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Supplementary Information
Design rules of pseudocapacitive electrode materials: ion Adsorption, diffusion and electron transmission over prototype TiO2
Lijing Wang#, Xiaolong Yao#, Da Chen, Jin Wang, Zhenzhou Zhang, Jieyu Liu, Tianquan Lin, Wei-Hua Wang, Zhanglian Hong, Fuqiang Huang, and Weichao Wang*
* Correspondence and requests for materials should be addressed to W. W (email:
weichaowang@nankai.edu.cn).
# These authors contributed equally to this work.
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1. Computational details of the transfer matrix method (TMM)
The width of depletion layer 𝑊 in semiconductor contacted with electrolyte is calculated by the following equations1:
𝑊 = √2𝜀0𝜀𝑟
𝑒𝑁𝐷 𝑉𝑏𝑖 (1)
where 𝜀0 is the vacuum dielectric constant, 𝜀𝑟 is the relative dielectric permittivity of TiO2, which is set to be 32, 𝑉𝑏𝑖 is the built-in voltage, and 𝑁𝐷 is the dopant concentration of semiconductor.
The value of 𝑁𝐷 is calculated to be 5.31×1021cm-3 (VO) and 2.66×1021cm-3 (VO+NO), respectively.
For pristine TiO2, the intrinsic carrier concentration is calculated based on the following formulas2:
𝑛𝑖2 = 𝑁𝑐𝑁𝑣𝑒𝑥𝑝 [−𝐸𝑔
𝑘𝑇] (2) 𝑁𝑐 = 2(2𝜋𝑚𝑛∗𝑘𝑇
ℎ2 )3/2 (3) 𝑁𝑣 = 2(2𝜋𝑚𝑝
∗𝑘𝑇
ℎ2 )3/2 (4)
where 𝑛𝑖 is the intrinsic carrier concentration, 𝑁𝑐 and 𝑁𝑣 is effective density of states in the conduction band and the valence band, respectively, 𝐸𝑔 is the bandgap, which is 3.30 eV for pristine TiO2.
After derivation, the 𝑁𝑐 and 𝑁𝑣 are calculated as following:
𝑁𝑐 = 2.51× 1019×(𝑚𝑛∗
𝑚0)3/2𝑐𝑚−3 (5) 𝑁𝑣 = 2.51× 1019×(𝑚𝑝
∗
𝑚0)3/2𝑐𝑚−3 (6)
With the effective mass 𝑚𝑛∗ = 0.0468𝑚0 , 𝑚𝑝∗ = 0.6614𝑚0, the effective density of states could be obtained:
NC = 2.54×1017 cm-3.
NV = 1.35×1019 cm-3.
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The intrinsic carrier concentration of pristine TiO2 is calculated to be ni = 5.09×10-10 cm-3
Finally, the width of the depletion layer in three configurations is obtained.
𝑊𝑉𝑂 = 4.47× 10−10𝑚 𝑊𝑉𝑂+𝑁𝑂 = 6.42× 10−10𝑚
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2. Calculation details of electrical double layer (EDL) capacitance and quantum capacitance
The EDL capacitance is calculated by the formula of 𝐶𝐸𝐷𝐿 =0𝑟
𝑑
where 0 is the vacuum dielectric constant, 𝑟 is the dielectric constant of the electrolyte solution which is set to be 6.0, and d is the width of the Helmholtz layer which is defined as the radius of the hydronium ion 2.8 Å.
The gravimetric EDL capacitance is obtained by 𝐶𝐸𝐷𝐿,𝑔 =0𝑟𝑆
𝑑𝑀
where S and M are the area and mass of the supercell, respectively.
The gravimetric EDL capacitance of pristine and defective TiO2 are calculated to be 47 F/g and 48 F/g, respectively.
The following is the calculation details of the quantum capacitance.
𝐶𝑄 is defined as dσ/dΦ, where dσ and dΦ are the differential charge density and potential, respectively.
The excess charge density above the Fermi level is obtained by 𝜎 = −𝑒 ∫ 𝐷(𝐸)[𝑓(𝐸 + 𝑒Ф) − 𝑓(𝐸)]𝑑𝐸
+∞
−∞
where 𝐷(𝐸) is the density of states, 𝑓(𝐸) is the Fermi−Dirac distribution function, 𝐸 is the energy relative to the Fermi level. By the derivative of charge with respect to potential, 𝐶𝑄 = 𝑒2∫ 𝐷(𝐸)𝐹𝑇(𝐸 + 𝑒Ф)𝑑𝐸
+∞
−∞
𝐹𝑇(𝐸) = 1
4𝑘𝑏𝑇𝑠𝑒𝑐ℎ2( 𝐸 2𝑘𝑏𝑇)
𝐹𝑇(𝐸)is the thermal broadening function.
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3. Figures and Tables
Supplementary Fig. 1 Calculated relative energy of the second O vacancy in the presence of VO9.
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Supplementary Fig. 2 Defect formation energy of TiO2 withthe chemical potential of N in N-poor condition (TiN).
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Supplementary Fig. 3 Configurations of H insertion in pristine TiO2 (101), TiO2 with one oxygen vacancy (VO), and TiO2 with one oxygen vacancy and one substitutional nitrogen (VO+NO). The number x represents the number of hydrogen adsorption.
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Supplementary Fig. 4 Density of states of TiO2 bulk in the anatase phase.
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Supplementary Fig. 5 Normalized charge transfer number of pristine and defective TiO2. Orange, blue and green lines indicate pristine TiO2, TiO2-VO and TiO2-VO+NO, respectively.
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Supplementary Fig. 6 The average electrostatic potential along the c axis of pristine and defective TiO2.
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Supplementary Fig. 7 The average electrostatic potential along the c axis of TiO2 with H adsorption. The three columns of figures with different colors are pristine TiO2 (orange), VO
(blue), and VO+NO (green), respectively. The gray dotted line represents the Fermi level.
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Supplementary Fig. 8 Hydrogen migration energy barrier from O6 to O12 in VO.
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Supplementary Fig. 9 Hydrogen migration energy barrier from O6 to O12 in VO+NO.
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Supplementary Fig. 10 Hydrogen migration energy barrier from O5 to O9 in VO.
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Supplementary Fig. 11 Hydrogen migration energy barrier from O5 to O9 in VO+NO.
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Supplementary Fig. 12 The amount of charge of H in the initial state(IS), transition state(TS) and final state(FS) in pristine TiO2, TiO2-VO and TiO2-VO+NO.
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Supplementary Fig. 13 Band structure of pristine anatase TiO2 (101).
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Supplementary Fig. 14 Band structures of defective TiO2 (101): (a) TiO2-VO, (b)TiO2- VO+NO.
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Supplementary Fig. 15 Quantum capacitances of pristine TiO2 (101), TiO2 with one oxygen vacancy (VO), and TiO2 with one oxygen vacancy and one substitutional nitrogen (VO+NO).
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Supplementary Fig. 16 The density of states of defective TiO2 with 1×2 supercells.
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Supplementary Fig. 17 The defect formation energy of one oxygen vacancy in different supercells versus oxygen chemical potential.
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Supplementary Table 1 Test of the thickness of the vacuum layer by the structure with 3 TiO2 atomlayers. dvac is the thickness of the vacuum layer, WF and WFH represent the work function of pristine TiO2 (101) before and after one hydrogen atom adsorption, respectively.
QH is the Bader charge of adsorbed H.
dvac (Å) WF (eV) WFH (eV) QH (e)
15 6.97 3.19 0.34
20 30
6.97 6.97
3.19 3.19
0.36 0.38
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Supplementary Table 2 Test of the atom layer of the TiO2 surface model using a 15Å vacuum layer. N is the number of TiO2 atom layers.
N WF (eV) WFH (eV) QH (e)
3 6.97 3.19 0.34
4 7.01 3.17 0.31
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Supplementary Table 3 Sites of one substitutional nitrogen (NO) on the surface of anatase TiO2 (101). Numbers represent the sites of oxygen substituted by nitrogen.
NO ΔE (eV)
1 0.25
4 0.23
5 0.15
7 0.17
9 0
11 0.26
14 0.19
15 0.21
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Supplementary Table 4 Sites of substitutional nitrogen on the surface of anatase TiO2 (101) with one oxygen vacancy. Numbers represent the sites of oxygen substituted by nitrogen.
VO+NO ΔE (eV)
1 0.57
2 0.92
3 0.56
4 0.45
5 0.60
6 0.65
7 0
8 0.32
10 0.58
13 0.44
11 0.66
15 0.27
14 0.54
12 0.61
16 0.52
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Supplementary Table 5 The charge states change of each Ti atom in pristine and defective TiO2 with increasing hydrogen adsorption concentration. When the hydrogen concentration is different, the values in brackets are the adsorption concentrations of defective TiO2.
H concentration
Charge states change
TiO2 TiO2-VO TiO2-VO+NO
37.5% (35.3%) 0.04 0.05 0.05
50% (47.1%) 0.09 0.11 0.10
62.5% (64.7%) 0.16 0.14 0.13
75% (76.5%) 0.22 0.20 0.19
87.5% (88.2%) 0.28 0.27 0.24
100% 0.36 0.32 0.30
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Supplementary Table 6 Sites of the oxygen vacancy on the surface of anatase TiO2
supercells. The five sites, s1 to s5, represent the positions of oxygens from the surface to the bulk.
Supercells Sites ΔE (eV)
1×2
s1 0.12
s2 1.37
s3 0.77
s4 0.50
s5 0
1×3
s1 0.15
s2 1.12
s3 0.57
s4 0.15
s5 0
1×4
s1 0.20
s2 0.64
s3 0.44
s4 1.05
s5 0
2×3
s1 0.17
s2 1.18
s3 0.54
s4 0.17
s5 0
28 References
[S1] Tan, M. X., Laibinis, P. E., Nguyen, S.T., Kesselman, J. M., Stanton, C. E., Lewis, N. S.
Progress in Inorganic Chemistry (John Wiley & Sons, Inc., 2007).
[S2] S. Sze, Physics of Semiconductor Devices (Wiley, New York 1981).
