Figure 3: (Color online) Comparison between magnon disper- sion curves along the [001] direction for a ML Fe on W(110). The dots are experimentally obtained data (Ref. 1), whereas the thick purple line represents our numerically obtained data. For comparison, the experimental spin wave spectrum of bulk bcc Fe (corresponding to a spin wave stiffness constant of 280 meV ˚ A $^{2}$) (Ref. 37) is also displayed. ried out an atomistic spin dynamics simulation based on the obtained exchange interaction parameters. The cal- culated magnon spectrum is displayed in Fig. 3, together with the experimental data of Prokop et al. $^{1}$, and the experimentally determined spectrum for bulk bcc Fe $^{37}$. The agreement with the SPEELS data is good, particu- larly for wave vectors around q = 1 ˚ A − $^{1}$. Furthermore, the softening of the magnons in the monolayer system is captured. The estimate for the spin wave stiffness from our spin dynamics simulation at T = 120 K with partial DLM exchange interaction parameters is 105 meV ˚ A $^{2}$, which compares well with the experimental fit of 74 meV ˚ A $^{2}$. In our calculations, relativistic effects have only been taken into account in the form of the magnetic anisotropy energy. It has recently been demonstrated that the Financial support from the Swedish Research Coun- cil (VR), the European Research Council (ERC), the European Comission (EC), the Swedish Foundation for International Cooperation in Research and Higher Ed- ucation (STINT) and the Carl Tryggers Foundation is acknowledged. Computer simulations were performed at the Swedish National Computer centre (NSC). We grate- fully acknowledge L. Nordstr¨ om, A. Delin, E. Papaioan- nou and R. B. Muniz for valuable discussions. In conclusion, we have shown that combining first prin- ciples calculations with atomistic spin dynamics simula- tions provides a powerful tool for studies of magnetic ex- citations in low dimensional systems. Applied to the ref- erence system of a monolayer Fe on W(110), the method captures a significant magnon softening compared to bulk Fe, a finding which is in agreement with recent measure- ments. The microsopic reason why the magnon curve of this system is much softer than bulk bcc Fe can be found in three intertwined mechanisms. Firstly, the chemical relaxation between the Fe and W atoms influences the hy- bridization between Fe and W states, which modifies the exchange interactions. Secondly, the magnetic relaxation suggests that a partial DLM configuration is appropriate for the experimental situation at hand, and this config- uration also reduces the exchange interaction strenghts. Finally, the dynamic effects also soften the magnon curve, which is captured by finite temperature simulations. Dzyaloshinskii-Moriya interaction is also important for the present system, most notably by giving rise to asym- metric magnon spectra around the Γ point in the Brillu- oin zone $^{13,38}$. Although we have reproduced this result by inputting physically reasonable Dzyaloshinskii-Moriya vectors in the second term of Eq. (2) (data not shown), we do not find that the DMI gives rise to any signifi- cant change in the overall agreement between theory and experiment shown in Fig. 3. 4 Acknowledgments ∗ Electronic address: anders.bergman@fysik.uu.se 1 J. Prokop, W. X. Tang, Y. Zhang, I. Tudosa, T. R. F. Peixoto, Kh. Zakeri, and J. Kirschner, Phys. Rev. Lett. 102 , 177206 (2009). 2 R. B. Muniz and D. L. Mills, Phys. Rev. B 66 , 174417 (2002). 3 A. T. Costa, R. B. Muniz, J. X. Cao, R. Q. Wu, and D. L. Mills, Phys. Rev. B 78 , 054439 (2008). 4 A. Grechnev, I. Di Marco, M. I. Katsnelson, A. I. Lichten- stein, J. Wills, and O. Eriksson, Phys. Rev. B 76 , 035107 (2007). 5 I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Sieg- mann, J. St¨ ohr, G. Ju, B. Lu, and D. Weller, Nature (Lon- don) 428 , 831 (2004). 11 M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O. Pietzch, 10 T. Moriya, Phys. Rev. 120 , 91 (1960). 9 I. A. Dzyaloshinskii, Sov. Phys. JETP 5 , 1259 (1957). 8 H. Kubota, A. Fukushima, K. Yakushiji, T. Naga- hama, S. Yuasa, K. Ando, H. Maehara, Y. Nagamine, K. Tsunekawa, D. Djayaprawira, et al., Nat. Phys. 4 , 37 (2008). 7 J. C. Sankey, Y.-T. Cui, J. Z. Sun, J. C. Slonczewski, R. A. Buhrman, and D. C. Ralph, Nat. Phys. 4 , 67 (2008). 6 G. D. Fuchs, N. C. Emley, I. N. Krivorotov, P. M. Bra- ganca, E. M. Ryan, S. I. Kiselev, J. C. Sankey, D. C. Ralph, R. A. Buhrman, and J. A. Katine, Appl. Phys. Lett. 85 , 1205 (2004).