The effective spin dependent Debye temperature of Gd(0001)

Physics Letters A 324 (2004) 242–246 www.elsevier.com/locate/pla

The effective spin dependent Debye temperature of Gd(0001) Hae-Kyung Jeong a , R. Skomski a , C. Waldfried a , Takashi Komesu a , P.A. Dowben a,∗ , E. Vescovo b a Department of Physics and Astronomy, and the Center for Materials Research and Analysis, Behlen Laboratory, University of Nebraska,

Lincoln, NE 68588-0111, USA b National Synchrotron Light Source, Brookhaven National Laboratory, Upton, NY 11973, USA

Received 17 February 2004; accepted 26 February 2004 Communicated by R. Wu

Abstract The collective vibrational motions along the surface normal direction of the expansively strained Gd(0001) surface has been investigated. The effective Debye temperature, indicative of the dynamic motion of lattice normal to the surface, is 137 ± 35 K from X-ray photoemission. With angle-resolved spin polarized photoemission spectroscopy, the Debye temperature is determined to be 79 ± 35 K in spin majority, and 125 ± 35 K for spin minority. Residual spin-mixing contributions to bands with Stoner like behavior as well as contributions from the spin dependent plasmon modes for Gd(0001) are implicated.  2004 Elsevier B.V. All rights reserved. PACS: 68.35.Ja; 63.20.Kr; 73.90.+f Keywords: Surface Debye temperature; Rare earth; Lattice vibrations and electron–phonon interaction

It has been known for decades that electron– phonon interactions modify the electronic structures of solids [1]. There is also an effect of the electronic degrees of freedom on the elastic behavior, for example in photoemission experiments. In magnetic solids, these effects are all spin-dependent, as epitomized by phenomena such as spin–lattice relaxation and magnetic polarons. Here we are concerned with the response of an electron–phonon system to a spindependent perturbation.

* Corresponding author.

E-mail address: [email protected] (P.A. Dowben). 0375-9601/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.02.071

The addition or removal of an electron in ferromagnets modifies the local spin-dependent potential and exerts a mechanical force on the lattice, and the effect depends on the relative orientation of the photoelectrons spin moment and the net moment of the magnetic solid. As in the nonmagnetic case, the small ratio of electron mass m to ionic mass M means the problem amounts to the interaction between fast plasmonic modes and much slower phononic modes. When an electron is added or subtracted to the metal, the electrons redistribute to screen the transient photohole or added electron [2]. There is no splitting of the phonon band(s) due to spin [3], but spin–lattice coupling and spin–lattice relaxation are nonetheless expected in the photoemission or inverse photoemis-

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sion process. Spin dependent lattice dynamic effects as a result of the addition or removal of a spin polarized electron, should be most profound when the Debye temperature is low relative to the Curie temperature, but the Curie temperature must be easily accessible in experiment. To enhance the observation of these expected lattice contributions to photoemission final state effects, we choose to investigate strained gadolinium, known to have a higher Curie temperature than unstrained gadolinium, of about 340 K [4,5], but exploit the low Debye temperature of 137 ± 35 K [6], in agreement with the values determined from heat capacity (119 K) [7] and the elastic constants (128 K) [8,9] of gadolinium. We have investigated the Debye temperature, vibrational motion of the strained Gd(0001) surface, by the angle-dependent spin-polarized photoemission spectroscopy and X-ray photoemission spectroscopy. While the true surface Debye temperature, containing the in-plane and anharmonic motions, is difficult to measure in most surface spectroscopies [10], the effective surface Debye temperature can be readily obtained using LEED, XPS, EELS (electron energy loss spectroscopy), IPES (inverse photoemission spectroscopy) and other surface sensitive techniques [4,10–20]. Due to increases in the thermal vibrations, the intensity of an emitted or scattered electron beam exponentially decays, with increasing temperature as I = I0 exp(−2W ). We can calculate the Debye temperature with careful analysis of the intensity change as a function of temperature [10–13,19,20] noting 2W =

3h¯ 2 T ( k)2 , 2 2mkB ΘD

(1)

where W is the Debye–Waller factor, T is the temperature of the sample (in Kelvin), h( k) is the elec¯ tron momentum transfer, m is the mass of the scattering center, kB is the Boltzmann constant, and ΘD is the Debye temperature. The calculated effective surface Debye temperature is most indicative that the dynamic motion of vibrational modes normal to the surface [10–12]. Strained thin films of gadolinium with an increased lattice constant of approximately 4% as compared to Gd(0001), and a well ordered hexagonal surface unit cell were obtained by growing Gd on the corrugated surface of Mo(112) [4,5,21]. Spin- and angle-resolved photoemission spectra on the Gd films of 15 to

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40 monolayers (ML) thickness were acquired in a UHV system at the new U5UA undulator beamline at the national synchrotron light source (NSLS), as described in detail elsewhere [22]. The photon energy was 35.2 eV, incident at an angle of 65◦ relative to the surface normal. The combined energy and angular resolution were better than 0.15 eV and ±1◦ , respectively. For XPS, we used the Mg Kα line, 1256.3 eV, and a GAMMA DATA SCIENTA SES-100 hemispherical electron energy analyzer. To calculate the effective Debye temperature, from spin dependent photoemission or inverse photoemission features, only the Stoner like bands can be used, otherwise the spin minority intensity will increased as temperature increases due to the spin mixing behavior, while the spin majority intensity can decrease without any dynamical scattering considerations [23]. The problem presented by gadolinium is that the band structure deviates dramatically from Stoner behavior both for the bulk bands [24–26] and surface bands [23, 27,28], particular near the Curie temperature, and spin mixing, not Stoner, behavior dominates the unoccupied band structure [29–31]. Careful selection of the Stoner like bands is therefore very important. Fig. 1 shows spin-polarized photoemission spectra on the strained Gd(0001) as a function of temperature, as published elsewhere [4]. Several band can be identified, but the most evident Stoner like behavior is observed for the bands denoted G2, M3 and M4 in deconvolution of the spin polarized photoemission spectra at Γ¯ (right panel) and M¯ (left panel) [4]. Using these bands, because of their dominant Stoner like behavior, we have extracted the intensities as a function of temperature, as plotted in Fig. 2, minus the lowest temperature of the maximum intensity (151 K), for both spin majority and spin minority. From fitting the experimental intensities with temperature, abstracted from the data in Fig. 2, Debye temperatures of spin majority and minority bands, 79 ± 35 K and 125 ± 35 K, respectively, are obtained. A spin integrated Debye temperature is 137 ± 35 K from X-ray photoemission and corresponds to the Debye temperature of 119 K [7] and 128 K [8,9] determined from heat capacity and elastic constant, respectively. As there is no spin dependent phonon splitting [3], why is the apparent spin majority Debye temperature lower than spin minority Debye temperature?

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Fig. 1. The temperature dependence of the valence band of strained Gd near the zone edge (left) and at the zone center (right). Spin-majority and spin-minority components are indicated by up and down triangles. TcB is 340 K. Deconvolution of the spin majority valence bands of Gd for zone center Γ¯ (right) and zone edge M¯ (left) are indicated. G2, M3 and M4 are Stoner-like bands while the other bands have some clear spin-mixing behavior.

Fig. 3 shows the model used to investigate the phenomenon of an effective spin-dependent Debye temperature. Aside from the displacement xi of the ith nucleus, we must consider two degrees of freedom associated with the atomic spin-up and spin-down electron clouds. For free atoms, eigenmode analysis of the electron-cloud displacements yields an ordinary plasmonic mode pi , closely related to the dielectric susceptibility, and an out-of-phase magnetic mode si . (In principle, the force constant of the magnetic mode can be measured by applying a strong magnetic field

gradient, but typical magnetostatic field gradients are much weaker than the exchange-field gradients operative and investigated in the present system.) Using a force-constant approach [32] and diagonalizing the equations of motion for modes k = kex yields    ω2   0 qsx s s s   p  . (2) ωp2 qpx ω2  p  =  0 x x qxs qxp ω02 1−cos(ka) 2

Here ωs and ωp are the free-atom magnetic and dielectric resonance frequencies, respectively, ω0 is

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Fig. 4. Modes for free atoms (dashed lines) and atoms in a solid (solid lines).

Fig. 2. Logarithm of the intensities of the photoelectron obtained in SPPES, after background (Ibg ) subtraction and normalization to the value (I0 ) at the lowest temperature for (a) spin majority (79 K) and (b) spin minority (125 K). (c) ln[(I − Ibg )/I0 ] versus temperature from Gd 4f peaks using XPS. The direction of the emitted electrons was along the surface normal. The spin-integrated Debye temperature is 137 K.

Fig. 3. Magnetic phonon–electron interaction model. The nuclei are in the very center of the atoms (gray).

the maximum phononic frequency (Debye frequency), and the qab are interatomic matrix elements. The nondiagonal matrix elements describe the inter-atomic

interactions of the spin-up and spin-down charge clouds shown in Fig. 3. Of course, the eigenvalues of Eq. (2) depend very strongly on the involved parameters. Fig. 4 shows how the resonance frequencies change due to the nondiagonal terms. The dielectric (plasmonic) and magnetic modes have a different though usually small effect on the phononic mode, but there is no splitting into two different phononic modes. However, when an electron of a given spin leaves or enters the solid, then the number of phonons and their wave vector distribution depends on the electron’s spin direction. The differences in Debye temperature extracted from the spin majority (79 ± 35 K) versus the spin minority (125 ± 35) bands is far greater than can be accounted for by a plasmonic model of electron cloud displacements. In spite of every effort to choose bands that exhibit largely Stoner like band ferromagnetism, we must confront the possibly that the Stoner model [33] is an inadequate description of the all of the 5d/6s bands of strained Gd(0001) on Mo(112). The situation is then like for the much less strained gadolinium, Gd(0001) grown on W(110), where spin mixing behavior contributions are observed to even the apparently Stoner like bands [24,25]. Given that we would expect greater intra-atomic overlap of the 5d wave function with the highly localized 4f levels with increasing expansive strain, avoiding spin-mixing [26, 34,35] contributions should be more difficult than for the less strained gadolinium film.

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In conclusion, we investigated the spin-dependent Debye temperature on strained Gadolinium using spin-resolved photoemission spectroscopy. The experimentally extracted Debye temperature was 79 ± 35 K for spin majority and 125 ± 35 K for spin minority. When the spin polarized electron enters or leaves the gadolinium, it will locally polarize the charge cloud giving rise to a local plasmon–phonon coupling. While final state effects are expected in photoemission, particularly in the case of gadolinium [36], this is perhaps one of the more unusual final state effects possible in ferromagnetic systems. The effect is dramatic in gadolinium, perhaps in part because of the very low Debye temperature for a refractory metal, and because of the very low plasmon energy.

Acknowledgements The support of the Office of Naval Research, the NSF “QSPINS” MRSEC (DMR 0213808), and the Nebraska Research Initiative are gratefully acknowledged.

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