Solution of two-dimensional Dirac equation by sectioned series expansion method

SUN Songyang; ZHU Jialin

doi:10.7498/aps.74.20250302

Abstract

With the discovery of two-dimensional materials like graphene, the relativistic two-dimensional Dirac equation has received increasing attention from researchers. Accurately solving the Dirac equation in electromagnetic fields is the foundation for studying and manipulating quantum states of Dirac electrons. Sectioned series expansion method is successful and accurate in solving Schrödinger equation under complex electromagnetic fields. Dirac equation is a system of coupled first-order differential equations with undermined eigenvalues, and it is more difficult to solve. By applying the sectioned series expansion principle to Dirac equation and conducting series expansions in regular, Taylor and irregular regions, we obtain an accurate method with wide applicability. With the method, a universal criterion for bound states of Dirac electrons in electromagnetic fields is derived and the energy levels and wave functions of bound states can be accurately calculated. The criterion given in the main text body shows that the magnetic field and mass field help to confine Dirac electrons while the electric field tends to deconfine them due to Klein tunneling. When the highest power of the electric potential is equal to that of the magnetic vector potential or the mass field, confined-deconfiend states depend on the comparison of their coefficients. We apply the method to two cases: one is massive Dirac electron in Coulomb electric potential (relativistic two-dimensional hydrogen-like atom) and the other is Dirac electron in uniform mangetic field (mangetic vector potenial is A = 1/2Br) and linear electric potential V = Fr. The energy levels of the hydrogen-like atom are calculated and compared with analytical solutions, demonstrating the exceptional accuracy of the method. By solving Dirac equation under uniform magnetic field and linear electric potential, the method proves to be broadly applicable to the solutions of Dirac equation under complex electromagnetic fields. Under uniform magnetic field B and V = Fr, as the F increases, level orders of negative energy states change and at the critical point F = 0.5B, the bound states of positive ones still exist while only certain negative ones can exist on condition that their energies exceed zero. The sectioned series expansion method provides an effective computational framework for Dirac equation and it deepens our understanding of relativistic quantum mechanics.

Keywords:

Dirac equation /
sectioned series expansion method /
bound states /
relativistic two-dimensional hydrogen-like atom

Authors and contacts

SUN Songyang¹,
ZHU Jialin²

1.
College of Physics, Changchun University of Science and Technology, Changchun 130022, China
2.
Department of Physics, Tsinghua University, Beijing 100084, China

Corresponding author: SUN Songyang, sunsongyang@cust.edu.cn

Funds: Project supported by the Scientific Research Foundation of the Education Department of Jilin Province, China (Grant No. JJKH20250474KJ) and the Scientific Research Program of the Higher Education Association of Jilin Province, China (Grant No. JGJX24B10).

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References

[1]	Semenoff G W 1984 Phys. Rev. Lett. 53 2449 Google Scholar
[2]	Sun S, Zhu J L 2014 Phys. Rev. B 89 155403 Google Scholar
[3]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666 Google Scholar
[4]	Feng B, Ding Z, Meng S, Yao Y, He X, Cheng P, Chen L , Wu K 2012 Nano Lett 12 3507 Google Scholar
[5]	Gomes K K, Mar W, Ko W, Guinea F, Manoharan H C 2012 Nature 483 306 Google Scholar
[6]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A V 2005 Nature 438 197 Google Scholar
[7]	Elahi M M, Vakili H, Zeng Y, Dean C R, Ghosh A W 2024 Phys. Rev. Lett. 132 146302 Google Scholar
[8]	陈晓娟, 徐康, 张秀, 刘海云, 熊启华 2023 物理学报 72 237201 Google Scholar Chen X J, Xu K, Zhang X, Liu H Y, Xiong Q H 2023 Acta Phys. Sin. 72 237201 Google Scholar
[9]	冉扬强, 薛立徽, 胡嗣柱 2002 物理学报 51 2435 Google Scholar RanY Q, Xue L H, Hu S Z 2002 Acta Phys. Sin. 51 2435 Google Scholar
[10]	Silbar R R, Goldman T 2010 Eur. J. Phys. 32 217
[11]	Kandemir B S, Mogulkoc A 2010 Eur. Phys. J. B 74 535 Google Scholar
[12]	Lee C M , Lee R C H, Ruan W Y, Chou M Y 2010 Appl. Phys. Lett. 96 212101 Google Scholar
[13]	Zhang Y H, Tang L Y, Zhang X Z, Shi T Y, Mitroy J 2012 Chin. Phys. Lett. 29 063101 Google Scholar
[14]	Giavaras G, Maksym P A, Roy M 2009 J. Phys. : Condens. Matter 21 102201 Google Scholar
[15]	Zhu J L, Xiong J J, Gu B L 1990 Phys. Rev. B 41 6001 Google Scholar
[16]	Ezawa M 2012 New J. Phys. 14 33003 Google Scholar
[17]	Fishbane P M, Gasiorowicz S G, Johannsen D C, Kaus K 1983 Phys. Rev. D 27 2433 Google Scholar
[18]	Batchelor M T, Henry R A, Lu X 2023 AAPPS Bulletin 33 29 Google Scholar
[19]	Zhu J L, Sun S 2012 Phys. Rev. B 85 035429 Google Scholar
[20]	王竹溪, 郭敦仁 2012 特殊函数概论(北京: 北京大学出版社) 第39—58页 Wang Z X, Guo D R 2012 Special Functions (Beijing: Peking University Press) pp39–58
[21]	Giavaras G , Nori F 2010 Appl. Phys. Lett. 97 243106
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[24]	Lukose V, Shankar R, Baskaran G 2007 Phys. Rev. Lett 98 116802 Google Scholar

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图 1 分区级数解法示意图, 径向波函数在不同区间分别用级数精确表示出来, 第0个区间是正则区(regular region), 最后是非正则区(irregular region), 中间都是泰勒区(Taylor region)

Figure 1. The sketch of the sectioned series expansion method, the radial wavefunction is exactly expressed by different series expansions in separate regions: The zeroth region is regular one, the last is irregular one and others between them are Taylor ones.

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图 2 在库仑电势中的α = –0.1时用分区级数解法计算相对论氢原子能级的径向波函数上下分量(R_A和R_B), 在计算时取M = 500 meV, 粗线(细线)分别代表上下分量, 右上方的图是对局部波函数的放大, (a), (b)分别是j = –0.5和j = 0.5时最低的3个能级的波函数

Figure 2. The upper and lower components (R_A和R_B) of the radial wave function of the relativistic hydrogen atom calculated by sectioned series expansion method when α = –0.1 in the Coulomb electric potential and M = 500 meV, bold (thin) lines represent the upper (lower) components and the insets in the top right are the zooms of parts of the functions: (a), (b) The three lowest energy levels of j = –0.5 and j = 0.5, respectively.

DownLoad: Full-Size Img PowerPoint

图 3 在均匀磁场B = 10 T时, 总角动量量子数j = –1/2, –3/2, 1/2, 3/2的能级随着线性电势V = Fr的变化, 其中(n, j)前面的负号表示负能态, F = 5时的垂直虚线是为了观察每个能级是否存在束缚态能级

Figure 3. The energy levels of the total angular momentum quantum number j = –1/2, –3/2, 1/2, 3/2 as a function of the linear electric potential, the minus signs in front of (n, j) indicates the negative energy states, and the vertical dashed line at F = 5 is to observe whether there is a bound state for every energy level.

DownLoad: Full-Size Img PowerPoint

图 4 Dirac电子在F = 0.5B时均匀磁场和线性电势下j = –1/2的径向波函数的上下分量(R_A和R_B) (a), (b)分别是原来负能态(0, –1/2)和(1, –1/2)在F = 0.5B时转变为正能态的波函数; (c), (d) 分别是原来正能态(0, –1/2)和(1, –1/2)在F = 0.5B时的波函数

Figure 4. The upper and lower components (R_A和R_B) of the radial wave function of Dirac electron at F = 0.5B in homogenous magnetic field and linear electric potential when j = –1/2: (a), (b) The original negative energy levels (0, –1/2) and (1, –1/2); (c), (d) the original positive energy levels (0, –1/2) and (1, –1/2).

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表 1 在库仑电势中的α = –0.1时用分区级数解法计算相对论氢原子能级的数值解与(53)式的解析解比较, 计算时取M = 500 meV, 能级均除以M

Table 1. Comparison between numerical energy levels by sectioned series expansion method and analytical energy levels by Eq. (53) when α = –0.1 in the Coulomb electric potential and M = 500 meV, with all energy levels divided by M.

(n, j)	E_{n, j} 数值解	E_{n, j} 解析解	相对误差	(n, j)	E_{n, j} 数值解	E_{n, j} 解析解	相对误差
0, –1.5	0.9991988238614	0.99919882386158	1.8×10^–13	0, 0.5	0.9797958971133	0.97979589711327	2.9×10^–14
1, –1.5	0.9995913079808	0.99959130798090	9.9×10^–14	1, 0.5	0.9977551312156	0.99775512256234	8.7×10^–9
2, –1.5	0.9997528114902	0.99975281149021	1.4×10^–14	2, 0.5	0.9991944716637	0.99919446965227	2.0×10^–9
3, –1.5	0.9998345510564	0.99983455105648	7.9×10^–14	3, 0.5	0.9995897240939	0.99958972295987	1.1×10^–9
0, –0.5	0.9977551536827	0.99775512256234	3.1×10^–8	0, 1.5	0.99777530313970	0.99777530313972	1.8×10^–14
1, –0.5	0.9991944768698	0.99919446965227	7.2×10^–9	1, 1.5	0.99919882386160	0.99919882386158	2.2×10^–14
2, –0.5	0.9995897256360	0.99958972295987	2.7×10^–9	2, 1.5	0.99959130798070	0.99959130798090	2.0×10^–13
3, –0.5	0.9997520675873	0.99975206631993	1.3×10^–9	3, 1.5	0.99975281149010	0.99975281149021	1.1×10^–13
0, 2.5	0.9991996881893	0.99919967974374	8.5×10^–9	2, 2.5	0.99975295828120	0.99975295828121	9.3×10^–15
1, 2.5	0.9995916199728	0.99959161997285	4.7×10^–14	3, 2.5	0.99983463144880	0.99983463144884	4.0×10^–14

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[1]	Semenoff G W 1984 Phys. Rev. Lett. 53 2449 Google Scholar
[2]	Sun S, Zhu J L 2014 Phys. Rev. B 89 155403 Google Scholar
[3]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666 Google Scholar
[4]	Feng B, Ding Z, Meng S, Yao Y, He X, Cheng P, Chen L , Wu K 2012 Nano Lett 12 3507 Google Scholar
[5]	Gomes K K, Mar W, Ko W, Guinea F, Manoharan H C 2012 Nature 483 306 Google Scholar
[6]	Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A V 2005 Nature 438 197 Google Scholar
[7]	Elahi M M, Vakili H, Zeng Y, Dean C R, Ghosh A W 2024 Phys. Rev. Lett. 132 146302 Google Scholar
[8]	陈晓娟, 徐康, 张秀, 刘海云, 熊启华 2023 物理学报 72 237201 Google Scholar Chen X J, Xu K, Zhang X, Liu H Y, Xiong Q H 2023 Acta Phys. Sin. 72 237201 Google Scholar
[9]	冉扬强, 薛立徽, 胡嗣柱 2002 物理学报 51 2435 Google Scholar RanY Q, Xue L H, Hu S Z 2002 Acta Phys. Sin. 51 2435 Google Scholar
[10]	Silbar R R, Goldman T 2010 Eur. J. Phys. 32 217
[11]	Kandemir B S, Mogulkoc A 2010 Eur. Phys. J. B 74 535 Google Scholar
[12]	Lee C M , Lee R C H, Ruan W Y, Chou M Y 2010 Appl. Phys. Lett. 96 212101 Google Scholar
[13]	Zhang Y H, Tang L Y, Zhang X Z, Shi T Y, Mitroy J 2012 Chin. Phys. Lett. 29 063101 Google Scholar
[14]	Giavaras G, Maksym P A, Roy M 2009 J. Phys. : Condens. Matter 21 102201 Google Scholar
[15]	Zhu J L, Xiong J J, Gu B L 1990 Phys. Rev. B 41 6001 Google Scholar
[16]	Ezawa M 2012 New J. Phys. 14 33003 Google Scholar
[17]	Fishbane P M, Gasiorowicz S G, Johannsen D C, Kaus K 1983 Phys. Rev. D 27 2433 Google Scholar
[18]	Batchelor M T, Henry R A, Lu X 2023 AAPPS Bulletin 33 29 Google Scholar
[19]	Zhu J L, Sun S 2012 Phys. Rev. B 85 035429 Google Scholar
[20]	王竹溪, 郭敦仁 2012 特殊函数概论(北京: 北京大学出版社) 第39—58页 Wang Z X, Guo D R 2012 Special Functions (Beijing: Peking University Press) pp39–58
[21]	Giavaras G , Nori F 2010 Appl. Phys. Lett. 97 243106
[22]	Pereira V M, Nilsson J, Neto A H C 2007 Phys. Rev. Lett. 99 166802 Google Scholar
[23]	Novikov D S 2007 Phys. Rev. B 76 245435 Google Scholar
[24]	Lukose V, Shankar R, Baskaran G 2007 Phys. Rev. Lett 98 116802 Google Scholar

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