Mapping the band structure of GeSbTe phase change alloys around the Fermi level

Abstract

Phase change alloys are used for non-volatile random-access memories exploiting the conductivity contrast between amorphous and metastable, crystalline phase. However, this contrast has never been directly related to the electronic band structure. Here we employ photoelectron spectroscopy to map the relevant bands for metastable, epitaxial GeSbTe films. The constant energy surfaces of the valence band close to the Fermi level are hexagonal tubes with little dispersion perpendicular to the (111) surface. The electron density responsible for transport belongs to the tails of this bulk valence band, which is broadened by disorder, i.e., the Fermi level is 100 meV above the valence band maximum. This result is consistent with transport data of such films in terms of charge carrier density and scattering time. In addition, we find a state in the bulk band gap with linear dispersion, which might be of topological origin.

Introduction

Phase change alloys are the essential components for optical data storage (DVD-RW, Blu-ray Disc) and for electrically addressable phase-change random-access memories (PC-RAM)^1,2. The latter are envisioned to become more energy efficient using interfacial phase-change memories, whose phase change has been related to a topological phase transition³. Phase change alloys are typically chalcogenides consisting of Ge, Sb, and Te (GST) with Ge₂Sb₂Te₅ (GST-225) being the prototype^1,4. They exhibit three different structural phases: an amorphous, a metastable rock salt, and a stable trigonal phase. Switching the system from amorphous to metastable leads to a large contrast in electrical conductivity and optical reflectivity, which is exploited for data storage^5,6. Such switching favorably occurs within nanoseconds^7,8 and at an energy cost down to 1 fJ for a single cell⁹.

The technologically relevant, metastable phase¹⁰, usually obtained by rapid quenching from the melt, has a rock salt like structure with Te atoms at one sublattice and a mixture of randomly distributed Ge, Sb, and vacancies (Vcs) on the other sublattice (Fig. 1a, b)^11,12,13,14. The stable phase consists of hexagonally close-packed layers of either Ge, Sb, or Vcs with hexagonal layers of Te in between. Hence, the Vc layers bridge adjacent Te layers^15,16. The stable phase has trigonal symmetry and is distinct in stacking of the hexagonal layers from the regular ABC stacking within the rock salt like metastable phase (Fig. 1c, d).

Fig. 1

Structural models of Ge₂Sb₂Te₅ (GST-225). a Metastable rock salt structure: red circles: Te, striped circles: Ge, Sb, or vacancy (Vc); (111) planes are marked with alternating colors (red: Te, turquoise: Ge/Sb/Vc); b DFT-optimized structure of the disordered subsurface layer of a GST-225 slab, blue: Ge, green: Sb; c layer structure of one unit cell of metastable rock salt GST-225 exhibiting ABC stacking; the 2 × 3 layers in the unit cell result from the two chemically distinct layers and the three distinct stacking positions; d same as c, but for the cubic Petrov phase of GST-225 exhibiting a unit cell of 10 × 3 layers with 10 layers due to the alternating chemistry and a threefold repetition due to the ABC stacking

In the metastable phase, the disorder on the (Ge, Sb, and Vc) site leads to Anderson localization of the electrons¹⁷. The localization is lifted by annealing due to the respective continuous ordering of the Ge, Sb, and Vcs into different layers^18,19,20. This is accompanied by a shift of the Fermi level E_F toward the valence band (VB)^17,21. However, the corresponding Fermi surface is not known, as well as the exact position of E_F, such that it is difficult to understand the electrical conductivity in detail.

Most of the electrical transport measurements so far were conducted using polycrystalline GST^17,22, such that many established tools requiring crystallinity of the samples could not be applied. Only recently, epitaxial films of single-crystalline quality have been achieved by molecular beam epitaxy (MBE)^{20,23,24,25,26}. These films have been probed so far by X-ray diffraction (XRD), electron microscopy^{20,23,24,25,26,27}, magnetotransport studies²⁰, Raman spectroscopy, and Fourier transform infrared spectroscopy^26,28. Most importantly, it was found that the epitaxial GST films are in the technologically relevant rock salt phase, but often exhibit ordering of the vacancies in separate layers²⁰.

Here we provide the first detailed measurement of the band structure of such epitaxial films by angular resolved photoelectron spectroscopy (ARPES). We focus on the nominal composition GST-225, and employ an ultrahigh-vacuum (UHV) transfer from the MBE system to prevent surface oxidation²⁹ (Methods section). Within the whole-Brillouin zone (BZ), we find an M-shaped bulk VB in all directions parallel to the surface. This is in qualitative agreement with density functional theory (DFT) calculations of the cubic adaption of the trigonal Petrov phase^15,30, sketched in Fig. 1d. For brevity, we call this structure the cubic Petrov phase. Connecting the VB maxima of the experimental data results in a hexagonal tube at an energy about 100 meV below E_F. Hence, the classical Fermi volume of a strictly periodic system would be zero, which contradicts the observation of metallic conductivity²⁰. This apparent contradiction is solved by the significant broadening of the E(k) states due to disorder, such that there is still considerable weight of the valence band states above E_F. The sum of these weights results in a charge carrier density n_eff consistent with the charge carrier density obtained from Hall measurements of the MBE films. The width of the states is, moreover, compatible with the scattering time deduced from the transport data. Such a detailed description of electrical transport provides a significant improvement over more simplistic models based on a parabolic and isotropic valence band as used so far^17,22.

Additionally, we find an electronic band within the fundamental bulk band gap of the metastable phase by two-photon ARPES. This band exhibits a largely isotropic, linear dispersion and circular dichroism, such as known for topological surface states (TSS)^31,32,33. We also find states close to the VB maximum with a strong in-plane spin polarization perpendicular to k by conventional ARPES again similar to TSSs. A non-trivial topology of GST-225 has indeed been predicted for certain stacking configurations by DFT calculations^{34,35,36,37,38} and has been conjectured from the M-type VB dispersion³⁹. Assuming that the Dirac-type state is indeed a TSS and, hence, cuts E_F, it would contribute to the electronic transport. It would even dominate the conductivity, if its mobility μ >0.1 m² (Vs)⁻¹. This is lower than the best TSS mobilities found in other topological insulators, such as Bi₂Se₃ and BiSbTeSe₂ films (μ ≃ 1 m² (Vs)⁻¹)^40,41.

Results

Constant energy surfaces

ARPES experiments were performed at 29 different photon energies hν = 16–31 eV with a step size of 0.5 eV. This allows a detailed determination of the k _z dispersion of the bands (k _z : wave vector perpendicular to the surface). Using the estimated crystal potential E_inner = 14 eV (Methods section), the chosen hν relate to k _z  = 2.55–3.09 Å⁻¹. The ARPES spectra show an inverted M-shaped VB in energy-momentum cuts (EMCs) taken along the surface plane (Fig. 2d). The independently measured Fermi level $E_{\mathrm{F}}^{{\mathrm{PES}}}$ is well above the VB maximum. Both are in line with earlier, less extensive results³⁹. We label $E_{\mathrm{F}}^{{\mathrm{PES}}}$ additionally with the superscript PES, since it differs from $E_{\mathrm{F}}^{{\mathrm{DFT}}}$ in DFT calculations with respect to the VB maximum. Figure 2a displays constant energy cuts (CECs) of the normalized photoelectron intensity (methods) at $E_{\mathrm{F}}^{{\mathrm{PES}}}$ for selected k _z . To determine peak positions, momentum distribution curves (MDCs) are extracted and fitted by Voigt peaks (Fig. 2b, c). The resulting peak momenta k form a hexagonal tube (Fig. 2e, f) called the pseudo Fermi surface of GST-225. We call it pseudo, since the peak energies E_peak(k) resulting from fits of energy distribution curves (EDCs) do not cross $E_{\mathrm{F}}^{{\mathrm{PES}}}$ for any k, as visible, e.g., in Fig. 2d. Consequently, there are no band centers at $E_{\mathrm{F}}^{{\mathrm{PES}}}$ as required for a conventional Fermi surface⁴². Only the tails of the broadened energy peaks cross E_F. The sizes of the hexagons of the pseudo Fermi surface slightly vary with hν, i.e., along k _z , with minimal diameter at hν = 21 eV (arrow in Fig. 2f). We conjecture (in accordance with DFT) that this minimum corresponds to the BZ boundary and, hence, use it to determine E_inner = 14 eV, unambiguously relating hν to k _z (Methods section). For the Fermi wave number in x (y) direction, we find k_F,x = 1.52 ± 0.3 nm⁻¹ (k_F,y = 1.43 ± 0.2 nm⁻¹), where the ± interval describes the full variation along k _z . Hence, with a precision of 20%, the pseudo Fermi surface is a hexagonal tube without dispersion along k _z .

Fig. 2

ARPES and DFT data for different photon energies hν. a Constant energy cuts (CECs) at the Fermi level $E_{\mathrm{F}}^{{\mathrm{PES}}}$ for different hν. We deduce k _z as marked on the left from hν using an inner potential E_inner = 14 eV (Methods section); b CEC at $E_{\mathrm{F}}^{{\mathrm{PES}}}$ for k _z  = 2.83 Å⁻¹ with direction equivalent to k _y marked by a dotted black line; c momentum distribution curve (MDC) along dotted black line in b; two Voigt peaks (red line) are fitted to the ARPES data (blue dots); d Energy-momentum cut (EMC) along the dotted line marked in b at k _z  = 2.83 Å⁻¹, dashed line marks $E_{\mathrm{F}}^{{\mathrm{PES}}}$; e Brillouin zone (BZ) of metastable, disordered rock salt GST-225 (structure in Fig. 1a) with principal k directions marked; the measured pseudo Fermi surface is shown in gold, including the missing part due to vanishing photoelectron intensity (Methods section); the Fermi lines cutting the BZ side planes in (111) direction are drawn in red; f Voigt peak positions at $E_{\mathrm{F}}^{{\mathrm{PES}}}$ from MDC fits as in c; colors are alternating in Δk _z  = 0.12 Å⁻¹, being the k _z size of the BZ of the cubic Petrov phase as used in the DFT calculations; the tentative BZ boundary of the metastable rock salt phase at k _z  = 2.72 Å⁻¹ is marked by a black arrow; it is used to determine E_inner; the additionally given photon energies (right scale) correspond to the average diameter of the hexagons; g E_peak,1 (k) at $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = −200 meV after backfolding along k _z into the BZ of the cubic Petrov phase (same color code as in f); h constant energy surfaces (CESs) of the valence bands (VBs) of the cubic Petrov phase according to DFT at the experimental $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = −200 meV, i.e., after shifting $E_{\mathrm{F}}^{{\mathrm{DFT}}}$ by 100 meV upward, such that the VB maxima in DFT and ARPES match

The EDCs consist of up to two peaks down to $E - E_{\mathrm{F}}^{{\mathrm{PES}}} = - 1$ eV for all probed k. These peaks are fitted by two Voigt peaks with peak centers E_peak,j(k) (j = 1, 2). The highest peak energy for all k, i.e., the VB maximum, is found at $E_{{\mathrm{peak}},1}({\bf{k}}) - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = −105 ± 10 meV with k = (0 ± 0.02, 1.53 ± 0.02, and 25.8 ± 0.2) nm⁻¹, as well as at equivalent k points. Projecting back to the first BZ, we get k = (0, 1.53, 1.6) nm⁻¹, i.e., the VB maximum is offset from Γ also in k _z direction.

Constant energy surfaces (CESs) of E_peak,j(k) are constructed below the VB maximum^42,43,44,45. They are compared with CESs from DFT calculations, which require periodic boundary conditions, i.e., a distinct order within the Ge/Sb/Vc layer. We have chosen the cubic Petrov phase (Fig. 1d) to represent the metastable ABC stacking of the rock salt structure employing chemically pure Sb, Ge, and Vc layers^30,39. Since the corresponding DFT BZ is reduced in k _z direction by a factor of 5 with respect to the disordered rock salt phase (Fig. 1a–c), the ARPES data have to be back-folded into a k _z range of Δk _z  = 0.12 Å⁻¹ for comparison. Therefore, the measured k _z data are divided into parts covering Δk _z  = 0.12 Å⁻¹ each (see color code in Fig. 2f) and projected accordingly. Results at $E - E_{\mathrm{F}}^{{\mathrm{PES}}} = - 200$ meV are shown in Fig. 2g, where each MDC has been fitted by four Voigt peaks as exemplary shown in Fig. 3g. The qualitative agreement with the DFT CESs (Fig. 2h) is reasonable, in particular, for the outer hexagon. Such agreement is also found at other energies as shown in Fig. 3a–f, where the different k _z values are projected to the (k _x , k _y ) plane. However, quantitative differences are apparent as discussed in Supplementary Note 1.

Fig. 3

Comparison between ARPES and DFT at different binding energies $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$. a–c Voigt peak positions E_peak,1(k), deduced from MDC fittings, projected to the (k _x , k _y ) plane; the marked $\overline {\mathrm{\Gamma }} \overline {\rm M}$ and $\overline {\mathrm{\Gamma }} \overline {\rm K}$ belong to a surface BZ projection; the color code is the same as in Fig. 2f–g; d–f corresponding CESs from DFT calculations exhibiting three Sb p-type VBs (different colors) within the BZ of the cubic Petrov phase; numberings in c and f label the different bands; g, h measured MDCs (black curves) along k_|| directions and at $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ as marked. Red fit curves consist of the displayed four Voigt peaks (blue, orange, yellow, and violet curve)

Effective charge carrier density from ARPES and magnetotransport

Next, we deduce the effective charge carrier density n_eff from the detailed mapping of the VBs by ARPES. Since the VB maximum is found 105 meV below $E_{\mathrm{F}}^{{\mathrm{PES}}}$ (Fig. 2d), one might conjecture the absence of a Fermi surface, i.e., n_eff = 0, at least close to the surface, i.e., at the origin of the ARPES signal. However, the bands are significantly broadened, such that their tails cut $E_{\mathrm{F}}^{{\mathrm{PES}}}$ (Fig. 2). Hence, the tails of the VB give rise to a non-vanishing n_eff. Accordingly, we replace the usual

$$n_{{\mathrm{eff}}} = \frac{2}{{8\pi ^3}}{\int}\!\!\!\!{{\int}\!\!\!\!{{\int}_{{\mathrm{Fermi}}\,{\mathrm{volume}}} {{\rm d}^3{\bf{k}}} } } ,$$

(1)

where the Fermi volume includes all occupied states, with:

$$n_{{\mathrm{eff}}} = \frac{2}{{8\pi ^3}}\mathop {\sum}\limits_j {\int}\!\!\!\!{{\int}\!\!\!\!{{\int}_{{\mathrm{BZ}}} {\alpha _j({\bf{k}})\,{\rm d}^3{\bf{k}}} } } .$$

(2)

The integral covers the whole BZ and includes the weight of each state above $E_{\mathrm{F}}^{{\mathrm{PES}}}$ (inset of Fig. 4a) according to

$$\alpha _j({\bf{k}}) = {\int}_{E_{\mathrm{F}}^{{\mathrm{PES}}}}^\infty \,p_{j,{\mathrm{norm}}}\left( {E,{\bf{k}}} \right){\rm d}E.$$

(3)

Here, p_j,norm (E, k) is the fitted EDC peak at k of band j, after normalizing its area to unity.

Fig. 4

Charge carrier density n_eff and curvature parameter $m^ \star$ deduced from ARPES. a EDC at k = (0.0, 0.13, 2.73) Å⁻¹ (black dots) with fit curve (red) consisting of two Voigt peaks (p₁ = green, p₂ = blue) multiplied by the Fermi distribution function f₀ (E, T = 300 K) (thin red line); inset: zoom around $E_{\mathrm{F}}^{{\mathrm{PES}}}$ displaying peak tails, after scaling each peak area to unity; only the colored tail of peak 1 (marked α₁) contributes to n_eff,1; b EMC at k_z = 2.73 Å⁻¹ with marked peak positions p₁ (green dots) and p₂ (blue dots) resulting from fits of EDCs as in a; parabolic fits to these points (accordingly colored full lines), used to determine $m^ \star$, are added; c histogram of resulting $m^ \star {\mathrm{/}}m_{\mathrm{e}}$ belonging to the band of p₁ using 90 different azimuths in (k _x , k _y ) direction for 29 different k _z -values; errors for individual $m^ \star {\mathrm{/}}m_{\mathrm{e}}$ are about the bin width; d red full line: n_eff as calculated from the ARPES data according to Eq. (2) for a hypothetically varying E_F with respect to the measured $E_{\mathrm{F}}^{{\mathrm{PES}}}$; different contributions from p₁ (green full line) and p₂ (blue full line) are labeled; error intervals are depicted by correspondingly colored dashed lines; charge carrier densities from Hall measurements (n_eff,H) of identically prepared GST films are added as a gray box; inset: α _j (k) (relative part of the Voigt peak p _j at k above $E_{\mathrm{F}}^{{\mathrm{PES}}}$) for the EMC of b

Figure 4a shows an exemplary EDC (black points) fitted with two Voigt peaks p _j (E, k) (blue and green line), which are multiplied by the Fermi distribution function f₀(E, T) at T = 300 K (thin, red line). This provides an excellent fitting result (thick red line). The weights of the two peaks above E_F are evaluated to be α₁(k) = 3% and α₂(k) = 1% (inset of Fig. 4a) (Methods section). Generally, we find α₁(k) ≤ 7% for 97% of the EDCs, where the largest α₁(k) are coincident with the maxima of E_peak(k). This is illustrated in the inset of Fig. 4d showing α₁(k) and α₂(k) for the EMC of Fig. 4b.

We only evaluate the contributions of the two upper VBs (p₁, p₂), since all other bands are >1 eV below $E_{\mathrm{F}}^{{\mathrm{PES}}}$. Then, the α _j (k) for different k are summed up and multiplied by 1.25 in order to compensate for the part of the BZ, which is not probed by ARPES. Finally, we normalize appropriately. This eventually leads to n_eff = (4.4 ± 1.1) × 10²⁶ m⁻³ with uncertainty resulting from the individual error bars of peak energies and peak widths within the Voigt fits. Taking only the contributions from the upper peak p₁, we get n_eff,1 = (2.6 ± 0.6) × 10²⁶ m⁻³. Since the surface might be influenced by band bending, we also calculated n_eff for an artificially varying E_F with respect to the measured $E_{\mathrm{F}}^{{\mathrm{PES}}}$ as displayed in Fig. 4d.

Next, we compare these n_eff with the results from Hall measurements, which yields the bulk charge carrier density n_eff,H = eσ _xy /B (σ _xy : Hall conductivity, B: magnetic field) varying between 1.3 × 10²⁵ m⁻³ and 3.0 × 10²⁶ m⁻³ for nominally identical samples (Table 1). The variation is probably caused by the known, strong sensitivity of GST transport properties to disorder¹⁷. The temperature dependence of σ _xy is small within the interval T = 4–300 K (changes <10%) demonstrating metallic conductivity. The interval of the n_eff,H data is marked in Fig. 4d. The larger n_eff,H excellently match $n_{{\mathrm{eff}}}\left( {E_{\mathrm{F}}^{{\mathrm{PES}}}} \right)$, while the smaller ones are compatible with an E_F shifted further upward. In any case, the tails of the VB provide enough density of states to host the charge carrier density n_eff,H. We conclude that E_F of GST-225 is indeed well above the VB maximum. In turn, we can estimate the required n_eff to locate E_F at the VB maximum ($E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = −105 meV) to be n_eff ≃ 3 × 10²⁷ m⁻³ (n_eff,1 ≃ 2.3 × 10²⁷ m⁻³), i.e., an order of magnitude larger than the highest values found by the Hall measurements. This excludes a significant downward band bending of the VB toward the surface.

Table 1 Magnetotransport data

In principle, one could argue that the peak width is not due to disorder, but due to the finite lifetime of the photo-hole produced by ARPES^43,44. However, the Voigt fits, which add up a Gaussian peak and a Lorentzian peak, exhibit, on average, 99% (97%) Gaussian contribution and 1% (3%) Lorentzian contribution for p₁ (p₂). Therefore, the lifetime broadening, encoded in the Lorentzian part, is negligible^43,44. Moreover, the average electron scattering time $\bar \tau$ detected by magnetotransport reasonably fits to the disorder-induced peak widths (see below).

Electron mean free path from ARPES and magnetotransport

Next, we deduce the average scattering lifetime of the electrons ($\bar \tau$) and the average mean free path $\bar \lambda _{{\mathrm{MFP}}}$ from the combination of ARPES and magnetotransport. In Supplementary Note 2, we show that the longitudinal conductivity σ _xx and σ _xy can be straightforwardly related to $\bar \tau$ for an isotropic, M-shaped parabolic band in (k _x , k _y ) direction with negligible dispersion in k _z direction and without peak broadening. Thus, in line with the ARPES data, we approximate the dispersion as

$$E_{{\mathrm{peak}}}({\bf{k}}) = E_{{\mathrm{peak}},0} - \frac{{\hbar ^2}}{{2m^ \star }} \left( {k_{||} - k_0} \right)^2$$

(4)

with (E_peak,0, k₀) being the cusp of the inverted parabola and $m^ \star : = \hbar ^2\left( {{\rm d}^2E{\mathrm{/}}{\rm d}k_{||}^2} \right)^{ - 1}$ representing the curvature in radial in-plane direction. This $m^ \star$ is different from a universal effective mass of the VB, since the band curvature differs for other k directions. We obtain (Supplementary Note 2):

$$\sigma _{xx} = \frac{{e^2n_{{\mathrm{eff}}}\bar \tau }}{{2m^ \star }}$$

(5)

$$\sigma _{xy} = en_{{\mathrm{eff}}}{\mathrm{/}}B,$$

(6)

with σ _xx being distinct by a factor of 1/2 from the standard Drude result, which is only valid for an isotropic, parabolic band centered at Γ. To determine $\bar \tau$, we have, hence, to deduce $m^ \star$ from ARPES, besides n_eff. Corresponding parabolic fits to E_peak,1 (k), exemplary shown in Fig. 4b, are executed for all EMCs at different azimuths in (k _x , k _y ) direction and different k _z . This leads to the histogram of $m^ \star$ values in Fig. 4c with mean $\bar m^ \star = (0.35 \pm 0.01) \cdot m_{\mathrm{e}}$ (m_e: bare electron mass, Table 2).

Table 2 VB parameter of epitaxial GST-225

During the same fit, we naturally get an average k₀ as given in Table 2 ($\overline{k}_0$) and an average E_peak,0 being $\overline E _{{\mathrm{peak,0}}} - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = −122 ± 3 meV. With the determined $\bar m^ \star$, we can use magnetotransport data and Eq. (5) to estimate $\bar \tau$. For the sample, where n_eff,H fits best to n_eff from ARPES (Table 1), we measured n_eff,H = (3.0 ± 0.2) × 10²⁶ m⁻³ and σ _xx  = (6 ± 1) × 10⁴ S m⁻¹ (at 300 K) leading to $\bar \tau = 5 \pm 1$ fs (Table 3). The variation between different samples grown with the same parameters (Methods section, Table 1) is negligible.

Table 3 Comparison of scattering time and connected parameters derived from simple and full model

Straightforwardly, we can determine other parameters of the dispersion of Eq. (4), including the Fermi wave vector $\bar k_{\mathrm{F}}$ and the Fermi velocity $\bar v_{\mathrm{F}}$, while still neglecting the peak broadening (Supplementary Note 2):

$$E_{\mathrm{F}} - \bar E_{{\mathrm{peak,0}}} = - \frac{{\hbar ^2}}{{8\bar m^ \star }}\left( {\frac{{\pi cn_{{\mathrm{eff}}}}}{{\overline k _0}}} \right)^2 = - \left( {12 \pm 1} \right)\,{\mathrm{meV}}$$

(7)

$$\left| {\bar k_{\mathrm{F}} - \bar k_0} \right| = \sqrt {\frac{{2\bar m^ \star \left| {E_{\mathrm{F}} - \overline E _{{\mathrm{peak,0}}}} \right|}}{{\hbar ^2}}} = \left( {0.34 \pm 0.02} \right)\,{\mathrm{nm}}^{ - 1}$$

(8)

$$\bar v_{\mathrm{F}} = \frac{\hbar }{{\bar m^ \star }}\left| {\bar k_{\mathrm{F}} - \bar k_0} \right| = \frac{{\pi \hbar }}{{2\bar m^ \star }} \cdot \frac{{cn_{{\mathrm{eff}}}}}{{\bar k_0}}$$

(9)

$$= (1.1 \pm 0.1) \times 10^5\,{\mathrm{m}}\,{\mathrm{s}}^{ - 1}$$

(10)

$$\overline \lambda _{{\mathrm{MFP}}} = \overline v _{\mathrm{F}} \cdot \overline \tau = \frac{{\pi \hbar c}}{{e^2\overline k _0}} \cdot \sigma _{xx} = (0.6 \pm 0.1)\,{\mathrm{nm}},$$

(11)

where c = 1.04 nm (Table 2) is the length of the unit cell of the metastable rock salt phase (structure model in Fig. 1a–c) perpendicular to the layers. The numerical values are again given for the sample with n_eff = (3.0 ± 0.2) × 10²⁶ m⁻³ and σ _xx  = (6 ± 1) × 10⁴ S m⁻¹ (Table 1). E_F is located in the band belonging to p₁ for all n_eff,H of our samples. Note that neither $\bar m^ \star$, as usual, nor n_eff, as typical for two-dimensional (2D) systems⁴⁶, enters the evaluation of $\overline \lambda _{{\mathrm{MFP}}}$, but only $\overline k _0$ does. This reflects the dominating 2D-type dispersion for GST-225.

We also used a more refined, numerical calculation, which considers the variation of $m^ \star$ across the BZ and the peak broadening, i.e., the fact that $E_{\mathrm{F}}^{{\mathrm{PES}}}$ is above the VB maximum, explicitly. Therefore, we use the low-temperature limit of Boltzmann’s relaxation model. We assume that the scattering time τ _j (k) does not depend on k and band index j, reading $\overline \tau : = \tau _j({\bf{k}})$ which leads to

$$\sigma _{xx} = \frac{{ - e^2}}{{4\pi ^3}}{\int\!\!\!\!\!\int\!\!\!\!\!\int} \mathop {\sum}\limits_{j = 1}^2 v_{x,j}^2({\bf{k}})\tau _j({\bf{k}})p_{j,{\mathrm{n}}}(E,{\bf{k}})\frac{{{\rm d}f_0(E({\bf{k}}),T)}}{{{\rm d}E}}d^3{\bf{k}}$$

(12)

$$= \frac{{\overline \tau e^2}}{{4\pi ^3}}{\int\!\!\!\!\!\int\!\!\!\!\!\int} \mathop {\sum}\limits_{j = 1}^2 v_{x,j}^2({\bf{k}})p_{j,{\mathrm{n}}}(E,{\bf{k}})\delta \left( {E({\bf{k}}) - E_{\mathrm{F}}} \right){\rm d}^3{\bf{k}}{\kern 1pt} .$$

(13)

The group velocity ${\bf v}_j({\bf{k}})$ is determined from the ARPES data as ${\bf v}_j({\bf{k}}) = \nabla _{\mathbf{k}}E_{{\mathrm{p}}eak,j}({\bf{k}}){\mathrm{/}}\hbar$ with the derivative taken at E_peak,j and not at $E_{\mathrm{F}}^{{\mathrm{PES}}}$. Since the results now depend critically on E_F, we restrict the analysis to the sample with n_eff ≃ n_eff,H as used in Eq. (7). Numerically, we obtain $\overline \tau = 3 \pm 1_{}^{}$ fs, which is nearly a factor of two smaller than within the simplified calculation. By the same numerical p_j,n (E, k) weighting, we determine the average group velocity at $E_{\mathrm{F}}^{{\mathrm{PES}}}$ as $\overline v _{\mathrm{F}} = (1.2 \pm 0.1) \times 10^5\,{\kern 1pt} {\mathrm{m}}\,{\mathrm{s}}^{ - 1}$ leading to $\overline \lambda _{{\mathrm{MFP}}} \simeq \overline v _{\mathrm{F}} \cdot \overline \tau = 0.4 \pm 0.1$ nm (Table 3).

We compare the results of the refined model and the simplified model (Eqs. (5) and (7)) in Table 3 revealing that the simplified model returns reasonable values, but deviates from the more exact, refined model by up to 40%. This must be considered for the interpretation of magnetotransport data, where Eqs. (5) and (7) provide only reasonable estimates for n_eff, $\overline v _{\mathrm{F}}$, $\overline \tau$, and $\overline \lambda _{{\mathrm{MFP}}}$ with an intrinsic error of about 40%.

Finally, we comment on the peak width, which is, on average, ${\mathrm{\Delta }}\overline E _1 = 0.20 \pm 0.03$ eV (FWHM) for the band p₁. This can be compared with ${\mathrm{\Delta }}\overline E _1\sim \hbar {\mathrm{/}}\tau$ (Supplementary Note 3). We find $\hbar {\mathrm{/\Delta }}\overline E _1 = 3.5$ fs in excellent agreement with $\overline \tau = 3$ fs as deduced from the transport data of the sample with highest conductivity (largest $\overline \tau$) (Table 3). This corroborates the assignment of the peak widths to disorder broadening, as already conjectured from its dominating Gaussian shapes. The fact that σ _xx increases by only 15% between room temperature and T = 4 K²⁰ additionally shows that $\overline \tau$ is dominated by disorder scattering. We conclude that disorder broadening is responsible for the peak widths within the E(k) spectral function of the upper valence band of GST-225. The relatively large peak widths (0.2 eV) allows the Fermi level to be well above all peak maxima, i.e., the charge carrier density fits into the tails of the bands. We finally stress that the peak broadening is not the origin of the p-type doping, which has been found previously to be dominated by excess vacancy formation^47,48.

In-gap surface state

Motivated by our previous finding, that an M-shaped VB with maxima away from high-symmetry points is only compatible with DFT calculations of GST exhibiting non-trivial topology³⁹, we searched for a surface state within the fundamental band gap. We found such a state by two-photon ARPES (2P-ARPES) exhibiting a linear, largely isotropic dispersion, as well as helical, circular dichroism. The state is probably connected to a strongly spin-polarized state at the VB maximum revealed by spin polarized ARPES (S-ARPES).

Optical measurements revealed a band gap of GST-225 of E_gap ≃ 0.5 eV in rough agreement with DFT data⁴⁹, which was recently corroborated by scanning tunneling spectroscopy (E_gap = 0.45 ± 0.05 eV)^39,50,51. Hence, we have to probe this energy interval above the VB maximum, which does not contain bulk states. We employed laser-based 2P-ARPES at pump energy hν = 1.63 eV and probe energy hν = 4.89 eV, hence, populating states in the bulk band gap and in the lower part of the conduction band by the pump pulse, which are subsequently probed by ARPES using the probe pulse. The time delay Δt = 1.33 ps is chosen to optimize the contrast of the states within the bulk band gap. The EMC in Fig. 5a reveals a strong band above $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = 350 meV, which we attribute to the bulk conduction band (CB) at 450 meV above the VB maximum. Below this CB, a mostly linearly dispersing band faintly appears (arrows). Corresponding CECs (Fig. 5b) exhibit a largely circular structure of this band in the (k _x , k _y ) plane increasing in diameter with increasing energy. The linear dispersion of the band is deduced by applying two Voigt fits to each MDC as shown for two examples in Fig. 5c. The resulting $E_{{\mathrm{peak}}}\left( {\left| {\overline {\bf{k}} } \right|} \right)$ (points), averaged along k _x and k _y , are fitted using $E_{{\mathrm{peak}}}\left( {\left| {\overline {\bf{k}} } \right|} \right) = E_{{\mathrm{peak,D}}} \pm \hbar v_{\mathrm{D}}k_{||}$ (red and blue line). This reveals a presumable band crossing at $E_{{\mathrm{peak,D}}} - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = 160 ± 10 meV and a band velocity v_D = (3.8 ± 0.3) × 10⁵ m s⁻¹.

Fig. 5

Two-photon ARPES. a EMC of unoccupied states with energy regions of conduction band (CB) highlighted and presumable topological surface state (state in gap) marked by arrows; b CECs within the bulk band gap for different $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ as indicated; c green lines: MDCs along k _y at $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ as marked (jagged lines) with fits consisting of two Voigt peaks (smooth lines); red, blue dots: peak positions of MDC Voigt fits averaged for the MDCs along k _x and k _y ; red, blue lines: linear fits to the red and blue dots; d circular dichroism (CD) intensity of the 2P-ARPES data of a with CB and bulk band gap region (gap) marked; e CD intensity of the VB states recorded by one photon ARPES; f CD intensity after energy integrating: left: $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = 200–350 meV from d, right: $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = −580−0 meV from e (VB)

Due to the relatively strong one-photon background (Methods section), we could not evaluate the 2P-ARPES signal at lower $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$, such that the presumable crossing point was not probed directly. However, all signatures of this band are compatible with a TSS with mostly linear dispersion.

In addition, we probed the circular dichroism (CD) by 2P-ARPES using a linearly polarized pump and a circular polarized probe pulse⁵². The CD intensity is the scaled difference of photoelectron intensity after clockwise and counterclockwise circular polarization of the probe. It is known that the CD cannot be directly assigned to a spin polarization of initial states³², but is likely related to an interplay of spin and orbital textures⁵³. In our case, it shows a sign inversion with the sign of k (Fig. 5d). The opposite inversion is found within the CB and the upper VB, the latter probed by CD measurements of conventional ARPES (Fig. 5e). The same sequence of CD inversions between VB, TSS, and CB has been found for the prototype strong topological insulators Bi₂Se₃³¹, Bi₂Te₃^31,32, and Sb₂Te₃³³, which is an additional hint that the linearly dispersing state within the bulk band gap is a TSS.

Another fingerprint of non-trivial surface states is spin polarization^54,55,56. Such spin polarized surface states have been predicted by DFT calculations of the cubic Petrov phase of GST-225, in particular, a TSS traversing the band gap and a Rashba-type surface state at $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = −300… −700 meV³⁹. To this end, we probe the spin polarization of the occupied states at a selected pair of in-plane wave vectors ±k_||,1 (Fig. 6). We choose hν = 30 eV such that the CEC of the bulk VB is large in diameter (Fig. 2f), thereby increasing the possibility to probe a surface state in the inner part of the BZ, where DFT of the cubic Petrov phase predicted the presence of a TSS³⁹. Moreover, we used $\left| {{\bf{k}}_{||,1}} \right|$ = 0.17 ± 0.04 Å⁻¹, large enough to avoid overlap of intensity from k_||,1 and −k_||,1, thereby getting along with the typically reduced angular resolution of S-ARPES. Indeed, we find strong in-plane spin polarization of 40% close to $E_{\mathrm{F}}^{{\mathrm{PES}}}$ (Fig. 6b, c). The spin polarization inverts sign with the sign of the in-plane wave vector and is perpendicular to k_||,1 within error bars (Fig. 6c, d). The other spin polarized state at lower $E - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = −0.5… −1.2 eV might be related to the Rashba state mentioned above, which has similarly been found, e.g., for Sb₂Te₃(0001)⁵⁷.

Fig. 6

Spin polarized ARPES. a ARPES spectrum along k _y (ΓK) at hν = 30 eV; labeled white lines mark the k _y positions for the spin resolved measurements in b–d; b recorded in-plane signal of the spin up (blue curve) and spin down (red curve) channel perpendicular to k_||,1 after scaling with the Sherman function; black line: sum of red and blue line; left: −k_||,1, right: +k_||,1; c resulting in-plane spin polarization perpendicular to k_||,1; d in-plane spin polarization parallel to k_||,1. Note the different scales in c, d

The peak energy of the spin polarized state close to $E_{\mathrm{F}}^{{\mathrm{PES}}}$ is $E_{{\mathrm{peak}}} - E_{\mathrm{F}}^{{\mathrm{PES}}}$ = −120 ± 40 meV (Fig. 6b), i.e., very close to the VB maximum, such that likely the spin-polarized state extends into the band gap. The error mostly comes from the different peak energies at + k_||,1 and −k_||,1. We cannot prove that this state is connected to the linearly dispersing state of Fig. 5, which would hit the VB maximum at $\left| {{\bf{k}}_{||}} \right|$ = 0.10 Å⁻¹, if perfectly linear in dispersion down to the VB, but we believe that this is likely.

One might ask why such a linearly dispersing state is not observed in the one-photon ARPES data. A possible explanation is the fact that a surface state will follow the roughness of the surface, which for our films amounts to angles of 0.5°–3° according to atomic force microscopy³⁹. Assuming, for the sake of simplicity, the same state dispersion on all surfaces, this results in a k_|| broadening of the surface state by (0.15–1.5) nm⁻¹ for E_kin = 12–27 eV. The widths of the 2P-ARPES peaks in Fig. 5c is 0.6 nm⁻¹ and, hence, well compatible with this analysis. Since the bulk states are not influenced by this broadening mechanism, it gets rather difficult to discriminate the TSS in the presence of bulk VB states at similar E(k) as within our one-photon ARPES data.

We did not reproduce the dispersion of the found state in the bulk band gap by the DFT calculations of slabs of the cubic Petrov phase, which revealed a less steep dispersion of its TSS and another E_D³⁹. We ascribe this discrepancy to the known, strong sensitivity of the TSS to details of GST’s atomic structure^{34,35,36,37,38,39}. However, besides these remaining questions, both, the strong spin polarization close to the VB maximum and the linear dispersion within the bulk band gap are compatible with a TSS. This corrobarates the previous conjecture of a topologically inverted band structure of metastable GST-225³⁹.

Possible contribution of the topological surface state to conductivity

The possible presence of a TSS at $E_{\mathrm{F}}^{{\mathrm{PES}}}$, naturally protected from backscattering^54,55,56, raises the question whether it would contribute significantly to the conductivity. To answer this, we firstly compare the charge carrier density of the presumable TSS n_2D,TSS with the measured charge carrier density of the epitaxial film, after projecting to 2D according to n_2D,H = n_eff,H  ·d (d: film thickness). The latter varies between n_2D,H = 7 × 10¹⁸ m⁻² (film of largest conductivity) and n_2D,H = 3 × 10¹⁷ m⁻² (Table 1). For the non-degenerate 2D band of a linearly dispersing TSS, we have⁴⁶:

$$n_{{\mathrm{2D,TSS}}} = \frac{{k_{\mathrm{F}}^2}}{{2\pi }}.$$

(14)

A reasonable assumption for k_F results from extrapolating the fitted linear dispersion of Fig. 5c to $E_{\mathrm{F}}^{{\mathrm{PES}}}$ leading to $\left| {k_{\mathrm{F}}} \right| = 6 \times 10^8\,{\mathrm{m}}^{ - 1}$. An upper estimate is $\left| {k_{\mathrm{F}}} \right| = 1.7 \times 10^9\,{\mathrm{m}}^{ - 1}$, i.e., the k_|| value of the spin polarized state at the VB maximum. Hence, we get n_2D,TSS ≃ 6 × 10¹⁶ m⁻², respectively n_2D,TSS ≤ 5 × 10¹⁷ m⁻².

Comparing with the sample exhibiting n_eff ≃ n_eff,H (n_2D,H = 7 × 10¹⁸ m⁻²), n_2D,H is more than an order of magnitude larger than n_2D,TSS. We conclude that the charge carrier density is dominated by the bulk VB.

However, the mobility of a TSS (μ_TSS) could be much larger than the mobility of the bulk VB (μ_bulk). Such a TSS conductivity dominates, if μ_TSS/μ_bulk > n_2D,bulk/n_2D,TSS (n_2D,bulk: charge carrier density in the bulk VB after 2D projection). Within the two band model, we have⁴⁶

$$n_{{\mathrm{2D,H}}} = \frac{{n_{{\mathrm{2D,TSS}}}^{\mathrm{2}} \cdot \mu _{{\mathrm{TSS}}} + n_{{\mathrm{2D,bulk}}}^{\mathrm{2}} \cdot \mu _{{\mathrm{bulk}}}}}{{n_{{\mathrm{2D,TSS}}} \cdot \mu _{{\mathrm{TSS}}} + n_{{\mathrm{2D,bulk}}} \cdot \mu _{{\mathrm{bulk}}}}}$$

(15)

$$\sigma _{xx} = \left( {n_{{\mathrm{2D,TSS}}} \cdot \mu _{{\mathrm{TSS}}} + n_{{\mathrm{2D,bulk}}} \cdot \mu _{{\mathrm{bulk}}}} \right) \cdot \frac{e}{d}.$$

(16)

We evaluate these equations for the sample with n_eff ≃ n_eff,H (σ _xx  = 6 × 10⁴ S m⁻¹) using the assumption of a linearly dispersing TSS down to $E_{\mathrm{F}}^{{\mathrm{PES}}}$ (n_2D,TSS = 6 × 10¹⁶ m⁻²). We solve Eq. (15) and n_2D,TSS · μ_TSS ≥ n _2D,bulk · μ_bulk for the three remaining unknowns (μ_TSS, μ_bulk, n_2D,bulk) leading to μ_TSS ≥ 0.1 m² (Vs)⁻¹. The threshold for dominating μ_TSS is even lower for the other samples (Table 1). We moreover assume that only the surface contains a highly mobile TSS. In turn, the threshold for dominating μ_TSS has to be divided by two, if the interface to the Si(111) contains a TSS with the same n_2D,TSS and μ_TSS. For comparison, the record mobilities found for TSS in other systems (Bi₂Se₃, BiSbTeSe₂) are μ_TSS ≃ 1 m² (Vs)^{−1 40,41}, i.e., significantly larger than the threshold.

We conclude with the encouraging possibility to prepare highly mobile, metastable GST-225 films, noting that polycrystalline films exhibit σ _xx  < 6400 S m^{−1 22}, which is an order of magnitude lower than for our best epitaxial film. Thus, one might boost the GST-225 conductivity by the combination of epitaxial films and adequate interface design leading to optimized TSS mobility^40,41. This could be exploited within innovative devices combining the fast^7,8 and energy efficient⁹ phase change with ultrahigh mobility of the on-state.

Discussion

We have mapped the 3D electronic bulk band structure E(k) close to E_F of epitaxial GST films in the metastable rock salt phase and have correlated the results with magnetotransport data of identically prepared samples. The constant energy surfaces of the valence band close to E_F are hexagonal tubes with little dispersion along k _z , the direction perpendicular to the chemically distinct layers. The valence band maximum is about 100 meV below E_F, such that only the tails of the disorder broadened E(k) states contribute to the conductivity. This is in line with the measured charge carrier densities from Hall measurements. We use the mapped band structure in combination with magnetotransport to determine the elastic scattering time (3 fs) and the mean free path (0.4 nm), the former being compatible with the peak widths found in ARPES.

Our detailed modeling reveals that variations of the band structure across the BZ. i.e., different band curvatures and peak broadenings, modify the deduced scattering time and average mean free path by about 40%, such that simplified models, as typically used for the interpretation of magnetotransport data, cannot provide a better accuracy.

Besides, we find a linearly dispersing state within the bulk band gap which might have a topological origin. We estimate that this state would dominate the longitudinal conductivity at a mobility above 0.1 m² (Vs)⁻¹, which is lower than the best mobilities of topological surface states so far (μ ≃ 1 m² (Vs)⁻¹)^40,41. Currently, topological conductivity is not expected to be dominant, but by surface or interface optimization one might exploit it in future GST devices providing an ultrahigh mobility on-state.

Methods

Sample preparation

The GST films are grown by MBE at base presure 10⁻⁸ Pa on a Si(111) substrate using elementary sources of Ge, Sb, and Te and a substrate temperature of 250 °C. The growth rate was 0.05 ± 0.02 nm s⁻¹ and the pressure increased to 2 × 10⁻⁷ Pa during growth. XRD reveals that the films grow epitaxially in the single crystalline, metastable rock salt phase with [111] surface. The surface is Te terminated as evidenced by DFT calculations (not shown). The film thickness is determined by XRD fringes or by X-ray reflectometry to be 25, 18, and 13 nm for the samples used for ARPES, 2P-ARPES and S-ARPES, respectively. Twin domains are found, i.e., adjacent areas of ABC and CBA stacking of the hexagonal layers³⁹. A peak indicating the formation of a vacancy layer is observed by XRD, hinting to more ordered samples than in the purely disordered rock salt phase²⁰. The XRD data recorded after the ARPES measurements show variations in the (222) peak position by up to 1.5%²⁰ and in the height of the vacancy layer peak by 15–25%. However, the ARPES data of these samples are quite similar, i.e., peak positions of the VB vary by less than the peak widths. Samples are transferred in UHV between the MBE and the three different analyzing ARPES systems using a UHV shuttle with background pressure of p = 5 × 10⁻¹⁰ mbar. This prevents oxidation and surface contamination as cross-checked by X-ray photoelectron spectroscopy (XPS), such that no further preparation steps are required. The UHV transfer is crucial, since surface oxidation starts already at ~1 Pa ⋅ s of O₂²⁹.

Photoelectron spectroscopy

The ARPES measurements of the valence band are recorded at a sample temperature T = 300 K at BESSY II (beam line UE112-lowE-PGM2 (1²)) using a Scienta R8000 analyzer with energy resolution 20 meV and angular resolution 0.2–0.3°. Linearly p-polarized light with photon energies hν = 16–31 eV and an incidence angle of ~45° is applied, which enabled a three-dimensional mapping of the band structure in momentum space (k _x , k _y , k _z ). The Fermi energy $E_{\mathrm{F}}^{{\mathrm{PES}}}$ of the ARPES setup has been determined on polycrystalline Cu with 5 meV precision.

The data set contained 998 × 666 × 49 × 29 pixels, i.e., 998 different photoelectron energies E_kin, 666 different azimuthal angles ϕ, 49 polar angles θ, and 29 photon energies hν. The energy interval E_kin ∈ [E_F + 0.042 eV, E_F + 0.104 eV] is used for background subtraction for each energy distribution curve (EDC) at a particular (ϕ, θ, hν). Subsequently, the data are smoothed along E_kin and ϕ by a 5-point averaging. Accordingly, the data set is reduced to 256 × 256 × 49 × 29 pixels. Finally, all EDCs are scaled to the same average value for each (ϕ, θ, hν).

In order to deduce band centers E_peak,j (k) of band j, MDCs and EDCs at constant hν are extracted from the data and fitted by two or four Voigt peaks with variable intensities, widths, and relative contributions of the Gaussian and the Lorentzian. This leads to an excellent fit quality with negligible residuals as exemplarily shown in Figs. 2c, 3g, h, and 4a. The resulting up to four (k _x , k _y ) _i of MDC fits are then attributed to the preselected $E - E_{\mathrm{F}}^{{\mathrm{PES}}} = E_{{\mathrm{peak}}}$, respectively, the resulting E_peak,j of EDC fits are attributed to the preselected (k _x , k _y ) values. The resulting E_peak,j(k _x , k _y ) curves deduced from the two methods vary by Δk_|| ≃ 0.01 Å⁻¹, respectively, by $\left| {{\mathrm{\Delta }}E_{{\mathrm{peak}}}} \right|$ = 5–15 meV, except for extreme k-values (see main text). The small deviations contribute straightforwardly to the error of the determined effective charge carrier densities n_eff and curvature parameters $m^ \star$ (Fig. 4).

Displaying the upper E_peak,1(k _x , k _y ) at selected E_peak for different hν, as shown for energy $E_{{\mathrm{peak}}} = E_{\mathrm{F}}^{{\mathrm{PES}}}$ in Fig. 2f, consistently reveals a minimum diameter of the resulting constant energy lines at hν ≃ 21 eV. Since DFT finds the minimum diameter of the upper VBs at the BZ boundary (e.g., Fig. 2h), we assume that the minimum at hν ≃ 21 eV corresponds to the BZ boundary in k _z direction. This assumption is used to determine the inner potential E_inner with respect to the vacuum level for the final state electrons in the crystal according to $E_{{\mathrm{final}}} = \hbar ^2\left| {\bf{k}} \right|^2{\mathrm{/}}2m_{\mathrm{e}} - E_{{\mathrm{inner}}}$. Restricting E_inner between 10 and 25 eV leaves us with the only possibility of E_inner = 14 eV. However, if the minimum diameter is in the center of the BZ, we would get E_inner = 20 eV. Since these differences are not important for our main conclusions, we select the most reasonable assumption that the smallest diameter is at the BZ boundary. Using the inner potential, we calculate k _z according to $k_z = \frac{1}{\hbar} \sqrt {2m_eE_{{\mathrm{kin}}}\,{\mathrm{cos}}(\theta )^2 + E_{{\mathrm{inner}}}}$. For the CECs and CESs in Fig. 2, we use an average value of θ to relate hν to k _z .

The ARPES data cover only 80% of the BZ, i.e., a small part in k _z direction is missing (Fig. 2e). This is due to the fact, that at lower and higher hν, the ARPES intensity drops drastically, such that fits become unreliable. However, in line with the DFT results, we do not believe that the hexagons change strongly within the remaining 20%.

Fit procedures and fit errors

All peaks of MDC and EDC curves are fitted by several Voigt peaks, i.e., by a combination of a Gaussian and a Lorentzian peak with the same maximum each. Comparing the results of MDC fits and EDC fits for energies below the VB maximum reveals only small differences between deduced E(k) values by 0.1 nm⁻¹ or 10 meV on average, except for the extreme cases k_|| < 0.5 nm⁻¹ and k_|| > 2.5 nm⁻¹ $\left( {k_{||}: = \sqrt {k_x^2 + k_y^2} } \right)$. The small discrepancies set a lower bound for error margins.

In order to extract n_eff from the fitted peaks of EDCs, the peak areas of p _j (E, k) (j = 1, 2) are normalized to one leading to p_j,norm (E, k), from which we evaluate the relative part of the peaks above $E_{\mathrm{F}}^{{\mathrm{PES}}}$ (inset of Fig. 4a), being α _j (k), the unoccupied percentage of the corresponding E(k) state.

The error of $\bar m^ \star$ is only slightly smaller than the error of individual $m^ \star$, being 5% on average, which is due to the considerable variation of $m^ \star$ across the BZ (Fig. 4c). The deviation of individual curves from the parabola is negligible (Fig. 4b), i.e., the average energy distance of individual E_peak,1 (k) from the parabola (≈8 meV) is less than the average fit error from the determination of E_peak,1 (k) by Voigt fits (≈30 meV).

Spin polarized photoelectron spectroscopy

Spin resolved ARPES measurements are conducted at BESSY II, too, using the electron analyzer SPECS PHOIBOS 150 and linearly p-polarized synchrotron radiation at hν = 30 eV and incidence angle 45° at T = 300 K, providing an energy resolution of 100 meV and an angular resolution of 1–2°. Spin analysis is performed with a Rice University Mott polarimeter operated at 26 kV resulting in a Sherman function of S = 0.118.

Two photon photoelectron spectroscopy

Angle-resolved bichromatic 2P-ARPES and additional, conventional ARPES experiments are conducted using the first, third and fourth harmonic of a titanium:sapphire oscillator, i.e., hν = 1.63 eV, hν = 4.89 eV, and hν = 6.2 eV, within a home-built setup^52,58. The repetition rate of the laser is 80 MHz and the pulse length is 166 fs. The beam is initially p-polarized with an incidence angle of 60°. The photon energy hν = 1.63 eV is used for the pump pulse followed by the probe pulse at hν = 4.89 eV, which hits the sample at a time delay Δt after the pump. Due to time restrictions, hν of the probe pulse has not been changed such that we probe only a single k _z . The photon energy hν = 6.2 eV is used for conventional ARPES to cross check the results obtained at BESSY II. Circular polarization, necessary for circular dichroism (CD) experiments, is obtained using a λ/4 wave plate. Two-dimensional momentum distribution patterns at constant E_kin are recorded using an ellipsoidal ‘display-type’ analyzer exhibiting an energy and angular resolution of 55 meV and 3°, respectively^52,59. The work function of GST-225 turned out to be 5.1 ± 0.1 eV leading to a strong one-photon photoemission background from the probe pulse. In order to discriminate 2P-ARPES data from this background, the intensity of measurements at Δt = −2.67 ps is subtracted from the data recorded at Δt = +1.33 ps. Subsequently, the data are normalized to compensate for inhomogeneities of the channel plates. CD intensity displays the difference of photoelectron intensity using clockwise and counterclockwise polarized probe pulses divided by the sum of the two intensities.

Magnetotransport

Magnetotransport measurements are performed ex situ. Since Hall measurements require insulating substrates while ARPES requires a conducting sample, the Hall data are from samples with lower substrate doping, but grown with identical parameters. After growth they are capped by Te to protect from oxidation. The samples are cut in square shapes of 5 × 5 mm² and, after decapping by a HF dip, are contacted by In and Au bond wires in a four-contact van der Pauw geometry. Magnetotransport measurements are conducted at T = 4–300 K, current I = 10 mA, and magnetic field B = 0.25 T perpendicular to the surface. This leads to charge carrier densities n_eff,H and longitudinal conductivities σ _xx as displayed in Table 1 for T = 300 K. It is likely that the upper 5 nm of the sample are oxidized^29,60,61 resulting in a systematic error of 25%. We find a relatively broad statistical distribution of n_eff,H and σ _xx , but due to correlations between the two values, the variation of the mobility μ = σ _xx /n_eff,He is relatively small.

Band structure calculations

Density functional theory (DFT) calculations are performed within the generalized gradient approximation. We employ the full-potential linearized augmented plane-wave method in bulk and thin-film geometry as implemented in the FLEUR code. According to ref. ³⁰, the Petrov stacking sequence (Te-Sb-Te-Ge-Te-Vc-Te-Ge-Te-Sb-)¹⁵ is assumed for the metastable rock salt phase by tripling the Petrov-type unit cell containing 10 layers in order to realize the ABC stacking of the rock salt phase (Fig. 1d). The resulting BZ of the unit cell of 30 layers, is a factor of five smaller in k _z direction (Δk _z  = 0.12 Å⁻¹) than the BZ of the disordered metastable rock salt phase, relevant for the ARPES data (6 layers in a unit cell, Fig. 1c). Hence, we use fivefold backfolding of the experimental data (Fig. 2g) to compare with the DFT data (Fig. 2h). Additional DFT calculations are performed for disordered slabs (Fig. 1b) with methodology otherwise similar to ref. ⁶². We simulate maximum disorder by occupying each cationic plane randomly with Ge:Sb:Vc in a 2:2:1 ratio. These planes are parallel to the (111) surface, and include the disordered subsurface layer, whereas the surface itself is terminated by Te⁶². Different structure models of the cationic plane were randomly generated, and after relaxation showed a standard deviation of 3 meV/atom in total energies. The computed surface energies range from 12 to 17 meV Å⁻² in a Te-poor environment, which can well be reconciled with previous results for ideally ordered GST⁶².

More details including atomic coordinates are given in Supplementary Note 4.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.