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Energy gap of the spin density wave at the Cr(110) surface
Surface Science 454–456 (2000) 885–890 www.elsevier.nl/locate/susc
Energy gap of the spin density wave at the Cr(110) surface J. Scha¨fer a,b, *, Eli Rotenberg a, S.D. Kevan b, P. Blaha c a Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA b Department of Physics, University of Oregon, Eugene, OR 97403, USA c Institut fu¨r Physikalische und Theoretische Chemie, Technische Universita¨t Wien, 1060 Wien, Austria
Abstract Angle resolved photoemission is used to directly monitor the energy gap of the spin density wave of Cr at a (110) surface. Density functional calculations, including a Fermi surface contour, predict an energy gap in the surface proximity. Surface states are identified, and a near-surface gap of ~200 meV at room temperature is found, using spectroscopy, extending above the Fermi level. The gap exhibits an increased critical temperature of ~440 K and extends isotropically around the C point. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Chromium; Density functional calculations; Magnetic phenomena (cyclotron resonance, phase transitions, etc.); Magnetic surfaces; Photoelectron spectroscopy
1. Introduction The well-known incommensurate spin density wave (SDW ) state in Cr [1] (and references therein) mediates the interlayer coupling observed in Fe:Cr multilayers, which exhibit the giant magnetoresistance effect [2]. In weak coupling mean field theory [3], the formation of an SDW leads to an energy gap at T=0 K of D=3.53kT , where N T is the Ne´el temperature. With the established N bulk transition temperature of 311 K [1], mean field theory predicts a gap of 95 meV. Data on the actual bulk energy gap come from the infrared measurements of Machida et al. [4] which determine a low temperature band gap of ~120 meV. However, some crucial points have not been resolved yet — where exactly the gap forms in k-space, what magnitude it assumes in the proximity of a surface, and its behaviour with temper* Corresponding author. Fax: +1-510-486-7588. E-mail address: [email protected] (J. Scha¨fer)
ature. Attempts at determining the band structure from photoelectron spectroscopy [5–7] suffered from surface contaminants and insufficient resolution, thus rendering them inconclusive in this respect. In the present work, we present surface band structure calculations, as well as angle resolved photoemission measurements with high momentum and energy resolution on clean Cr(110) thin films. The full range of the energy gap extending above the Fermi level, E , has been documented F by using an ultra-bright synchrotron light source, thus enabling us to follow the electron occupancy above E for many hundreds of meV. F
2. Band structure calculation For the band structure calculations, with the SDW spanning ~27 lattice constants [8], the alternating magnetization of neighbouring atoms
0039-6028/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 00 ) 0 02 4 8 -X
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J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
can be described approximately in a doubled unit cell: a body centered cubic Brillouin zone (BZ ) for the paramagnetic (PM ) phase, and a half-size simple cubic (sc) BZ for the antiferromagnetic (AFM ) phase which effectively leads to backfolding of the electronic band structure [9]. The need
in the present work is to identify the SDW gap in those near-surface layers which we probed by photoemission (exponential attenuation with mean ˚ ), and in an experimentally escape depth of ~5 A suitable direction. We have calculated the band structure for the bulk and a nine-layer slab at the
Fig. 1. (a) Calculated Fermi surface cross-section from the GGA method. SDW nesting occurs between the electron jack at C and the hole jack at H. (b) GGA band structure calculations for bulk ( left) and nine-layer surface slab (right), including surface states (SS ). The PM band structure is drawn backfolded into the sc BZ. The SDW energy gap D is consistently predicted at 0.4 C−S.
J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
surface, using density functional theory within the generalized gradient approximation (GGA), and a relativistic self-consistent full potential linearized augmented plane wave procedure [10]. The Fermi surface nesting wave vector relevant to SDW formation, which amounts to q ~0.95 C–H as SDW known from neutron scattering [8], occurs between the electron ‘octahedron’ at C and the hole ‘octahedra’ at the six H points. The structures are of very similar shape and curvature, as obtained from our
887
Fermi surface calculation in Fig. 1a. The intersecting C–S direction (notation of superimposed surface BZ ) is expected to give a very clear cut view of the gap formation. The GGA band calculations for this direction are shown in Fig. 1b. The bulk gap formation occurs at a distance of ~40% away from C towards S. The gap obtained for the subsurface layers is located at the same momentum as the bulk gap. The gap magnitude is D~0.3 eV for the actual
Fig. 2. (a) Coarse overview band map along C–S taken at 180 K showing the states and location of the SDW gap at the Fermi level crossing near C. (b) Band maps taken at 300 K ( left) and 570 K (right) with the Fermi distribution divided out.
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J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
was prepared in ultra-high vacuum ( UHV ) by evaporation of ~100 layers of Cr onto a clean W(110) crystal and subsequent annealing. From a measurement of the size of the BZ, the Cr film surface was found to be strain free. From a scan of the photon energy, we chose hn=119.6 eV so that the bulk BZ was intersected approximately at ˚ −1). C (k ~6A ) 4. Results and discussion
Fig. 3. Temperature series of SDW energy gap spectra, extracted from wider k-window band maps (Fermi distribution divided out). The gap closes at 440 K judged from the individual band maps.
surface state, and ~1.0 eV for the subsurface layers and the bulk. We obtained magnetic moments for the topmost three layers which are enhanced by factors of 1.72, 1.14, and 1.05 relative to the bulk moment of 1.15 m . The latter is B significantly larger than the experimental value [1] of 0.62 m , which we also obtain in local spin B density approximation (LSDA), a discrepancy attributed to the particular exchange-correlation potential.
3. Experimental Angle resolved photoelectron spectroscopy (ARPES) was carried out at undulator beamline 7.0. at the Advanced Light Source, Berkeley. The energy resolution used was ~50 meV, the momen˚, tum resolution was ~0.03×2p/a, with a=2.88 A the lattice constant of Cr. The Cr(110) surface
In the experimental band maps with fixed k ) and with k varied along C–S, the surface states d (SS ) in Fig. 2a appear weak compared to the more bulk-like states of the deeper layers, and their intensity deteriorates quickly even under UHV conditions. The identification of the SDW gap is achieved by a high resolution close-up measurement (Fig. 2b), taken far below (T=300 K ) and above (T=570K ) the measured surface Ne´el temperature (440 K, see below). The raw data were divided by the Fermi distribution f(E ) to allow observation of thermally excited states above the Fermi level. At low temperature we observe: (i) a second branch emerging which rises from S; and (ii) a diffuse intensity above E which we identify F with the band above the SDW gap. This interpretation is confirmed by detecting the dispersion of the upper band in constant energy slices (see Fig. 4). The band structure is thus exhibiting an almost direct gap of 200±20 meV at E . The F additional branch is identified with the aid of the calculation as the ‘backfolded’ branch. The gap formation occurs at a distance of ~0.40 away from C towards S as predicted by the calculation, although the model overestimates the energy gap D. The observed D is larger than the bulk value of 120 meV reported from infrared measurements [4]. We also measured the temperature dependence of the energy gap. The result of this series is shown in Fig. 3. The edge of the lower band clearly tracks the gradual closure of the gap. From examination of the numerous individual band maps, a surface transition temperature of T S =440±10 K is N determined, considerably above the bulk Ne´el temperature of T Bulk=311 K. The temperature depenN dence of the lower section of the SDW energy gap
J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
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Fig. 4. Constant binding energy ARPES without energy integration other than the experimental resolution (intensity calibrated with Fig. 2b). In the SDW-nested AFM state, (a) and (b) represent the edges of the bands which define the virtually isotropic SDW gap seen in (c) of ~200 meV, which is filled in the PM state (d).
in Fig. 3 (using a Lorentzian peak fit), is found to follow a power law (1−T/T )b with b~0.4. N Towards the surface, on the basis of the calculated increase of the magnetic moments, SDW mean field theory lets us qualitatively expect a larger energy gap and a higher transition temperature, as observed. Although short range magnetic order might persist above T Bulk [11], from bulk N studies this is not known to lead to an energy gap [1,4]. The existence of a gap near the surface up to a sharply defined T S rather has the signature N of a phase transition. The bulk transition exhibits
a very small discontinuity at T Bulk [1], but is N otherwise continuous. SDW weak coupling theory yields a critical exponent of 0.5 in a 3-D system. A reduction of dimensionality and exponent may come from the near-surface enhancement of the magnetic moments, similar to ferromagnetic thin films [12]. In Fig. 4 we present constant energy contours obtained from ARPES in the vicinity of E in the F AFM and PM states. The data represent k-space maps of states at energies corresponding to the lower and upper edges of the gap in the SDW
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J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
phase (T=300 K ), as well as the middle of the gap region in the SDW and PM phases (T= 570 K ) respectively. These data demonstrate impressively that the energy gap opens almost isotropically and with constant magnitude of 200±20 meV around this section of the electron octahedron at C. This finding of an isotropic energy gap is consistent with our band structure calculations.
5. Summary This spectroscopic study of the SDW phase transition at the Cr(110) surface shows an increased transition temperature of 440 K, with the approach of the order parameter being continuous and power-law like. The gap reaches a magnitude of ~200 meV at room temperature, and its extent covers the entire measured section of the electron octahedron. This might have important implications for interfacial magnetic coupling and resulting electronic properties.
Acknowledgements This work was supported by the US DOE under grant DE-FG06-86ER45275, and at the ALS under grant DE-AC03-76SF00098.
References [1] E. Fawcett, Rev. Mod. Phys. 60 (1988) 209. [2] D. Li et al., Phys. Rev. Lett. 78 (1997) 1154. [3] P. Fazekas, Electron Correlation and Magnetism, World Scientific, Singapore, 1999. [4] K. Machida, M.A. Lind, J.L. Stanford, J. Phys. Soc. Jpn. 53 (1984) 4020. [5] P.E.S. Persson, L.I. Johansson, Phys. Rev B 34 (1986) 2284. [6 ] L.E. Klebanoff et al., J. Magn. Mag. Mater. 54–55 (1986) 728. [7] Y. Sakisaka et al., Phys. Rev. B 38 (1988) 1131. [8] S.A. Werner, A. Arrott, H. Kendrick, Phys. Rev. 155 (1967) 528. [9] S. Asano, J. Yamashita, J. Phys. Soc. Jpn. 23 (1967) 714. [10] P. Blaha et al., WIEN97 code, Comput. Phys. Commun. 59 (1990) 339. [11] B. Sinkovic, B. Hermsmeier, C.S. Fadley, Phys. Rev. Lett. 55 (1985) 1227. [12] Y. Li, K. Baberschke, Phys. Rev. Lett. 68 (1992) 1208.
Energy gap of the spin density wave at the Cr(110) surface J. Scha¨fer a,b, *, Eli Rotenberg a, S.D. Kevan b, P. Blaha c a Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA b Department of Physics, University of Oregon, Eugene, OR 97403, USA c Institut fu¨r Physikalische und Theoretische Chemie, Technische Universita¨t Wien, 1060 Wien, Austria
Abstract Angle resolved photoemission is used to directly monitor the energy gap of the spin density wave of Cr at a (110) surface. Density functional calculations, including a Fermi surface contour, predict an energy gap in the surface proximity. Surface states are identified, and a near-surface gap of ~200 meV at room temperature is found, using spectroscopy, extending above the Fermi level. The gap exhibits an increased critical temperature of ~440 K and extends isotropically around the C point. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Chromium; Density functional calculations; Magnetic phenomena (cyclotron resonance, phase transitions, etc.); Magnetic surfaces; Photoelectron spectroscopy
1. Introduction The well-known incommensurate spin density wave (SDW ) state in Cr [1] (and references therein) mediates the interlayer coupling observed in Fe:Cr multilayers, which exhibit the giant magnetoresistance effect [2]. In weak coupling mean field theory [3], the formation of an SDW leads to an energy gap at T=0 K of D=3.53kT , where N T is the Ne´el temperature. With the established N bulk transition temperature of 311 K [1], mean field theory predicts a gap of 95 meV. Data on the actual bulk energy gap come from the infrared measurements of Machida et al. [4] which determine a low temperature band gap of ~120 meV. However, some crucial points have not been resolved yet — where exactly the gap forms in k-space, what magnitude it assumes in the proximity of a surface, and its behaviour with temper* Corresponding author. Fax: +1-510-486-7588. E-mail address: [email protected] (J. Scha¨fer)
ature. Attempts at determining the band structure from photoelectron spectroscopy [5–7] suffered from surface contaminants and insufficient resolution, thus rendering them inconclusive in this respect. In the present work, we present surface band structure calculations, as well as angle resolved photoemission measurements with high momentum and energy resolution on clean Cr(110) thin films. The full range of the energy gap extending above the Fermi level, E , has been documented F by using an ultra-bright synchrotron light source, thus enabling us to follow the electron occupancy above E for many hundreds of meV. F
2. Band structure calculation For the band structure calculations, with the SDW spanning ~27 lattice constants [8], the alternating magnetization of neighbouring atoms
0039-6028/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 00 ) 0 02 4 8 -X
886
J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
can be described approximately in a doubled unit cell: a body centered cubic Brillouin zone (BZ ) for the paramagnetic (PM ) phase, and a half-size simple cubic (sc) BZ for the antiferromagnetic (AFM ) phase which effectively leads to backfolding of the electronic band structure [9]. The need
in the present work is to identify the SDW gap in those near-surface layers which we probed by photoemission (exponential attenuation with mean ˚ ), and in an experimentally escape depth of ~5 A suitable direction. We have calculated the band structure for the bulk and a nine-layer slab at the
Fig. 1. (a) Calculated Fermi surface cross-section from the GGA method. SDW nesting occurs between the electron jack at C and the hole jack at H. (b) GGA band structure calculations for bulk ( left) and nine-layer surface slab (right), including surface states (SS ). The PM band structure is drawn backfolded into the sc BZ. The SDW energy gap D is consistently predicted at 0.4 C−S.
J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
surface, using density functional theory within the generalized gradient approximation (GGA), and a relativistic self-consistent full potential linearized augmented plane wave procedure [10]. The Fermi surface nesting wave vector relevant to SDW formation, which amounts to q ~0.95 C–H as SDW known from neutron scattering [8], occurs between the electron ‘octahedron’ at C and the hole ‘octahedra’ at the six H points. The structures are of very similar shape and curvature, as obtained from our
887
Fermi surface calculation in Fig. 1a. The intersecting C–S direction (notation of superimposed surface BZ ) is expected to give a very clear cut view of the gap formation. The GGA band calculations for this direction are shown in Fig. 1b. The bulk gap formation occurs at a distance of ~40% away from C towards S. The gap obtained for the subsurface layers is located at the same momentum as the bulk gap. The gap magnitude is D~0.3 eV for the actual
Fig. 2. (a) Coarse overview band map along C–S taken at 180 K showing the states and location of the SDW gap at the Fermi level crossing near C. (b) Band maps taken at 300 K ( left) and 570 K (right) with the Fermi distribution divided out.
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J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
was prepared in ultra-high vacuum ( UHV ) by evaporation of ~100 layers of Cr onto a clean W(110) crystal and subsequent annealing. From a measurement of the size of the BZ, the Cr film surface was found to be strain free. From a scan of the photon energy, we chose hn=119.6 eV so that the bulk BZ was intersected approximately at ˚ −1). C (k ~6A ) 4. Results and discussion
Fig. 3. Temperature series of SDW energy gap spectra, extracted from wider k-window band maps (Fermi distribution divided out). The gap closes at 440 K judged from the individual band maps.
surface state, and ~1.0 eV for the subsurface layers and the bulk. We obtained magnetic moments for the topmost three layers which are enhanced by factors of 1.72, 1.14, and 1.05 relative to the bulk moment of 1.15 m . The latter is B significantly larger than the experimental value [1] of 0.62 m , which we also obtain in local spin B density approximation (LSDA), a discrepancy attributed to the particular exchange-correlation potential.
3. Experimental Angle resolved photoelectron spectroscopy (ARPES) was carried out at undulator beamline 7.0. at the Advanced Light Source, Berkeley. The energy resolution used was ~50 meV, the momen˚, tum resolution was ~0.03×2p/a, with a=2.88 A the lattice constant of Cr. The Cr(110) surface
In the experimental band maps with fixed k ) and with k varied along C–S, the surface states d (SS ) in Fig. 2a appear weak compared to the more bulk-like states of the deeper layers, and their intensity deteriorates quickly even under UHV conditions. The identification of the SDW gap is achieved by a high resolution close-up measurement (Fig. 2b), taken far below (T=300 K ) and above (T=570K ) the measured surface Ne´el temperature (440 K, see below). The raw data were divided by the Fermi distribution f(E ) to allow observation of thermally excited states above the Fermi level. At low temperature we observe: (i) a second branch emerging which rises from S; and (ii) a diffuse intensity above E which we identify F with the band above the SDW gap. This interpretation is confirmed by detecting the dispersion of the upper band in constant energy slices (see Fig. 4). The band structure is thus exhibiting an almost direct gap of 200±20 meV at E . The F additional branch is identified with the aid of the calculation as the ‘backfolded’ branch. The gap formation occurs at a distance of ~0.40 away from C towards S as predicted by the calculation, although the model overestimates the energy gap D. The observed D is larger than the bulk value of 120 meV reported from infrared measurements [4]. We also measured the temperature dependence of the energy gap. The result of this series is shown in Fig. 3. The edge of the lower band clearly tracks the gradual closure of the gap. From examination of the numerous individual band maps, a surface transition temperature of T S =440±10 K is N determined, considerably above the bulk Ne´el temperature of T Bulk=311 K. The temperature depenN dence of the lower section of the SDW energy gap
J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
889
Fig. 4. Constant binding energy ARPES without energy integration other than the experimental resolution (intensity calibrated with Fig. 2b). In the SDW-nested AFM state, (a) and (b) represent the edges of the bands which define the virtually isotropic SDW gap seen in (c) of ~200 meV, which is filled in the PM state (d).
in Fig. 3 (using a Lorentzian peak fit), is found to follow a power law (1−T/T )b with b~0.4. N Towards the surface, on the basis of the calculated increase of the magnetic moments, SDW mean field theory lets us qualitatively expect a larger energy gap and a higher transition temperature, as observed. Although short range magnetic order might persist above T Bulk [11], from bulk N studies this is not known to lead to an energy gap [1,4]. The existence of a gap near the surface up to a sharply defined T S rather has the signature N of a phase transition. The bulk transition exhibits
a very small discontinuity at T Bulk [1], but is N otherwise continuous. SDW weak coupling theory yields a critical exponent of 0.5 in a 3-D system. A reduction of dimensionality and exponent may come from the near-surface enhancement of the magnetic moments, similar to ferromagnetic thin films [12]. In Fig. 4 we present constant energy contours obtained from ARPES in the vicinity of E in the F AFM and PM states. The data represent k-space maps of states at energies corresponding to the lower and upper edges of the gap in the SDW
890
J. Scha¨fer et al. / Surface Science 454–456 (2000) 885–890
phase (T=300 K ), as well as the middle of the gap region in the SDW and PM phases (T= 570 K ) respectively. These data demonstrate impressively that the energy gap opens almost isotropically and with constant magnitude of 200±20 meV around this section of the electron octahedron at C. This finding of an isotropic energy gap is consistent with our band structure calculations.
5. Summary This spectroscopic study of the SDW phase transition at the Cr(110) surface shows an increased transition temperature of 440 K, with the approach of the order parameter being continuous and power-law like. The gap reaches a magnitude of ~200 meV at room temperature, and its extent covers the entire measured section of the electron octahedron. This might have important implications for interfacial magnetic coupling and resulting electronic properties.
Acknowledgements This work was supported by the US DOE under grant DE-FG06-86ER45275, and at the ALS under grant DE-AC03-76SF00098.
References [1] E. Fawcett, Rev. Mod. Phys. 60 (1988) 209. [2] D. Li et al., Phys. Rev. Lett. 78 (1997) 1154. [3] P. Fazekas, Electron Correlation and Magnetism, World Scientific, Singapore, 1999. [4] K. Machida, M.A. Lind, J.L. Stanford, J. Phys. Soc. Jpn. 53 (1984) 4020. [5] P.E.S. Persson, L.I. Johansson, Phys. Rev B 34 (1986) 2284. [6 ] L.E. Klebanoff et al., J. Magn. Mag. Mater. 54–55 (1986) 728. [7] Y. Sakisaka et al., Phys. Rev. B 38 (1988) 1131. [8] S.A. Werner, A. Arrott, H. Kendrick, Phys. Rev. 155 (1967) 528. [9] S. Asano, J. Yamashita, J. Phys. Soc. Jpn. 23 (1967) 714. [10] P. Blaha et al., WIEN97 code, Comput. Phys. Commun. 59 (1990) 339. [11] B. Sinkovic, B. Hermsmeier, C.S. Fadley, Phys. Rev. Lett. 55 (1985) 1227. [12] Y. Li, K. Baberschke, Phys. Rev. Lett. 68 (1992) 1208.