbit coupling (SOC, BOX 1) underpins the physics behind the properties that make tetradymites both good TEs and TIs. SOC is also the basis of the record‑high spin Hall angles19 (which quantify the efficiency of the spin–charge conversion) observed in Bi2Se3.

The relative importance of SOC compared with the effects of the electronegativity difference between X and M results, for many tetradymites, in an inverted bulk band structure that, together with time‑reversal symme‑try, gives rise to the topological protection of the surface states3. Conventional compound semi conductors built from a metal cation (M) and an anion (X) have a con‑duction band in which the orbital character is dominated by the s or p orbitals of M and a valence band in which the orbital character is dominated by the s or p orbitals of X. Inverted bands have an opposite character near

the bandgap, with an X‑like conduction band and an M‑like valence band. BOX 1 illustrates how the inversion of the orbital nature of the conduction and valence bands occurs. Group theory imposes restrictions on this pro‑cess20; symmetry arguments enforce the condition that a strict two‑band (conduction and valence band) model applied to a solid with band extrema at the Γ point in the presence of time‑reversal symmetry must always describe a metal. Bi2Se3 has a bandgap at the Γ point, and therefore, the minimum model for its conduction and valence bands must involve at least four bands; hybridi‑zation (influenced by the degree of ionicity of the bonds) of the energy levels near the Γ point must be present. As illustrated in BOX 1, the shape of the bands depends on the relative strength of the SOC with respect to this hybridization. Tetradymites can develop two types of inverted bands, both of which are conducive for top‑ologically protected surface states; however, only the Bi2Te3‑type band structure produces good TEs, because it results in a high band degeneracy, as discussed later.

Angular‑resolved photoemission spectroscopy (ARPES) data21–23 for the Fermi surfaces of Bi2Se3, Bi2Te3, Sb2Te3 and Bi2Te2Se are displayed in FIG. 2. Besides the bulk Fermi surfaces, the data reveal topologically protected metallic  surface‑state bands. For binary

Nature Reviews | Materials

F

L

Z

K

Г

K M

c Stibnite

a Crystallographic phases of M2X3 compounds b Tetradymite

Bi

Sb

Se Te

As

S

Stibnite

Orpiment

Tetradymite

M

X

d Orpiment

X2MX1

Figure 1 | Crystallography of M2X3 compounds with M = Bi, Sb or As and X = Te, Se or S. a | Schematic showing the crystallographic phases of M2X3 compounds. As the difference in electronegativity between M and X increases from the top right corner to the bottom left corner, these materials crystallize in progressively more distorted lattices. The structure evolves from tetradymite, with a trigonal R3

–m crystal structure, to stibnite (Pnma), to orpiment (P21/n). b | Crystal structure

of tetradymite compounds (such as Bi2Se3) with the unit cell indicated and the corresponding first Brillouin zone. There are two types of chalcogen atoms in the tetradymite structure: X1 atoms have bonds with metal atoms (M) and van der Waals bonds with X1 atoms in neighbouring layers, whereas X2 atoms have bonds only with M atoms. c,d | The stibnite (panel c, as in Sb2Se3) and orpiment (panel d, as in As2S3) structures.

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tetradymites, the bulk bands are in good agreement with calculated band structures1 and with the Fermi surfaces shown in BOX 1. Because Bi2Te2Se is a line compound, it can be made more resistive than the binary compounds. The ARPES spectra24 of two samples with high bulk resis‑tivity (FIG. 2d) show that the chemical potential is in the surface‑state bands, with no bulk electronic band occu‑pancy. Transport experiments have proved more sensitive to the location of the chemical potential; thus, the obser‑vation of a bulk transport gap is the true measure that can demonstrate whether a tetradymite is a bulk insulator. The energy width of the surface‑state bands is reported to be temperature‑ independent, a possible indication of topological protection.

Besides inducing band inversion, SOC results in band structures with small energy gaps, small effective masses and high group velocities. TABLE 2 summarizes the energy gaps and static dielectric constants (parallel and perpendicular to the c axis) of binary tetradymites, along with other properties of these compounds. Whereas the optical frequency permittivity, ε∞, is an electronic prop‑erty, the DC permittivity, εdc, contains a phonon con‑tribution (discussed later), which makes it anisotropic. These properties promote high electron mobilities and good electrical conductivity.

It is easier to develop an intuition about thermopower and resistivity using the concept of effective mass than using band‑structure calculations and hypotheses about electron scattering. It is possible to view the components of the Fermi surfaces of any binary tetradymite as ellip‑soids, at least for sufficiently low carrier concentrations, for which the Fermi surfaces have not yet merged (BOX 1, FIG. 2). This enables a parameterization of the bands in

terms of effective mass, m* (the second derivative of the energy with respect to the wave vector, k); m* is a ten‑sor expressed in terms of the coordinates corresponding either to the crystallographic axes or the principal axes of the ellipsoid. The ellipsoids in question are centred either on Γ, on some point along the Γ–Z or Z–F directions in the first Brillouin zone, or on both. The number of degen‑erate ellipsoids at the same energy and equivalent points of the Brillouin zone is represented by N. Only three mass values (m1, m2 and m3), which correspond to the three principal axes of each ellipsoid, are necessary to charac‑terize the bands. The volume of the ellipsoid is therefore a function of the product of the three principal masses, giv‑ing the density of states effective mass mDOS = √m1m2m3

3 for each ellipsoid or mDOS = N 2/3 √m1m2m3

3 for the whole solid. The relevant values for the binary tetradymites are given in TABLE 2. Because the ellipsoidal model is over‑simplified and the shape of the Fermi surface changes incongruently with doping level (BOX 1), the carrier con‑centration at which the effective masses are defined must be specified.

In addition to ARPES, Fermi surfaces are mapped using the Shubnikov–de Haas25 and the de Haas–van Alphen25 effects, which give the cross‑section of the Fermi surfaces in the direction normal to an external applied magnetic field. From these maps and their temperature dependence, it is possible to deduce the values for m1, m2 and m3, or the cyclotron masses mω// and mω⊥, which are the geometric averages of the mass tensor elements: mω// in the (x, y) plane and mω⊥ in the (x, z) plane. The density of states effective mass, mDOS, is derived from the volume of the ellipsoids or from simultaneous measure‑ments of the thermopower, S, and the Hall coefficient, RH,

Table 1 | Summary of the (Bi,Sb)2(Te,Se,S)3 family of solid solutions and alloys

Alloy Phase Solid solution range

Bi–Sb

Bi2Te3/Sb2Te3 R3–

m Solid solution over the whole range

Bi2Se3/Sb2Se3 R3–

m From Bi2Se3 to Bi34.5Sb5.5Se60

Pnma From Sb2Se3 to Bi19Sb21Se60

Bi2S3/Sb2S3 Pnma Solid solution over the whole range

S-Se-Te

Bi2Te2/Bi2Se3 R3–

m Solid solution over the whole range

R3–

m Bi2Te2Se line compound

Bi2Te2/Bi2S3 R3–

m and Pnma Two‑phase mixture, except for Bi2Te2S

R3–

m Compound (actual composition closer to Bi40Te36S24)

Bi2Se3/Bi2S3 R3–

m Solid solution from Bi2Se3 to Bi40Se56S4

Pnma Bi2Se2S line compound (actual composition range Bi40Se39S21 – Bi40Se40.2S19.8)

Pnma Solid solution from Bi2S3to Bi40Se32S28

Sb2Te3/Sb2S3 R3–

m Solid solution from Sb2Te3 to Sb40Te25Se35 + Sb2Te2Se line compound (not firmly established)

Pmma Sb2Se3 dissolves perhaps 1% Te

Sb2 Te3/Sb2S3 R3–

m and Pnma Two‑phase mixture

Sb2Se3/Sb2S3 Pnma Solid solution over the whole range

Summary of the (Bi,Sb)2(Te,Se,S)3 family of solid solutions and alloys that comprise the V–VI tetradymites (space group R3–

m), indicating the competing phases (stibnite, Pnma, and orpiment, P21/n), the ranges of solubility and the known ordered variants.

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of a sample. The transport integrals relate S to the carrier concentration (n = 1/RHe) in the Pisarenko relation, which includes mDOS as an independent variable26. Solving the Pisarenko relation for mDOS requires some assumptions about the details of the scattering mechanism, often assumed to be either energy‑independent or dependent on E−1/2, as is characteristic for acoustic phonon scatter‑ing. The resulting values of the ‘transport mass’ mDOS are also reported in TABLE 2; the differences between the val‑ues of mDOS obtained from transport and those obtained from Shubnikov–de Haas measurements can be attrib‑uted to the effects of doping and to assumptions made about the scattering mechanisms.

Criteria for making tetradymite TIsThe primary criterion for having topologically pro‑tected surface states on TIs is the presence of band inversion and time‑reversal symmetry. To best charac‑terize the transport properties of the topological sur‑face states, transport in the bulk must be suppressed and the energies of the surface states must be distinct from those of the bulk states. This leads to two addi‑tional criteria: the bulk must be as electrically insu‑lating as possible, and the Dirac point (where the Dirac cones meet in the energy–momentum space, FIG. 2) in the dispersion of the surface states must fall within the bandgap of the bulk semiconductor.

Nature Reviews | Materials

a c

b d

0.7

–0.1–0.05

0.000.05

0.10

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0

0

0.1

–0.1

–0.2 0.0 0.2–0.2 0.0 0.2

–0.2 0.0 0.2 –0.2 0.0 0.2

0.4

0.2

0.0

0.4

0.2

0.0

0.4

0.2

0.0

0.4

0.2

0.0

Bi2Te

2Se (sample 1)

Bi2Te

2Se (sample 2)

Bind

ing

ener

gy (e

V)

Bind

ing

ener

gy (e

V)

Bind

ing

ener

gy (e

V)

Bind

ing

ener

gy (e

V)

k (Å–1) k (Å–1)

k (Å

–1)

k (Å–1) k (Å–1)

k (Å–1)k (Å–1)–0.2 0.0 0.2

–0.2 0.0 0.2–0.2 0.0 0.2

–0.2 0.0 0.2

0.4

0.2

0.0

0.2

0.0

–0.2

0.0

0.2

–0.2

0.0

0.2

n-Type Bi2Te

3

p-Type Bi2Te

3

Bi2Se

3p-Type Sb

2Te

3

Bind

ing

ener

gy (e

V)

Bind

ing

ener

gy (e

V)

Bind

ing

ener

gy (e

V)

k (Å

–1)

Bind

ing

ener

gy (e

V)

k (Å

–1)

k (Å–1)

k (Å–1)

k (Å–1)

k (Å–1)

–0.2 0.0 0.2

–0.2

0.0

0.2

k (Å–1)

k (Å

–1)

–0.2 0.0 0.20.5

0.4

0.3

0.2

0.1

0.0

k (Å–1)

0.1–0.1

SSB

SSB

SSB

SSB

BCB

BCB

BCB

BVB

BVB

BVB

ГK K

ГK MГK K ГK K

ГM

MГ

MM

ГK KГM M

ГK

K

ГK K

ГM M

ГM M

BCB

BVBSSB

SSB

SSB

BVB

BVB

EF

EB

ED 25 eV 51 eV

21 eV 23 eV

Dirac point

ГK

KГ

KK

ГK KГM M

ГK KГM M

Figure 2 | Angular-resolved photoemission spectroscopy data on the binary tetradymites and line compounds. a | Energy dispersion for Bi2Se3 and Fermi surface maps at the photon energies indicated on the frames as obtained from the k(Γ‑to‑M) versus k(Γ‑to‑K) dependence of the angular‑resolved photoemission spectroscopy (ARPES) intensity, where k is the wave vector. b | Energy dispersion of the ARPES intensity and k(Γ‑to‑M) versus k(Γ‑to‑K) dependence of the Fermi surface for n‑type and p‑type Sn‑doped (0.9 at% Sn) Bi2Te3. c | Energy dispersion and Fermi surface for heavily doped p‑type Sb2Te3. d | Dispersion relations measured by ARPES on two high‑resistivity samples of the line compound Bi2Te2Se. Samples 1 and 2 reached approximately 5 and 0.15 Ω cm, respectively, at 4 K. BCB, bulk conduction band; BVB, bulk valence band; EB, bottom of the BCB; ED, energy of the Dirac point; EF, Fermi energy; SSB, surface‑state band. Panel a is adapted with permission from REF. 22, AAAS. Panel b is adapted with permission from REF. 21, AAAS. Panel c is adapted with permission from REF. 23, Macmillan Publishers Limited. Panel d is adapted with permission from REF. 24, American Physical Society.

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Table 2 | Measured properties of binary tetradymites

Property Units Bi2Se3 Bi2Te3 Sb2Te3

Relative dielectric permittivity

ε⊥dc – 113 (REF. 115) 290 (REF. 115) 168 (REF. 116)

ε//dc – – 75 (REF. 115) 36.5 (REF. 116)

ε⊥∞ – 25 (REF. 117) 85 (REF. 115) 50 (REF. 118)

ε//∞ – 17 (REF. 117) 50 (REF. 115) 30 (REF. 118)

Energy gap

Eg indirect meV (at 293 K) – 130 (REF. 119) 210 (REF. 119)

Eg direct meV (at 77 K) 160 (REF. 120), 35 (REF. 121) – –

meV (at 10 K) ≥300 (REF. 32) −0.15 (REF. 119) –

dEg/dT meVK−1 −0.20 (REF. 120) – –

Electrons

Number of pockets N 1 6 –

mω// me 0.124 (REF. 122) 0.06 (REF. 123) –

mω///mω⊥ – 4 (REF. 117) – –

m1 me – 0.021 (REF. 124) –

m2 me – 0.32 (REF. 124) –

m3 me – 0.081 (REF. 124) –

mDOS (transport) me 0.16 (REF. 125) 0.32(a)126 –

mDOS (SdH) me – 0.27 (REF. 123) –

ncrit cm−3 3 × 1015 1013 –

aΒ⊥* nm 50 250 –

aB//* nm 2 25 –

Holes

Number of pockets N 1 6 6‡, 6§

mω// me – 0.09 (REF. 127) 0.049‡ (REF. 128), 0.14§ (REF. 128)

mω⊥ me 0.25 (REF. 26) – –

m1 me – 0.031 (REF. 124) 0.084‡ (REF. 129), 0.034§ (REF. 128)

m2 me – 0.44 (REF. 124) 1.24‡ (REF. 129), 0.34§ (REF. 128)

m3 me – 0.086 (REF. 124) 0.127‡ (REF. 129), 0.54§ (REF. 128)

mDOS (transport) me 0.25 (REF. 26) 0.46(b)126 0.34(c) ‡ (REF. 126)

mDOS (SdH) me – 0.35 (REF. 127) 0.3‡ (REF. 129), 0.97§ (REF. 128)

ncrit cm−3 3 × 1016 3 × 1013 10‡ (REF. 14), 10§ (REF. 15)

aΒ⊥* nm 24 170 180‡, 60§

aB//* nm 1 20 5‡, 5§

Experimental values for the relative dielectric permittivity, ε, both in the low‑frequency (εdc) and high‑frequency (ε∞) limit, and in the directions parallel and perpendicular to the c axis of the binary tetradymite compounds. Also reported are the energy gap (Eg), Shubnikov–de Haas (SdH) and transport effective masses in units of me, the free electron mass, taken at the band edge, except for the valence band of Bi2Se3, which covers hole concentrations from 6 to 18 × 1018 cm−3 (REF. 26). The transport masses are reported at 5 × 1018 cm−3 (a), 8 × 1018 cm−3 (b) and 7 × 1019 cm−3 (c). For each type of carrier, we report a rough estimate of the Bohr radius aB*, both perpendicular and parallel to the c axis, and the critical value (ncrit) of the carrier concentration that corresponds to the Mott criterion for the metal–insulator transition. aB//* and aB⊥ *, effective Bohr radius in the direction parallel and perpendicular to the c axis; m1, m2 and m3, effective mass values measured along the three principal axes; mDOS, density of states effective mass; mω,  cyclotron mass; T, temperature. ‡These values refer to the upper valence band. §These values refer to the lower valence band.

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It is at the Dirac point that various unusual effects are expected in the surface‑state charge transport; thus, its accessibility in real materials is important. If the Dirac‑point energy is outside the bulk bandgap, there will be states from the bulk valence or conduction band at the same energy, and charge carriers in those states will interfere with the surface‑state characterization. It then becomes impossible to unambiguously probe the surface‑state electrons near the Dirac point by chemical control of the Fermi level or electronic gating.

It is also necessary to make the bulk of the sample resistive. The Mott criterion requires that to stop band electrons from conducting by freezing them at a low temperature — that is, to avoid metallic conduction at 0 K — the Bohr radius, aB*, of the charge carriers in a band must be smaller than the carrier Thomas–Fermi screening length, rTF (REF. 27).

From the relative permittivity, εdc, and mω (TABLE 2), it is possible to determine the effective Bohr radius of electrons and holes, aB* = aBεdc/m*, where the effective mass, m*, is expressed as a fraction of the free electron mass, and aB is the Bohr radius of the hydrogen atom (0.053 nm). Because both εdc and mω are anisotropic, order‑of‑magnitude esti‑mates for the Bohr radius values are reported in TABLE 2 for both the parallel and perpendicular direction to the c axis (aB//* and aΒ⊥*). Perturbations of the atomic potentials at length scales smaller than aB* have a negligible effect on the motion of free electrons or holes. In tetradymi‑tes, this includes most interstitial defects, foreign atoms, dopant atoms and anti‑site defects. The high mobility of the charge carriers in these materials and the relative lack of influence of defects or alloying on the mobility, which are important advantages in TE materials, result from the fact that aB* is larger than the in‑plane unit cell. The same properties set a stringent criterion for the formation of TIs. Because rTF depends on the charge‑ carrier concentration, the Mott criterion27, rTF = aB, to reach the metal–insulator transition sets a minimum value for the concentration of free electrons, ncrit, above which the bulk of a semicon‑ductor sample will remain electrically conducting even at the lowest temperature. Expressing rTF in terms of the carrier concentration leads to an expression for ncrit that reads aB*n1/3

crit = 1/4(π/3)1/3. Because the Bohr radii are ani‑sotropic, whereas ncrit, is scalar, the values for the order of magnitude of ncrit are calculated from both aB//* and aB⊥*, and TABLE 2 reports the geometrical average (REF. 1 reports the in‑plane value). Thus, producing bulk TI materi‑als requires extraordinary measures to reduce the bulk carrier concentration to values of ~1015–1016 cm−3 in tet‑radymite‑type materials. Remarkably, this has occasionally proven possible, in particular when TI researchers have used resonant‑level impurities — developed to under‑stand and enhance the TE properties of tetra dymites — to control the carrier concentration.

Bulk topological insulatorsBinary materialsThe quest to meet the criteria for obtaining good TIs has motivated a resurgence of theoretical and experimental work focused on the defect chemistry from binary to quaternary tetradymites. An overview of the progress of

this difficult research is given in BOX 2. In this section, we illustrate the essential points of defect chemistry, start‑ing from binary alloys, of which (with the exception of Sb2Te3) the point defect energies have been well‑studied theoretically28–30, which is also the case for Bi2Te2Se.

Bi2Te3. The wide range of reported Bi2 ± δTe3 ± ε stoichi‑ometries1 is an indication of the presence of defects and of the ease with which they form under normal synthetic conditions. In fact, the stoichiometric com‑position appears to melt incongruently, meaning that crystals grown at the congruent melting composition should be highly defective. Bi2Te3 grown in Bi‑rich conditions is p‑type because the dominant defect is a charged Bi atom on a Te site, which is a Bi anti‑site defect (BiTe˙ in Kröger–Vink notation) and an electron acceptor. Alternatively, Bi2Te3 grown in Te‑rich condi‑tions is n‑type because the dominant defect is a Te atom on a Bi site, which is a Te anti‑site defect (TeBi˙) and an electron donor. Experimentalists can control the rela‑tive abundance of these defects at the level required for optimal TEs, in the 1019–1020 cm−3 range. Thus, Bi2Te3 forms the basis for excellent n‑ or p‑type TEs. However, Bi2Te3 was abandoned as a TI owing to the difficulty of making it insulating. It has the lowest ncrit value among tetra dymites (TABLE 2), but reaching carrier concentra‑tions near ncrit has proven impossible because of the low formation energies of the defects. An additional disad‑vantage of Bi2Te3 as a TI is that the Dirac point of the sur‑face states lies below the energy of the top of the valence band (FIG. 2b); thus, surface electrons near the Dirac point cannot be isolated from bulk electrons.

Sb2Te3. Even though it is an important component of the best p‑type TEs, Sb2Te3 has not been studied exten‑sively as a TI. Contrary to Bi2Te3, its Dirac‑point energy is calculated23 to be well‑separated from the bulk valence band. The calculated value for ncrit is higher than for Bi2Te3. Unfortunately, Sb2Te3 is always very strongly p‑type because of a large fraction of a dominant charged defect, although modern electronic structure calcula‑tions to support the inferred nature of the defect have not been performed31.

Bi2Se3. Intuitively, the binary alloy Bi2Se3 should have a defect chemistry that is easier to control because of the larger difference in electronegativity between Bi and Se and its larger bandgap. However, this material is not truly insulating, as it is intrinsically strongly n‑type. The defect chemistry of Bi2Se3 is different from that of Bi2Te3 but is still not amenable to creating a bulk insulator. Under Bi‑rich conditions, the dominant defect is a vacant Se site, which is a doubly charged defect, VSe˙˙, donating two electrons per defect. This problem is exacerbated by the fact that Se is a highly volatile element and leaves Bi2Se3 crystals during growth, filling the growth vessel with Se vapour. Unfortunately, even back‑filling the growth vessels with a heavy gas, such as Ar, opposing the Se vapour pressure and minimizing the free volume of the chamber, fails to create high‑resistivity Bi2Se3 crystals. In addition, under Se‑rich conditions, the dominant

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defect has been calculated to be a Se anti‑site, SeBi˙, which is a charged defect that donates an electron. Thus, we do not have sufficient control of the stoichiometry to obtain an insulating material from undoped Bi2Se3.

Even aliovalent doping of Bi2Se3 has not led to a mat erial cleanly hosting topological surface states. The

typical acceptor‑type Bi2Se3 aliovalent dopants, Cu and Pb, are unfortunately ambipolar and thus ineffec‑tual in making the system insulating. Therefore, more ionic species have been used to induce a dominant car‑rier crossover from n‑type to p‑type. In recent years, Ca, Mg and Cd dopants have been used to generate

Nature Reviews | Materials

Bi2Te

2Se

SeTe

Bi

(Bi,Sb)2Se

3

(Bi,Sb)2Te

3

Se(Te)

Bi/Sb

Bi2(Te, Se)

3

Bi

Te/Se

Bi

Se(Te)

Sn:Bi2Te

2Se

TeSe

Bi:Sn

(Bi,Sb)2(Te, Se)

3

Bi/Sb

Te/Se

Sn:Bi1.1

Sb0.9

Te2S

Te

S

Bi/Sb:SnSn:Bi1.1

Sb0.9

Te2S

Te

S

Bi/Sb:Sn

ab

c

The Dirac point can be in the gap, and low carrier concentrations are possible at low temperature but composition is difficult to control.

The mixture of Se and Te inhibits the formation of anti-site defects and Se vacancies, but the Dirac point is not isolated from the bulk states.

Partial substitution of Bi for Sb

Incorporation of S in the middle

of the QL

Partial substitution of Bi for Sb

Partial substitution of Te for Se

Partial substitution of Te for Se with Se

confined to the middle layer in the QL

Partial substitution of Te for Se, random

solid solution

Sn doping

Bi2Se

3

Bi2Te

3

The Dirac point is in the gap, and low carrier concentrations are possible at low temperature, with good stability. Resonant-level doping is used to compensate for the last native defects.

Qui

ntup

le la

yer (

QL)

Resonant-level Sn doping allows better control of the chemical potential to better study the transport properties of the surface states. The Dirac point is still not isolated from the bulk states.

Box 2 | Chemical tuning of tetradymites to obtain better topological insulators

To obtain tetradymite semiconductors with topologically protected surface states, two primary requirements for the bulk crystals must be satisfied. The first is a very high bulk resistivity, allowing the study of the charge transport properties of the surface states without the overwhelming influence of the bulk states. The second is having a Dirac point in the surface-state energy–wave vector dispersion lying within the bulk bandgap. In other words, the energies of all possible bulk energy states must be far enough away from the energy at which the surface-state Dirac cones meet in k space, such that the Dirac point can be reached by applying a gate voltage or through chemical doping. A variety of important experiments have been performed on tetradymites, even for systems in which this latter criterion has not been satisfied.

A third material requirement — that tetradymites satisfying both the criteria above can be grown reproducibly as large bulk single crystals — has generally not been met. Again, however, important experiments have

been done on materials in which the defect chemistry has not been controlled, but excellent bulk properties have been nonetheless achieved.

For simple binary bulk tetradymites, Bi2Se3, Sb2Te3 or Bi2Te3, these requirements have not been satisfied because of the defects present in the bulk crystals, which require an increase in the chemical complexity to obtain the desired properties. The figure shows a schematic representation of the evolution of tetradymite-type topological insulators as obtained through two approaches, anion substitution and cation substitution. The arrows represent the progress made in achieving better materials. As an example, a ball-diagram schematic of a quintuple layer for the Sn-doped Bi1.1Sb0.9Te2S tetradymite is shown; the Bi/Sb atoms are coloured in a fraction that reflects the composition obtained by crystallographic refinement. In this material, which has good stability, the Dirac point lies in the gap of the bulk bands, making Bi1.1Sb0.9Te2S a very good TI.

Figure is adapted with permission from REF. 37.

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p‑type conductivity in Bi2Se3 (REFS 32–34). For exam‑ple, when using Ca, the resulting defect CaBi˙ acts as an acceptor, compensating for the donor defects present. Experimentally, Ca and Mg dopants have resulted in the formation of p‑type Bi2Se3, and the TE properties of Ca‑doped Bi2Se3 with p‑type carriers in the 1018–1019 range are good at low temperatures32. However, control of the chemical system resulting in carrier concentra‑tions below ncrit through bulk crystal growth has not yet been achieved.

Ternary and quaternary materialsFortunately, it is possible to grow ternary and quaternary tetradymite bulk crystals with high resistivity, which ena‑bles the study of the surface state. For example, research‑ers mixed Bi and Sb on M sites and Te and Se on X sites, aiming to produce non‑thermodynamically stable atomic orderings to define the stoichiometry. In this way, (Bi1 − xSbx)2(Te1 − ySey)3 electrically insulating solid solu‑tions were produced, with x ≈ 0.27 and y ≈ 0.43 (REF. 35). Notably, these crystals were also used to observe a funda‑mental consequence of SOC: the arising of the spin Hall effect. Although it is logical to mix tetradymites that are dominantly p‑type and n‑type to achieve an insulating crystal through defect‑balancing compensation, this does not work in practice. It is difficult to create and reproduce the correct balance between donors and acceptors in a double‑random alloy system in which the target mat‑erial is thermodynamically unfavourable. Bi2Te2Se is an alternative tetradymite that is easier to control5,7,12,13.

Bi2Te2Se. In Bi2Te2Se, a Se atom is thermodynamically preferred for the middle X2 layer, resulting in an intrin‑sically lower concentration of charged defects. Increased low‑temperature bulk resistivities can be reached, ena‑bling an improved study of the topological surface states compared with what is possible for binary compounds. The defect energies in Bi2Te2Se have been studied explic‑itly from a theoretical standpoint30: the dominant defects under all conditions are Se and Te anti‑sites (SeTe and TeSe). These defects are electrically neutral and have no negative effects on the bulk insulating behaviour. The next most stable defects are Bi vacancies (VBi) or BiTe anti‑sites, which are acceptors, and Te vacancies (VTe) or TeBi anti‑sites, which are donors30. Chemically, inserting Se between the Bi layers in Bi2Te2Se increases the ener‑getic cost of the Se vacancy, as well as the energetic cost of creating donor defects. Bi2Te2Se works best as a low‑ temperature bulk insulator when a small amount of Sn is added; this is a critical dopant for both TIs and TEs.

Sn is a resonant‑level dopant in Bi2Te3, as identi‑fied through theory and experiment36. Resonant‑level dopants are a special case of dopant metals or semi‑conductors. In small quantities (2% or lower), such dopants lead to a narrow peak in the electronic density of states, either at the top of the valence band or at the bottom of the conduction band. Thus, besides acting as an acceptor or donor impurity, they can pin the Fermi energy in a nearly compensated semiconductor, create a state in which the delocalized charge carriers resonate between the band and impurity states, and/or act as

resonant electron scatterers. In Bi2Te3, the Sn levels are at the top of the valence band. Small levels of Sn doping lead to an unexpected increase in the p‑type thermopower, which is attributed to an increase in the density of states at the resonant energy. Similarly, Sn is postulated to be a valence band, resonant‑level dopant30 in Bi2Se3, although a level of p‑type carrier control sufficient to bring the Fermi energy through that resonant level has not yet been achieved. For Bi2Te2Se, two groups have used Sn as a resonant‑level dopant to localize any naturally present holes or electrons to obtain an unexpectedly high bulk resistivity, on the order of 10 Ω cm, at low tempera‑tures12,13. Presumably, the resonant level overcomes the ncrit limitation in the bands by creating a local excess in the density of states at the resonant energy level, where electrons have a smaller Bohr radius than in the bands of the host semi conductor. This allows the bulk of the samples to reach the Mott insulator transition and the conductance of the surface states to dominate at tem‑peratures below ~50 K. ARPES measurements deter‑mined that the Bi2Te2Se bulk bandgap is ~300 meV12,13. Furthermore, at temperatures above those at which the surface states start to affect the results, the resistivity dis‑plays activation behaviour with energy in the range of 100–140 meV. The lowest value — lower than half the gap — suggests that the excitation of defects present in some of the crystals dominates the activated behaviour in the gap. Thus, Sn‑doped Bi2Te2Se is a genuine TI with a large bandgap and the ability to become insulating in the bulk at low temperatures.

Tuning of the Dirac-point energy. Bi2Te2Se is not an ideal TI because the Dirac point of the surface‑state electrons has an energy that is 60 meV below the energy of the top of the bulk valence band (FIG. 2d); thus, it cannot be reached without inducing bulk conduction. Researchers working on the quaternary solid solution (Bi1 − xSbx)2(Te1 − ySey)3 reported getting close to the Dirac point using gating35. Because the Dirac‑point energy in Sb2Te3 is known to be well above the top of the valence band, substituting Sb for Bi can enable the tuning of the Dirac‑point energy in quaternary tetradymites. A new complex tetradymite, Sn‑doped BiSbTe2S (REF. 37), based on the original tetradymite mineral, is a low‑tem‑perature bulk insulator (with a bulk resistance higher than 100 Ω cm) that can be grown in large crystals by the vertical Bridgman method. This material displays quantum oscillations from the surface‑state electrons at low temperatures and has an activated resistivity with an activation energy that is roughly half of the meas‑ured bulk bandgap energy, implying intrinsic behaviour. Moreover, this material has a Dirac‑point energy that is well separated from both the conduction band and the valence band energies, and is thus a very promising TI.

Tetradymite thin filmsUsing a thin‑film sample geometry is the most intuitive way to reduce the influence of bulk conduction relative to surface conduction. The development of tetradymite thin films, obtained by exfoliation from bulk crystals or by epi‑taxial growth, has brought about several new questions in

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the study of 3D TIs. The first is how thin‑film geometries can be used to manipulate the helical spin texture of Dirac surface states using quantum confinement and strain. This is akin to the questions addressed in the successful era of ‘band‑structure engineering’ in semi conductor quantum wells and superlattices, except that the focus here is on the interplay between spin and momentum in the sur‑face states. The second question is whether it is possible to discover emergent quantum pheno mena in hetero‑structures interfacing the Dirac states of a 3D TI with the symmetry‑breaking environment of a ferro magnet, antiferromagnet or super conductor. Several theoretical proposals have been advanced concerning the resulting ‘exotic broken‑symmetry surface states’, heralding phe‑nomena such as image magnetic monopoles, Chern insu‑lators, the topological magneto‑optical Kerr effect and Majorana modes3. Finally, there is the important question of whether it is possible to exploit the momentum–space spin texture of helical Dirac electrons to design ‘topolog‑ical spintronic’ devices for memory and logic. Thin films of TIs are important in this context because they can be interfaced with ferromagnetic thin films in hybrid hetero‑structures and can be patterned into device geometries such as electrostatically gated field‑effect transistors.

This potential of TIs for emergent physics and tech‑nology is predicated on the ability to synthesize thin films of sufficiently high quality. Advances have been made in this context, but the direct epitaxial synthesis of tetra‑dymite thin films still does not result in the crystalline quality that is routinely achieved in the epitaxy of more mature semiconductor thin films, such as those from group III–V, II–VI and IV materials.

Synthesis and imperfections in thin filmsThe van der Waals bonding between the X1 atoms at the edges of each tetradymite quintuple layer has a direct influence on the synthesis of these materials in thin‑film form. First, it allows the exfoliation of thin films from high‑quality bulk crystals, resulting in the observation of clean, surface‑dominated quantum transport38. Despite the skill required to exfoliate tetradymite thin films from bulk crystals, flakes with micrometre areas and thick‑nesses ranging from 5 to 100 quintuple layers can be pre‑pared and measured in electrostatically gated electrical transport devices39. Recently, the electrostatic gating of exfoliated thin films has enabled sophisticated studies of the fully developed integer quantum Hall effect of helical Dirac fermions on opposite surfaces40.

Compared with exfoliation, the epitaxial growth of tetradymite thin films and heterostructures offers a far broader landscape of scientific and technological pos‑sibilities. Several groups have pursued the epitaxy of tetradymite thin films on various substrates and the for‑mation of heterostructures using molecular beam epitaxy (MBE) and metal–organic chemical vapour deposition (MOCVD). The van der Waals nature of the tetradymite crystals relaxes the stringent lattice‑matching constraints that restrict the range of substrates for the epitaxial syn‑thesis of other classes of materials (for example, group III–V semiconductors). Thus, Bi‑ and Sb‑chalcogenides have been successfully grown in thin‑film form on

various substrates, including Si(111)41,42, GaAs(111)43, InP(111)44,45, SrTiO3(111)46,47, α‑Al2O3 (sapphire)48,49, graphene‑terminated 6H‑SiC(0001)50, CdS(0001)51 and Y3Fe5O12 (yttrium iron garnet, YIG)52,53. The in‑plane lat‑tice mismatch covers an astonishing range of values: for example, 0.24% and 25% for Bi2Se3 on InP and α‑Al2O3, respectively. Generally, regardless of the substrate used, the resulting films have a textured surface that is charac‑terized by pyramidal single‑crystal domains with quintu‑ple‑layer steps. The size and relative proportion of these pyramidal domains vary with factors that include film thickness, lattice mismatch with the substrate and inter‑facial chemistry‑dependent film nucleation. The largest domains reported in the literature are approximately 1–2 μm in Bi2Se3 grown on InP(111)A (REF. 44). Structural measurements, such as those performed by transmission electron microscopy and scanning tunnelling microscopy, show crystalline perfection within the limited spatial extent of such domains. However, classifying these tetra‑dymite thin films as high‑quality single crystals would not be appropriate. Although the domains can be orien‑tated coherently over macroscopic regions under certain circumstances44, the extent of each individual domain is orders of magnitude smaller than that attainable in the epitaxy of other semiconductor thin films, such as GaAs and ZnSe.

X‑Ray double‑crystal rocking curves are an impor‑tant macroscopic measure of thin‑film quality. Bi2Se3 and (Bi,Sb)2Te3 thin films can display rocking‑curve widths ranging from 0.07 to 0.1°, which are surprisingly inde‑pendent of large differences in lattice mismatch54. The primary imperfections contributing to the observed width are twinning and anti‑phase domain boundaries, as well as mosaic twists and tilts. For potential tech‑nological applications, an improvement of an order of magnitude in the rocking‑curve width is needed. There is evidence that this can be achieved by paying careful attention to the mode of nucleation and to the nature of the substrate: for example, for Bi2Se3 on InP(111)B, the rocking‑curve width is as narrow as 0.0036° (REF. 45).

A creative approach to virtual substrate engineering (the use of a strain‑relaxed buffer on which materials with a lattice constant different from that of the original sub‑strate can be grown) has been developed to grow Bi2Se3 films with low defect density by MBE55. The synthetic protocol involves the growth of an In2Se3/Bi2Se3/Al2O3 heterostructure, followed by annealing to remove the thin (three quintuple layers) Bi2Se3, presumably through diffusion and subsequent evaporation. This leaves an In2Se3 buffer capped with an additional film of (Bi,In)2Se3. This heterostructure forms a ‘virtual’ substrate, allow‑ing the deposition of Bi2Se3 with sufficiently high low‑ temperature mobility (~1.6 × 104 cm2 V−1 s−1) to exhibit the fully developed integer quantum Hall effect associated with its surface states. Approaches of this type could lead to improvements in the growth of other TI thin films, such as Bi2Te3 and (Bi,Sb)2Te3.

Thin‑film synthesis does not resolve the stoichiometry challenges described for the bulk growth of tetradymite crystals. At first, it could be expected that control over beam flux ratios during MBE growth might provide a

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solution to the problem of the formation of chalcogen vacancies. Unfortunately, this is not the case. This may be partly due to the high vapour pressures of the constitu‑ent elements of this material family. Thus, as‑grown films of binary tetradymites are always extrinsically doped as n‑type (Bi2Se3, Bi2Te3) or p‑type (Sb2Te3). Consequently, the chemical potential of as‑grown films lies in the bulk bands, complicating the interpretation of electrical transport measurements. This can be partly resolved by electrostatically gating tetradymite thin films in metal–oxide–semiconductor devices. The first modest success in this context arose from the electrostatic gating of Bi2Se3 thin films, either by back‑gating using a high dielectric substrate (SrTiO3) or by top‑gating using an oxide die‑lectric over‑layer. However, the high level of extrinsic doping (n2D >1013 cm−2) resulting from the defect chem‑istry has so far prevented the full tuning of the chemi‑cal potential through the Dirac point in MBE‑grown Bi2Se3 thin films56. Better results were obtained through a compensation process, similar to what was attempted in bulk crystals; because MBE‑grown Bi2Te3 thin films are inherently n‑doped, whereas Sb2Te3 thin films are p‑doped, (Bi,Sb)2Te3 ternary films provide an opportunity to reduce the carrier density through self‑compensation. The compensation point is as difficult to reach in films as

it is in the bulk, but in thin films, the Fermi level can be more finely tuned with electrostatic gating. Indeed, the epitaxial growth of (Bi,Sb)2Te3 thin films on InP(111) or SrTiO3(111) readily yields thin films with a carrier density <1013 cm−2, allowing the chemical potential to be varied through the Dirac point by gating57.

The ultrathin limitThe epitaxial growth of semiconductors in thin‑film form has significantly affected the development of opto‑electronics and high‑mobility transistors, because it enables band‑structure engineering through quantum confinement at heterointerfaces and in quantum wells. Similarly, the epitaxial growth of tetradymite thin films enables control over the electronic structure of topologi‑cal surface states. For example, ARPES measurements of the hybridization of Dirac states on two opposite surfaces of a very thin Bi2Se3 film (<6 nm) show the opening of a hybridization gap and the disruption of the Dirac point50 (FIG. 3). Quantum corrections to diffusive transport also indirectly demonstrate the presence of this gap, where weak anti‑localization corrections are replaced by weak localization corrections58. Spin‑resolved ARPES meas‑urements provided additional insight into the effects of the hybridization by probing changes in the helical spin

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k ≈ 0.05 Å–1Energy levels Spins Energy levels Spins

Figure 3 | Spin-resolved angular-resolved photoemission spectroscopy in ultrathin Bi2Se3 films. a | Angular‑resolved photoemission spectroscopy (ARPES) measurements for a series of Bi2Se3 films with different thicknesses (expressed in quintuplet layers, QLs), demonstrating the opening of a gap in the Dirac cone due to the hybridization from opposite surfaces and showing the modification of the spin texture caused by this hybridization. b | Net spin polarization as a function of the Bi2Se3 layer thickness shown at two different wave vectors, k. For a sample of a given thickness, the spin polarization of the charge current in a TI spin device can be potentially tuned by using an electrical gate to vary the chemical potential (and hence wave vector). c | Schematic representation of the spin texture of the surface states observed with and without the hybridization gap. Figure is adapted with permission from REF. 59, Macmillan Publishers Limited.

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texture59,60 (FIG. 3c). In the absence of hybridization (film thickness >6 nm), the spin polarization of helical Dirac electrons does not vary much with energy moving away from the Dirac point. By contrast, once the Dirac cone is gapped, the spin polarization at the Γ point is quenched, even though it recovers its original value at higher ener‑gies. The helical spin texture of the surface states in TIs is preserved for modest hybridization. This makes it pos‑sible to envisage topological spintronic devices in which the current spin polarization in an ultrathin 3D TI with a gapped surface state can be controlled by varying the chemical potential using an electrostatic gate.

The quantum anomalous Hall effectTheoretically, in a TI, if time‑reversal symmetry is bro‑ken by introducing a magnetization perpendicular to the surface, a gap opens at the Dirac point because the massless Dirac Hamiltonian acquires a ‘mass’ term. In principle, the magnetization can be introduced in two ways: by doping the TI with a sufficiently high density of magnetic dopants with appropriate magnetic aniso‑tropy to create long‑range magnetic order with an out‑of‑plane easy axis61, or by interfacing the TI with a magnetically ordered material62. If this is done success‑fully and the chemical potential is placed within the ‘magnetic gap’, theory predicts that the gapped helical Dirac surface states are replaced by a ballistic chiral edge state63. Consequently, the longitudinal conductivity van‑ishes (σxx = 0), and the transverse conductivity is quan‑tized (σxy = e2/h, where e is the electron charge and h is the Planck constant). This phenomenon — the quantum anomalous Hall effect — is similar to the integer quan‑tum Hall effect but does not involve any Landau levels and can be realized even without an external magnetic field. The only requirement is remnant magnetization in the TI itself or in a vicinal ferromagnet. In a sense, it was the first physical realization of Haldane’s Nobel Prize‑winning concept of the Chern insulator64.

The quantum anomalous Hall effect was experimen‑tally measured in electrostatically gated thin films of Cr‑doped and V‑doped (Bi,Sb)2Te3 (REFS 65,66). This result has been reproduced by several groups67–70; in particular, the quantization of the anomalous Hall conductance was demonstrated with a high level of precision, and near‑ly‑ballistic transport over millimetre lengths was meas‑ured in both V‑doped66 and Cr‑doped71 samples (FIG. 4). This precise quantization and nearly ballistic transport occur despite significant structural and magnetic disor‑der72. Although the quantum anomalous Hall effect has so far been restricted to very low temperatures (<150 mK), more complex tetradymite heterostructures have shown promise for an increased observation temperature73.

Topological spintronicsThe helical spin texture of topological surface states results in spin–momentum locking — that is, the direction of the momentum determines that of the spin — which is essential for possible topological spintronic devices. In principle, the idea is straightforward. The inherently spin‑polarized charge current, JC, carried by the sur‑face states results in a highly efficient charge‑to‑spin

conversion at the interface between a TI and a ferromag‑net, which enables the exertion of a spin‑transfer torque on proximal magnetic moments. In turn, this generates a spin current, JS, that is desirable for spintronics applica‑tions. The essential validity of this concept was demon‑strated through spin‑torque ferromagnetic resonance measurements at room temperature in bilayers of Bi2Se3 and permalloy (FIG. 5a), which revealed a large figure of merit for the charge–spin conversion, characterized in 3D systems by a spin‑torque ratio θST = (2e/ħ)JS/JC ≈ 1, where ħ is the reduced Planck constant19. A contempo‑raneous experiment also examined spin–charge conver‑sion at cryogenic temperatures in bilayers of (Bi,Sb)2Te3/(Cr,Bi,Sb)2Te3, yielding an effective spin‑torque ratio ranging from 140 to 425 (REF. 74). Subsequent measure‑ments of spin–charge conversion at metallic ferromagnet/TI interfaces have produced various values, ranging from ~0.1 to ~20 (REFS 75–78). Thus, the promised efficiency of ‘topological spintronics’ has not been convincingly real‑ized. Measurements of TI/metallic‑ferromagnet bilayers face the primary difficulty of current shunting through the more‑conducting magnetic layer. To avoid this prob‑lem, recent spin–charge conversion measurements have focused on heterostructures that interface TIs with an insulating ferrimagnetic layer of YIG53,79. Systematic spin‑pumping measurements of Bi2Se3/YIG bilayers (FIG. 5b) as a function of the Bi2Se3 layer thickness53 show a correlation between the spin–charge conversion efficiency and the development of the Dirac cone. The spin–charge conversion efficiency increases monotonically as the layer thickness is increased from two to six quintuplet layers but saturates for larger thicknesses when the Dirac point is stabilized53. Such measurements are being extended to electrostatically gated devices, wherein the chemical potential can be tuned. This will allow the disentangle‑ment of contributions to the spin–charge conversion from bulk states (which also involve large SOC) and surface states. Recently, the direct measurement of magnetization switching through spin‑transfer torque in TI/ferromagnet bilayer devices has been reported80,81.

ThermoelectricityTetradymites were identified82 in the 1950s as the class of semiconductors with the highest TE efficiency. The intrinsically high values for the TE figure of merit (ZT, BOX 3) in binary Bi2Te3 and in p‑type Sb2Te3 arise from the coincidence of a number of electronic and lattice properties in these materials, which were enhanced over decades of materials engineering through alloying, doping and nanostructuring. In this section, we first describe the intrinsic electronic band structures and emphasize the communality with TIs. We then describe the intrinsic phonon properties, which affect ncrit and thus the criteria to make a TI material. Finally, we sum‑marize the materials‑engineering steps that resulted in record values of ZT (1.85) in 2015 (REF. 83). There is a similarity between the progress in this research and that deployed to develop TIs.

The strong role of SOC in the band structures of tetra dymites results in small electronic effective masses, promoting high electron mobilities and electrical

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conductivity, because mobility scales with the effective mass as μ = eτ/m*, where τ is the average scattering time. By contrast, the thermopower of metals and degener‑ately doped semiconductors is given by

S = π2

3kB

ekBTEF

(where kB is the Boltzmann constant), assuming that electrons are mostly scattered by acoustic phonons, and is inversely proportional to the Fermi energy, EF. To simul‑taneously obtain a high electrical conductivity and high thermopower in a solid, a high carrier concentration and a low value of EF are needed; this condition is met in solids with a large mDOS. This appears to conflict with the small effective mass values that result from SOC. However, with increasing SOC, the bands become multi‑valleyed (BOX 1). The valley degeneracy, N, enables p‑type Sb2Te3 to have simultaneously high mobility from the small transport

mass and high thermopower for a given carrier con‑centration from the large mDOS ( mDOS = N 2/3 √m1m2m3

3 ). The thermopower is even higher in tetradymites that have other band extrema along the Z–F direction and can be further engineered (BOX 3).

Phonon propertiesAs expected for heavy p‑block semiconductors, the intrinsic phonon properties of tetradymites produce a very low intrinsic lattice thermal conductivity, κL. The total and lattice thermal conductivities measured in single crystals of binary tetradymites as a function of temperature1 are shown in FIG. 6a. The lattice com‑ponent κL = κ − κE (where κE is the electrical conduc‑tivity) is shown for Bi2Te3, but only κ is reported for Bi2Se3 and Sb2Te3; the value is dominated by κL up to at least 200 K. In these single crystals, the κL near room temperature is intrinsic and dominated by

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–1.0

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Figure 4 | Quantum anomalous Hall effect. a | Precise observation of the quantization in the quantum anomalous Hall effect in a magnetically doped topological insulator (Cr‑doped (Bi,Sb)2Te3). Ryx is the Hall resistance, Rxx is the longitudinal resistance, μ0 is vacuum permeability and H is the magnetic field. b | The magnification shows that Ryx = h/e2 with a precision of 3 parts in 104 for the indicated gating values. c | The atomic force microscopy image shows a rough surface, the texture of which arises principally from structural defects. This image demonstrates that, surprisingly, this precise quantization occurs in samples with significant structural disorder. d | Scanning superconducting quantum interference device (SQUID) images of a quantum anomalous Hall insulator sample (Cr‑doped (Bi,Sb)2Te3) close to switching at opposite magnetization states in a hysteresis loop. Note that the domains are strongly pinned by disorder, such that the two scans are almost mirror images of each other. Perhaps this is the best example of ‘topological protection’, but it only occurs when the 2D surface states have been converted into 1D edge states by breaking time‑reversal symmetry. Panels a and b are adapted with permission from REF. 71, AAAS. Panel d is adapted with permission from REF. 72, AAAS.

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Nature Reviews | Materials

µ0H (T)

–20

–10

0

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ix (µ

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DataSymmetric componentAntisymmetric component

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e–

IRF

Hext

Bi2Se

3

Permalloy

TI

YIG

τ⊥

τII

φ

HH

RF

JS

V

HRF

Figure 5 | Topological spintronics. a | Schematic geometry for spin‑torque ferromagnetic resonance (FMR) measurements of charge‑to‑spin conversion at the interface between a topological insulator and a ferromagnet (top). A microwave current, IRF, is driven through a topological insulator/ferromagnet bilayer (in this case, Bi2S3/permalloy) in the presence of an in‑plane magnetic field, Hext. A dc mixing voltage is measured to probe the torque, τ, exerted on the ferromagnet by the spin current. The bottom panel shows an exemplary spin‑torque FMR signal measured in a Bi2Se3/permalloy bilayer at 300 K. The data can be deconstructed into the contribution from an in‑plane spin‑transfer torque (symmetric signal) and an out‑of‑plane field‑like torque (anti‑symmetric signal). b | Schematic representation of a typical spin‑pumping experiment for measuring the spin‑to‑charge conversion at topological insulator/ferromagnet interfaces (top). The spin current, JS, generated by the precessing magnetization in the ferromagnet (in this example, yttrium iron garnet, YIG, a ferrimagnet that behaves like a ferromagnet) is detected as a dc voltage in the topological insulator (TI) through the inverse Rashba–Edelstein effect, in which a 3D spin current is converted to a voltage difference in a 2D system. The bottom panel shows an example of spin‑pumping data observed in a Bi2Se3/YIG bilayer at 300 K. The field‑ antisymmetric peaks are centred on the ferromagnetic resonance frequency of the YIG layer. The amplitude of each resonance peak is the spin‑pumping voltage signal, originating from the conversion of a spin current into a charge current. H, magnetic field; f, frequency; HRF, Oersted field; V, voltage. Panel a is adapted with permission from REF. 19, Macmillan Publishers Limited. Panel b is adapted with permission from REF. 53, American Physical Society.

anharmonic phonon–phonon scattering. The intrinsic value of κL is low for several reasons. First, Bi, Sb, Te and Se are some of the heaviest elements in the periodic table, and their chemical bonds are weak, resulting in low sound velocities. Second, the large unit cells result in small Brillouin zones and low phonon cut‑off wave vectors. Third, the bonds are anharmonic; the anhar‑monicity determines the scattering probability of phon‑ons during phonon–phonon interactions, as quantified by the Grüneisen parameter, which is the logarithmic derivative of the frequency, ω, with respect to the vol‑ume of the crystal, V (γi(ω) ≡ −V/ωi ∙ ∂ωi/∂V, where i is the index of the phonon mode). Intuitively, the deforma‑tion of the lattice induced by the presence of one phonon changes the local acoustic properties around it by the relative amount γi(ω); thus, a second phonon interacting

with the first sees a material region with an acoustic mismatch compared with the rest of the solid, and the degree of mismatch determines the probability that the second phonon is reflected84. The effect of anharmonic scattering in tetradymites is illustrated by the calcula‑tion85 of the linewidth (which is inversely proportional to the phonon–phonon scattering lifetime) of the phonon dispersion relations (FIG. 6b). At 77 K, the phonon dis‑persion lines are sharp and well defined. The significant broadening of the linewidths observed at 450 K indicates that the phonon lifetimes are reduced by the tempera‑ture increase. Only the low‑lying acoustic modes remain sharp. Several optical modes become diffuse beyond rec‑ognition. The phonons with ω <2 THz, which carry 60% of the heat85 at 300 K, have linewidths that are extremely affected by anharmonic scattering.

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Anharmonicities find their physical origin in a res‑onant bond86, which is a hybridization among different electronic configurations. Three valence p electrons on the pnictogen alternate their occupancy of six available covalent bonds that exist between a given atom and its octahedral neighbours. This occurs in tetradymites because of the octahedral coordination of atoms and weak sp hybridization (the s electrons form bands at ener‑gies well below those of the bands of the p orbitals). As a result, the bonds are highly polarizable, which correlates with high anharmonicity87. The deformation and polar‑izability of electron clouds resulting from atomic motion have other magnetic88 and electronic89 consequences. In particular, they must be related to the high static dielectric constants and to the large Bohr radius values reported in TABLE 2. The Lyddane–Sachs–Teller relation90 (εdc/ε∞ = ωL

2/ωT2, where ωL and ωT are the frequencies of

the longitudinal and transversal optical phonon modes, respectively) suggests a fairly large and anisotropic contribution to εdc due to phonons near the separation between longitudinal and transverse optical phonons of approximately 2 THz (FIG. 6). Te atoms in Bi2Te3 have a large charge shell89, suggesting that a motion of Te atoms in longitudinal and/or transverse modes is at the origin of the large value of εdc. However, further work is needed to identify the exact mechanism.

Optimizing the figure of meritWe now describe the engineering that resulted in the record ZT ≈ 1.85 in tetradymite alloys. The best p‑type TE tetradymites are Sb‑rich (Bi2Te3)–(Sb2Te3) alloys with compositions ranging from (Bi0.20Sb0.80)2Te3 to (Bi0.25Sb0.75)2Te3 (REF. 91). The n‑type material options are broader, but n‑type tetradymite alloys rarely attain the same ZT values as p‑type materials. The first family of n‑type tetradymites are Te‑rich (Bi2Te3)–(Bi2Se3) alloys with compositions between Bi2Te3 and the line com‑pound Bi2Te2Se (REF. 92), such as Bi2(Te0.8Se0.2)3, and often Bi2(Te0.98Se0.02)3 to Bi2(Te0.95Se0.05)3 (REF. 91). Quaternary alloys also have been studied93.

The TE power factor (S2σ) is enhanced by designing alloys with a valley degeneracy larger than that of the binary alloys. In p‑type materials, this is done by alloy‑ing Bi2Te3 with Sb2Te3. It is possible to find a composition near (Bi0.2Sb0.8)2Te3 for which the two valence band max‑ima have the same energy, such that the valley degeneracy of N = 6 becomes N = 12 (REF. 1). Therefore, compositions between (Bi0.2Sb0.8)2Te3 and (Bi0.25Sb0.75)2Te3 are typically used for p‑type materials.

There is an optimum concentration of electrons that maximizes ZT in TE materials, which can be reached by either controlling the defect chemistry or doping with aliovalent elements, as previously discussed. Particularly, the resonant acceptor Sn (REF. 36), which is instrumental in synthesizing electrically insulating Bi2Te2Se, falls in the valence band of Bi2Te3, contributing to conduction and increasing the p‑type Bi2Te3 power factor94.

Strong anisotropy91 is inherent in the R3–m crystal structure, but because phonons dominate the thermal conductivity, the anisotropy of S and of the resistiv‑ity, ρ, can be different from that of κ. Generally, with

Phonons

High valley degeneracy, N

Small effective masses

High polarizability

Small Brillouin zones

Octahedral bonds

Low group velocity

High S2n

Low lattice thermal conductivity, κL

High Bohr radius, aB*

High mobility, µ Defect insensitivity, μ/κL

High static dielectric constant, ε

0

High Grüneisen parameter, γ

Nature Reviews | Materials

ZT ≡S2σκT = eT(S2n) =

S2

L1

1 + κLκE

μκE

+ κL

⎞⎠

⎛⎝

Electrons (spin–orbit coupling)

Box 3 | Contribution to the figure of merit in tetradymites as thermoelectrics

The thermoelectric figure of merit of a semiconductor is written as ZT = TS2σ/κ, where S is the thermopower and σ and κ are the electrical and thermal conductivity of the material. ZT governs the thermal efficiency of thermoelectric generators and Peltier coolers. Because σ = neμ, where n is the carrier concentration, e is their charge and μ is their mobility, ZT can be re-written as the product of two mutually counter-indicated properties: the product S2n and the ratio μ/κ (see the figure). It is difficult to increase the value of S2n because, in solids, S generally decreases with increasing n. The same is true for the ratio μ/κ because defects that limit κ can also limit μ. Both phonons and electrons contribute to the thermal conductivity: κL is the lattice contribution, whereas the electronic contribution is related by the Wiedemann–Franz–Lorenz relation to the electrical conductivity, κE = LTσ, where L is the Lorenz ratio. For a free electron, L = 2.48 × 10−8 V2 K−2; in real solids, L can deviate from this value and is on the order of 2 ± 0.5 × 10−8 V2 K−2. ZT can be further re-written, as shown in the figure, to highlight the importance of the thermopower to ZT, and, in fact, it is not the total thermal conductivity that needs to be minimized but only the lattice contribution.

Spin–orbit coupling (SOC) results in small carrier effective masses, which have an ambiguous influence on S2σ. SOC increases the mobility, but in Bi2Se3, where there is only a single valley on the Fermi surface, SOC results in a low value of S2n. This is remedied in tetradymites with Bi2Te3-like bands, in which the increased valley degeneracy compensates for the problem: indeed, for a given total carrier concentration summed over all valleys, the Fermi level within each valley can be kept low enough to produce a high thermopower. The phonon structure in these materials is also very favourable for a high ZT. Intrinsically, the heavy atomic masses and weak bonds result in low phonon group velocities and small Brillouin zones. The high polarizability of some bonds and the octahedral bonding result in anharmonic phonon behaviour, intrinsically limiting the lattice thermal conductivity, κL. Combined with the high polarizability of specific chemical bonds (see main text), which gives rise to a high dielectric constant, the small effective masses due to SOC result in a large Bohr radius. This makes the electrons in tetradymites particularly impervious to mobility losses because of scattering on defects, especially point defects. Thus, a series of avenues can be exploited to induce phonon scattering but not electron scattering, such as nanostructuring and alloying. Classes of semiconductors in which all these favourable properties coincide are quite rare: perhaps, the only other examples are rocksalt IV–VI compounds (the lead salts).

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exceptions, the thermopower is isotropic. Typically, in Sb‑rich (Bi1 − xSbx)2Te3 alloys, the anisotropies of κ and ρ compensate each other, and ZT is isotropic. In the n‑type Te‑rich Bi2(Te1 − xSex) alloys there is no such compensa‑tion. The anisotropy of κL in TE tetradymites is roughly κ///κ ≈ 2, but the resistivity of n‑type Bi2Te3 is more aniso‑tropic than that of Sb2Te3; thus, n‑type Bi2(Te1 − xSex) alloys have a higher ZT in the trigonal plane than in the direc‑tion perpendicular to it. Because practical TE materials are polycrystalline, obtaining a good ZT in n‑type alloys requires preferential grain alignment.

Nanostructuring. Engineering TE materials requires adding phonon‑scattering mechanisms to the intrinsic anharmonic phonon scattering. This involves scattering phonons at various frequencies while monitoring the amount of heat carried by the phonons at such frequencies.

Point‑defect scattering occurs when foreign atoms are added at the atomic level. This is done by alloying, for example, by substituting Sb for Bi or Se for Te. These substitutions can also tune the defect chemistry and thereby the optimum doping level; other point defects include vacancies and isotopes. The length scale for point‑defect scattering is the interatomic spacing or the unit cell. Point defects scatter phonons at a frequency that follows the ω4 law for Raleigh scattering. Therefore, they are most effective in impeding the propagation of phon‑ons at the highest energies. The effective Bohr radius in the perpendicular direction to the c axis, aΒ⊥ *, typically extends over 10 unit cells, making the electron mobil‑ity less sensitive to point defects compared with κL. As a result, the decrease in mobility is less than the decrease in κL, making alloy scattering the oldest and most effective means of improving ZT.

Boundaries, grains and nanometre‑scale inclusions all limit the phonon mean free path roughly to their physical size. The scattering frequency scales with the phonon group velocity, which is constant at low fre‑quencies. Therefore, the scattering frequency scales as ω0. The group velocity decreases for phonons with larger momentum, and therefore, boundary scattering is effec‑tive for decreasing κL, particularly because of low‑energy phonons. The average size of obstacles must be opti‑mized to scatter phonons more than electrons, which is accomplished by calculating the thermal conductivity accumulation85 as a function of the mean free path. The thermal conductivity is an integral of the heat carried by phonons of different modes (index i), frequencies (ω) and mean free paths (ℓφ) under the influence of a tem‑perature gradient. The thermal conductivity accumula‑tion, κLC(ℓφ), is therefore the heat carried by phonons of a mean free path shorter or equal to ℓφ, calculated from phonon dispersions assuming anharmonic scattering85. The electronic mean free path, ℓe, is determined from the mobility and the band structure. The optimal nano‑structure size is larger than ℓe but has a maximal effect on κLC(ℓφ). In commercial alloys, ℓe is on the order of 6 ± 1 nm for electrons and 11 ± 1 nm for holes; therefore, structures with a diameter of ~20 nm should have a small effect on electrons but reduce the in‑plane κL by 30%1.

Nanostructures can be obtained in bulk materials through top‑down or bottom‑up synthesis. In top‑down synthesis, nanostructured powders are prepared by ball‑milling or melt‑spinning and are then sintered into a composite, typically by spark‑plasma sintering. Top‑down nanostructuring was used, for example, on p‑type (Bi,Sb)2Te3 ingots95. Nanostructured grains with dimensions ranging from 5 to 50 nm, but typically

0 100

100

10

1200 300 400

Freq

uenc

y (T

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1

Γ Z F Γ L

2

3

4

5

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Ther

mal

con

duct

ivit

y (W

m–1

K–1

)

Temperature (K)

Sb2Te

3 (κ)

Bi2Se

3 (κ)

Bi2Te

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Bi2Te

3 (κ

L = κ – κ

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a

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Γ Z F Γ L

2

3

4

5b77 K 450 K

Figure 6 | Phonons in tetradymites. a | The lattice thermal conductivity of the binary tetradymites, compiled as in REF. 1. b | The calculated phonon dispersion of Bi2Te3 at 77 K and at 450 K, illustrating the low phonon velocities and also the fact that the low‑lying optical branches are well within reach of kBT (where kB is the Boltzmann constant and T is temperature) at 300 K and contribute significantly to heat transport. The linewidth reflects the phonon lifetime, which is limited by intrinsic effects, mostly anharmonicity. The lifetime of the phonons that carry the most heat are severely limited at 300 K, even in intrinsic binary Bi2Te3. The spectra of Bi2Se3 and Sb2Te3 are similar to those of Bi2Te3. Panel a is Copyright 2016© from Materials Aspect of Thermoelectricity by C. Uher. Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa plc; and REF. 126, Elsevier. Panel b is adapted with permission from REF. 85, American Physical Society.

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near 20 nm, were obtained, with the target range deter‑mined above. Consistent with the above model, κL in this material decreased by 20% near room temperature. Moreover, the power factor increased owing to an un ex‑plained increase in thermopower. As a result, the ZT of the nanostructured material increased by 20% at room temperature.

In the bottom‑up approach, nanostructures grow intrinsically in the material following appropriate heat treatment, much like the case for pearlitic steels. In the (Bi,Sb)2Te3 alloy system, samples95 may also show evi‑dence of the presence of nanoprecipitates; Te precipitates with diameters between 5 and 30 nm were observed, as well as Sb‑rich nanodots with diameters between 2 and 10 nm. The pseudobinary [(Bi0.25Sb0.75)2Te3]–Te phase diagram (see the ASM Alloy Phase Diagram Database) shows the existence of a Te‑rich (~93 at% Te) eutectic phase with a melting point of ~420 °C. If the melt is deliberately made rich in Te, some of this phase will pre‑cipitate as inclusions with sizes that are dependent on the cooling history of the sample. With the appropriate heat treatment, this phase can give rise to nanometre‑scale inclusions. Because many binary and pseudobinary phase diagrams have a similar shape, with one or two eutectics to the left or to the right of the line compound, this technique is of general applicability.

Dislocations — lines of scattering centres that scatter mid‑energy phonons through most of the 109–1012 Hz fre‑quency range96 — have also been introduced into alloys. Dislocations can be charged or form electric dipoles, and they accumulate at grain boundaries and can catalyse an inhomogeneous distribution of vacancies. Although the preliminary results seem very favourable, the study of the use of dislocations in TEs and their interactions with other TE parameters is new. The scattering frequency of phonons on dislocations scales with ω2: this complements the boundary scattering in (or at) nanostructures, which scales as ω0, and the point defect scattering in alloys, which scales as ω4. A full‑spectrum phonon‑scattering approach that uses all of these mechanisms together was developed83 with p‑type (Bi0.25Sb0.75)2Te3. The thermal conductivity and TE transport properties were com‑pared for samples created with three preparation tech‑niques: a conventional stoichiometric ingot of Te‑rich Bi10Sb28.9Te61.1; the same material melt‑spun and sintered; and the melt‑spun material mixed with a very large (25%) Te excess and then sintered. In the latter technique, at 420 °C, the excess Te forms the eutectic phase, which is expelled as a liquid during the sintering process. The resulting material has a complicated morphology83 with small grains95 and a new feature: dense dislocation arrays at grain boundaries. The thermal conductivity is decreased, which can be attributed to a reduction in κL to just above 0.3 W m−1 K−1 in the 300–350 K tempera‑ture range, without affecting the electronic properties, which results in the current highest ΖΤ value (~1.85) in tetradymites (FIG. 7).

The value of κL in n‑type Bi2(Te1 − xSex) alloys with x = 0.07 has also been reduced97 by a multi‑scale, full‑spec‑trum approach based on nanostructuring and disloca‑tion scattering98. A Bi2(Te0.93 Se0.07)3 material, prepared

a

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Nature Reviews | MaterialsFigure 7 | Thermoelectric figure of merit. This figure summarizes some of the highest values of the figure of merit, ZT, reached over the years in p‑type (top) and n‑type (bottom) materials. a | Ball‑milled sintered (Bi0.25Sb0.75)2Te3 (REF. 100). b | Ball‑milled sintered (Bi0.2Sb0.8)2Te3 (REF. 101). c | Melt‑spun sintered (Bi0.24Sb0.76)2Te3 (REF. 102). d | Sintered nanopowders of (Bi0.25Sb0.75)2Te3 with 20–30 nm diameters103. e | Ball‑milled sintered chalcogen‑rich material of nominal composition Bi0.42 Sb1.51Te3.05Se0.02 (REF. 104). f | p‑Type Pb‑doped (Bi0.24Sb0.76)2(Te0.97 Se0.03)3 and n‑type SbI3‑doped Bi2(Te0.8 Se0.2)3 + δ with excess Te (REF. 105). g | p‑Type (Bi0.25Sb0.75)2Te3 and n‑type Bi2(Te0.9 Se0.1)3 (REF. 106). h | p‑Type (Bi,Sb)2Te3 and n‑type Bi2Te3 (REF. 107). i | n‑Type Bi2Te2Se and p‑type (Bi0.25Sb0.75)2Te3 (REF. 108). j | Highly plastically deformed n‑type Bi2(SeTe)3 and p‑type (BiSb)2Te3 (REF. 109). k | Te‑rich nanostructured Bi10Sb28.9Te61.1 with dislocation arrays83. l | Sintered nanopowders of Bi2Te3 prepared via a hydrothermal process110. m | Bi2Te3 with 10 vol% Bi2Se3 nanoparticles111. n | Sintered nanorods of Bi2Te3 with the nominal composition Bi2Te2.4 (REF. 112). o | SbBr3‑doped Bi2(Se0.2Te0.8)3 (REF. 113). p | n‑Type (Bi1 − xSbx)2Te3 alloys, which show a maximum ZT at x = 0.65–0.70 (REF. 114). q | Aligned and plastically deformed Bi2(Te0.93Se0.07)3 (REF. 97). Unless otherwise indicated, the materials were not doped with aliovalent donor or acceptor impurities.

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commercially through zone‑melting, was aligned by applying a hot deformation process that created a very high dislocation density97. After optimization, the in‑plane thermal conductivity in a sample showing highly distorted regions at the 10–50 nm scale was approximately 25% lower than that of the pristine zone‑melting material. Inside the distorted regions, arrays of dislocations with 3 nm spacing constituted a multi‑scale set of phonon scattering centres; the introduction of dislocations in the zone‑melting sample had an n‑type dopant effect. The effect of the hot deformation process on the electronic properties was much smaller than that on κL, and thus, a high ZT was achieved (FIG. 7).

The ZT values resulting from the optimization of p‑ and n‑type tetradymites1 are summarized in FIG. 7. The ZT of n‑type Te‑rich Bi2(Te1 − xSex) alloys is generally lower than that of p‑type Sb‑rich (Bi1 − xSbx)2Te3 alloys, because the higher band degeneracy increases the power factor of p‑type alloys, and the need for preferentially oriented n‑type alloys complicates the implementation of new research ideas in these compounds.

ConclusionsTetradymites have proven to be important materials from both a technological and fundamental perspective. Their application in practical thermoelectric devices has motivated the study of their defect structure. This study

also resulted in the development of an in‑depth knowl‑edge of their phonon structures and contributed to our understanding of the electronic properties of narrow‑gap semiconductors. Furthermore, their utility as hosts for topological surface states has motivated their growth as high‑quality TIs with low defect concentrations, which led to the observation of new electronic states of matter. In short, the effects of SOC dominate the band struc‑tures of the tetradymites and are central to both their TE and TI properties. By contrast, TEs must lie on the metal side of the Mott metal–insulator transition and TIs on its insulating side. This sets stringent requirements on the charge carrier concentrations that must be achieved in both classes of materials (<1016 cm−3 for TIs and 1019–1020 cm−3 for TEs). The techniques used to reach these goals are remarkably similar: control of the defect chem‑istry and aliovalent doping. Tetradymites are attractive for both theorists and experimentalists, because they are complex enough to be interesting to study yet not too complex to understand, as demonstrated by the num‑ber of publications based on tetradymites over the past 50 years. It is doubtful that the properties and potential of tetradymites have been fully uncovered. Current and future generations of scientists will find much to think about when considering tetradymites, and the rich lit‑erature of previous studies will form fertile ground for planting the discoveries of the future.

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AcknowledgementsThe research of R.J.C. and N.S. on topological insulators was supported by the Army Research Of f i ce (ARO) Multidisciplinary University Research Initiative (MURI) (Grant No. W911NF-12-1-0461) and the National Science Foundation (NSF) Materials Research Science and Engineering Center (MRSEC) (Grant No. DMR-1420451). N.S. acknowledges the Pennsylvania State University Two-Dimensional Crystal Consortium–Materials Innovation Platform (2DCC–MIP), which is supported by the NSF coop-erative agreement DMR-1539916. The work of J.P.H. and R.J.C. was supported by the Air Force Office of Scientific Research (AFOSR) MURI on thermoelectrics (Grant No. FA9550-10-1-0533). J.P.H. is currently supported by the NSF (Grant Nos EFRI-1433467 and DMR-142051); he

acknowledges useful discussions with B. Wiendlocha, W. Windl and T. M. McCormick.

Competing interests statementThe authors declare no competing interests.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

How to cite this articleHeremans, J. P., Cava, R. J. & Samarth, N. Tetradymites as thermoelectrics and topological insulators. Nat. Rev. Mater. 2, 17049 (2017).

DATABASESASM Alloy Phase Diagram Database: http://mio.asminternational.org/apd/index.aspxICSD database: http://icsd.fiz‑karlsruhe.de

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