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We present a theoretical analysis of the temperature-magneticfield-concentration phase diagram of the multiferroic Mn$_{1-x}$M$_{x}$WO$_{4}$(M=Fe, Zn, Mg), which exhibits three ordered phases, with collinear andnon-collinear incommensurate and with a commensurate magnetic order. The middlephase is also ferroelectric. The analysis uses a semi-phenomenological Landautheory, based on a Heisenberg Hamiltonian with a single-ion anisotropy. With asmall number of adjustable parameters, the Landau theory gives an excellent fitto all three transition lines, as well as the magnetic and the ferroelectricorder parameters. The fit of the magnetic and ferroelectric order parameters isfurther improved by including the effect of fluctuations near the transitions.We demonstrate the highly frustrated nature of these materials and suggest asimple explanation for the dramatic effects of doping with different magneticions at the Mn sites. The model enables an examination of different sets ofexchange couplings that were proposed by a number of groups. Smalldiscrepancies are probably a consequence of small errors in the experimentalmagnetic parameters. In addition, using the Ginzburg criterion we estimate thetemperature range in which fluctuations of the order parameters becomeimportant.

Landau theory for the phase diagram of the multiferroic Mn$_{1-x}$(Fe,Zn,Mg)$_{x}$WO$_{4}$