Indirect Exchange Interaction Mediated by Dirac Fermions
Based on the model of s-d interaction discussed in Chap. 6 , the indirect exchange interaction between a pair of magnetic impurities is calculated. The interaction carries the signature of the topological surface states. The chapter discusses possible ma
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Indirect Exchange Interaction Mediated by Dirac Fermions
The indirect exchange interaction carries the signature of the topological surface states. The most direct illustration of the relation between the topological order and the magnetic exchange can be seen in a sample where the Fermi level is tuned to the Dirac point: the magnetic field-induced violation of time reversal symmetry destroys the topological protection and turns massless fermions into massive ones. This topological phase transition drastically changes the character of the exchange range function: from one mediated by gapless fermions and scaled as R2 [1] to that mediated by massive excitations and then demonstrating the monotonic and short-ranged exponential character typical of an intrinsic semiconductor, the Bloembergen-Rowland (BR) interaction [2]. The spin–electron interaction, discussed in Chap. 6, serves as a starting point in calculations of the indirect exchange interaction between magnetic atoms mediated by surface excitations. Below we consider two typical settings where surface electrons are either degenerate, providing for a well-defined Fermi surface, or non-degenerate, with the chemical potential located in the top-bottom tunneling energy gap.
7.1
Generic Indirect Exchange via Conduction Electrons
The s-d interaction (6.3) can be extended for a two-band solid as the matrix generalization of single-band expression:
© Springer Nature Switzerland AG 2020 V. Litvinov, Magnetism in Topological Insulators, https://doi.org/10.1007/978-3-030-12053-5_7
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7 Indirect Exchange Interaction Mediated by Dirac Fermions
H ¼ H el þ H sd ,
H el ¼
X
aþ E ðkÞ I ak , k ½^
k
H sd
1 X X iqRi þ ^ ¼ e ak J σSi akþq , N kq i
ð7:1Þ
where ak is the 4-spinor in bands and spins, J^jl ¼ Juj ðRi Þul ðRi Þ is the 2 2 matrix operating in the space of two bands ( j, l ¼ 1, 2). The unit matrix I and the spin matrices σ act in the 2 2 electron spin space. Since the matrix products in (7.1) are comprised of matrices acting in different spaces, the product is understood as the Kronecker product. The indirect exchange interaction between two localized spins appears as a second-order energy correction with respect to Hsd. It is convenient to express the second-order correction by the diagram shown in Fig. 7.1: The diagram depicts the process in which an impurity spin creates the virtual electron-hole pair (solid lines) which then propagates to another impurity and annihilates there. Two localized spins exchange the virtual electron-hole pair, maintaining the indirect exchange interaction between them. The process, shown in Fig. 7.1, can be called vacuum polarization if the vacuum is understood to be either a Fermi sphere in a single-band metal (RKKY model), or fully occupied valence band and empty conduction band in an intrinsic semiconductor at zero absolute temperature. In a two-band solid, E1
0
!
J1
, J^ ¼ 0 E2 J 12 2 2 E1,2 ¼ h k =2mc, v þ E g =2
^E ðkÞ ¼
J 12 J2
! ,
ð7:2Þ
where Eg is the bandgap, the reference energy is
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