First-principles study of stabilities, electronic structures, and mechanical properties of <i>M</i>(Mn, Ti, Mo)-doped <i>α</i>-Fe

HUA Sunmingqiang; KONG Detong; HU Xiao; SHE Xu; WANG Xiao; WANG Yuan

doi:10.7498/aps.74.20251044

Abstract
Ferrite (α-Fe), as a fundamental phase of steel materials, plays a decisive role in determining their macroscopic mechanical behaviors through its microscopic properties, particularly in engineering applications involving resistance to plastic deformation and fracture, fatigue resistance, wear resistance, and low-temperature toughness. Therefore, alloying elements are commonly introduced to improve the performance of steel via mechanisms such as grain refinement strengthening and precipitation strengthening. However, in these strengthening mechanisms, the effects of doped alloying elements on the stability, electronic structure, and mechanical properties of ferrite itself have not been thoroughly investigated. In this study, orthogonal experimental design and first-principles calculations are employed to investigate the effects of ternary alloy doping with M (Mn, Ti, Mo) on the stabilities, electronic structures, and mechanical properties of a ferrite-based supercell model Fe_16-x-y-zMn_xTi_yMo_z (x, y, or z = 0, 1, or 2), aiming to provide both theoretical insight and experimental reference for improving the comprehensive performance of ferrite-based steels by modifying the properties of the matrix phase. The results of the formation enthalpy (H_form) calculations indicate that all solid solutions have negative formation enthalpies, suggesting that they can form spontaneously. Among them, Ti doping is the most favorable for solid solution formation, followed by Mn, with Mo being the least favorable. The Fe₁₃Ti₁Mo₂ configuration is the easiest to form spontaneously. The cohesive energy (E_coh) results demonstrate that all solid solutions exhibit structural stabilities. Fe₁₃Ti₁Mo₂ has the largest (most negative) cohesive energy of –477.96 eV, indicating that it possesses the highest structural stability. The contribution of Mo doping to stability enhancement is the greatest, followed by Ti, while the influence of Mn is the smallest. Electronic structure calculations reveal that M doping consistently reduces the density of states (DOS) at the Fermi level for Fe_16-x-y-zMn_xTi_yMo_z. The lowest DOS at the Fermi level is found to be 4.294 in Fe₁₃Ti₁Mo₂, indicating enhanced hybridization and overlap between Mn 3d, Ti 3d, Mo 4d, and Fe 3d states. This strong hybridization leads to a decrease in the Fermi level and contributes to the high stability of the Fe₁₃Ti₁Mo₂ phase. Mechanical property calculations indicate that M doping reduces the Young’s modulus (E) and Vickers hardness (H_V) of the solid solutions. However, the K values (K = G_H/B_H) are all greater than 1.75, and Poisson’s ratios (ν) exceed 0.26, implying that while stiffness and hardness decrease, the ductility of the materials is improved. This study provides valuable guidance for designing ductile and tough ferrite-based steel materials.

Keywords:
ferrite (α-Fe) /

Mn /

Ti /

Mo doping /

first-principles calculations /

mechanical properties
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图 1 Fe_16-x-y-zMn_xTi_yMo_z超胞模型　(a) Fe₁₆; (b) Fe₁₃Ti₂Mo₁; (c) Fe₁₃Ti₁Mo₂; (d) Fe₁₃Mn₁Ti₂; (e) Fe₁₃Mn₁Ti₁Mo₁; (f) Fe₁₃Mn₁Mo₂; (g) Fe₁₃Mn₂Ti₁; (h) Fe₁₃Mn₂Mo₁; (i) Fe₁₀Mn₂Ti₂Mo₂

Figure 1. Supercell model of Fe_16-x-y-zMn_xTi_yMo_z: (a) Fe₁₆; (b) Fe₁₃Ti₂Mo₁; (c) Fe₁₃Ti₁Mo₂; (d) Fe₁₃Mn₁Ti₂; (e) Fe₁₃Mn₁Ti₁Mo₁; (f) Fe₁₃Mn₁Mo₂; (g) Fe₁₃Mn₂Ti₁; (h) Fe₁₃Mn₂Mo₁; (i) Fe₁₀Mn₂Ti₂Mo₂.

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图 2 态密度图和分波态密度图　(a) Fe₁₆; (b) Fe₁₃Ti₂Mo₁; (c) Fe₁₀Mn₂Ti₂Mo₂; (d) Fe₁₃Ti₁Mo₂; (e) Fe₁₃Mn₁Ti₁Mo₁; (f) Fe₁₃Mn₁Ti₂

Figure 2. Density of states and partial density of states plots: (a) Fe₁₆; (b) Fe₁₃Ti₂Mo₁; (c) Fe₁₀Mn₂Ti₂Mo₂; (d) Fe₁₃Ti₁Mo₂; (e) Fe₁₃Mn₁Ti₁Mo₁; (f) Fe₁₃Mn₁Ti₂.

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图 3 差分电荷密度图　(a) Fe₁₆; (b) Fe₁₃Ti₂Mo₁; (c) Fe₁₀Mn₂Ti₂Mo₂; (d) Fe₁₃Ti₁Mo₂; (e) Fe₁₃Mn₁Ti₁Mo₁; (f) Fe₁₃Mn₁Ti₂

Figure 3. Differential charge density diagram: (a) Fe₁₆; (b) Fe₁₃Ti₂Mo₁; (c) Fe₁₀Mn₂Ti₂Mo₂; (d) Fe₁₃Ti₁Mo₂; (e) Fe₁₃Mn₁Ti₁Mo₁; (f) Fe₁₃Mn₁Ti₂.

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表 1 M掺杂α-Fe的正交试验设计及原子百分比

Table 1. Orthogonal experimental design and atomic percentage content of M-doped α-Fe.

超胞	元素M原子百分比/%
超胞	Mn	Ti	Mo
Fe₁₆	0	0	0
Fe₁₃Ti₂Mo₁	0	12.5	6.25
Fe₁₃Ti₁Mo₂	0	6.25	12.5
Fe₁₃Mn₁Ti₂	6.25	12.5	0
Fe₁₃Mn₁Ti₁Mo₁	6.25	6.25	6.25
Fe₁₃Mn₁Mo₂	6.25	0	12.5
Fe₁₃Mn₂Ti₁	12.5	6.25	0
Fe₁₃Mn₂Mo₁	12.5	0	6.25
Fe₁₀Mn₂Ti₂Mo₂	12.5	12.5	12.5

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表 2 Fe_16-x-y-zMn_xTi_yMo_z的ε, H_form和E_coh的计算值

Table 2. Calculated values of ε, H_form and E_coh for Fe_16-x-y-zMn_xTi_yMo_z.

超胞	晶胞体积	ε/%	H_form /(kJ·mol^–1)	E_coh /(kJ·mol^–1)
Fe₁₆	178.79	—	–7.51	–449.39
Fe₁₃Ti₂Mo₁	190.51	6.56	–16.52	–477.95
Fe₁₃Ti₁Mo₂	190.70	6.66	–10.73	–477.96
Fe₁₃Mn₁Ti₂	185.61	3.81	–15.48	–460.85
Fe₁₃Mn₁Ti₁Mo₁	187.73	5.00	–10.40	–461.58
Fe₁₃Mn₁Mo₂	188.92	5.67	–5.37	–462.35
Fe₁₃Mn₂Ti₁	182.89	2.29	–9.53	–444.06
Fe₁₃Mn₂Mo₁	184.48	3.18	–4.58	–445.51
Fe₁₀Mn₂Ti₂Mo₂	194.98	9.06	–12.97	–473.45

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表 3 Fe_16-x-y-zMn_xTi_yMo_z的ε, H_form和E_coh的正交试验分析

Table 3. Orthogonal test analysis of ε, H_form and E_coh of Fe_16-x-y-zMn_xTi_yMo_z.

指标	ε/%			H_form/(kJ·mol^–1)			E_coh/(kJ·mol^–1)
指标	Mn	Ti	Mo	Mn	Ti	Mo	Mn	Ti	Mo
K₀	13.22	8.85	6.1	–34.76	–17.46	–32.53	–1405.30	–1357.24	–1354.30
K₁	14.48	13.95	14.47	–31.25	–30.66	–31.50	–1384.78	–1383.60	–1385.04
K₂	14.53	19.43	21.39	–27.08	–44.97	–29.07	–1363.02	–1412.25	–1413.76
R	1.31	10.58	15.29	7.68	27.51	3.46	42.28	55.01	59.46
Ranking	Mo＞Ti＞Mn			Ti＞Mn＞Mo			Mo＞Ti＞Mn

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表 4 Fe_16-x-y-zMn_xTi_yMo_z的独立弹性常数和力学稳定性

Table 4. Independent elastic constants and mechanical stability of Fe_16-x-y-zMn_xTi_yMo_z.

试样	C₁₁	C₁₂	C₄₄	C₁₁-C₁₂	C₁₁+2C₁₂
Fe₁₆	296.273	159.033	126.763	137.240	614.339
Fe₁₃Ti₂Mo₁	292.506	158.281	117.403	134.225	609.068
Fe₁₃Mn₂Ti₂Mo₂	293.869	148.470	116.251	145.399	590.809
Fe₁₃Ti₁Mo₂	210.409	103.675	92.415	106.734	417.759
Fe₁₃Mn₁Ti₁Mo₁	291.402	158.680	123.704	132.722	608.762
Fe₁₃Mn₁Ti₂	275.790	150.536	112.712	125.254	576.862

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表 5 Fe_16-x-y-zMn_xTi_yMo_z的力学性能参数

Table 5. Mechanical property parameters of Fe_16-x-y-zMn_xTi_yMo_z.

超胞	B/GPa			G/GPa			E/GPa	K	v	H_V
超胞	B_V	B_R	B_H	G_V	G_R	G_H	E/GPa	K	v	H_V
Fe₁₆	204.779	204.779	204.779	103.506	94.675	99.091	255.983	2.067	0.292	13.737
Fe₁₃Ti₂Mo₁	210.383	210.383	210.383	103.675	92.403	98.039	254.573	2.146	0.298	13.206
Fe₁₀Mn₂Ti₂Mo₂	196.401	196.401	196.401	98.830	91.632	95.231	245.942	2.062	0.291	13.272
Fe₁₃Ti₁Mo₂	202.961	202.961	202.961	97.287	90.163	93.725	243.667	2.165	0.300	12.496
Fe₁₃Mn₁Ti₁Mo₁	202.921	202.921	202.921	100.767	91.929	96.348	249.548	2.106	0.295	13.168
Fe₁₃Mn₁Ti₂	192.287	192.287	192.287	92.678	85.395	89.036	231.393	2.159	0.299	11.933

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First-principles study of stabilities, electronic structures, and mechanical properties of M(Mn, Ti, Mo)-doped α-Fe

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First-principles study of stabilities, electronic structures, and mechanical properties of M(Mn, Ti, Mo)-doped α-Fe

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