This page presents a transmission line model for channels exhibiting spin-momentum locking (SML). Spin-momentum locking is observed in diverse classes of materials with spin-orbit coupling (SOC) e.g. topological insulator (Bi_{2}Se_{3}, Bi_{2}Te_{3}, etc.), transition metals (Pt, Ta, W, etc.), III-V semiconductor (InAs), perovskite oxides (LaAlO_{3}|SrTiO_{3}), etc. The model enables easy analysis of complex geometries involving these materials.

Each of the terminals of the module has two components: charge (*c*) and z-component of spin (*s*) as shown in the figure below. The circuit representation of the channel with external contact (with conductance *G*_{0}) is also shown. No external contact corresponds to *G*_{0} = 0. The external contact can be normal metal (NM) or ferromagnet if the contact polarization *p _{f}* = 0 or

The coupling between the charge and the spin is considered by the dependent voltage and current sources in the circuits which are proportional to the parameter 0 ≤ *p*_{0}* ≤ *1 indicating the degree of SML. The channel is normal metal if *p*_{0} = 0. A discussion on this parameter has been given below. The dependent source parameters are:

Note that a number of such modules should be connected together with proper boundary conditions to simulate a structure of interest. This module can’t be used as a single block as allowed by our prior spin-circuit modules.

Symbol |
Expression |
Description |
Unit |

C_{E} |
- | Electrostatic capacitance per unit length of the channel. | F-m^{-1} |

C_{Q} |
2/(R)_{B}|v_{x}| |
Quantum capacitance per unit length of the channel. | F-m^{-1} |

L_{M} |
- | Magnetic inductance per unit length of the channel. | H-m^{-1} |

L_{K} |
R/(2|_{B}v|)_{x} |
Kinetic inductance per unit length of the channel. | H-m^{-1} |

R_{c} |
R/_{B}λ |
Series charge resistance per unit length of the channel. | Ω-m^{-1} |

R_{s} |
α^{2}R/_{B}λ_{0} |
Series spin resistance per unit length of the channel. | Ω-m^{-1} |

G_{sh} |
4/(α^{2}R_{B}λ)_{s} |
Shunt spin conductance per unit length of the channel, taking into account the spin relaxation in the channel. | Ω^{-1}-m^{-1} |

p_{0} |
- | Degree of spin-momentum locking. (0 ≤ p_{0}≤ 1). |
- |

p_{f} |
- | Contact polarization. (0 ≤ p1)._{f }≤ |
- |

G_{0} |
- | Contact conductance per unit length. | Ω^{-1}-m^{-1} |

α |
2/π | Angular averaging factor. | - |

η_{c} |
2/(αλ)_{r} |
Spin to charge conversion coefficient. | m^{-1} |

η_{s} |
2α/λ_{0} |
Charge to spin conversion coefficient. | m^{-1} |

γ_{s} |
2/(αλ)_{t} |
Charge to spin conversion coefficient. | m^{-1} |

g_{m} |
2/(αR)_{B} |
Time varying charge voltage to spin current conversion coefficient. | Ω^{-1} |

r_{m} |
α / (|v|_{x}C)_{E} |
Time varying charge current to spin voltage conversion coefficient. | Ω |

λ |
- | Mean free path indicating back scattering length (determines R)._{c} |
m |

λ_{0} |
λ / (λ_{r}λ_{t}_{r}+λ_{t}) |
Mean free path indicating back scattering length for spin (determines R)._{s} |
m |

λ_{s} |
- | Mean free path indicating spin-flip scattering length for spin (determines G)._{sh} |
m |

λ_{r} |
- | Characteristic length for reflection without spin-flip. | m |

λ_{t} |
- | Characteristic length of transmission with spin-flip. | m |

R_{B} |
(h/q^{2}) (1/M)_{tot} |
Ballistic resistance of the channel. | Ω |

M_{tot} |
^{2}(wt)/(2π) (3D) |
Total number of modes for 2D and 3D channels with Fermi wave vector k, width _{F}w, and thinkness t. |
- |

|v|_{x} |
- | Magnitude of thermally averaged electron velocity. | m-s^{-1} |

Note that the parameters that we can control are: *p*_{0}, *p _{f}*,

However, in most of our simulations related to SML, we have assumed that the reflection with spin-flip is the dominant scattering mechanism in the channel. This results in:

Thus, the control parameters reduces to:

*Channel parameters: p*_{0},*R*, and_{B}*λ**.**Channel parameters for time-dependent characteristics:*|*v*|,_{x}*C*, and_{E}*L*._{M}*Contact parameters:**p*, and_{f}*G*_{0.}

Please see Ref. [1] for details on the parameters and the model.

0 ≤ *p*_{0}*≤ *1 quantifies the degree of spin-momentum locking observed in diverse classes of materials with spin-orbit coupling. *p*_{0} = 0 represents normal metal (NM) channel and *p*_{0} = 1 represents perfect topological insulator (TI) surface states. In real TI, *p*_{0} gets effectively lowered by the presence of parallel channels. Several experimental groups [2] have quantified their spin voltage measurements on TI using *p*_{0}. *p*_{0} has been estimated for a Rashba channel [3] using similar spin voltage measurements which is <<1. Recently, spin voltage measurements have been reported on heavy metals [4], which can be quantified by *p*_{0} as well. However, the underlying mechanism of heavy metals is a subject of active debate and could involve a bulk or interface Rashba-like channel.

Below we have listed rough numbers for *k _{F}* and

Material |
k_{F} [nm^{-1}] |
p_{0} |

LaAlO_{3}|SrTiO_{3} |
1.12 | 0.0612 |

Ag|Bi | 12 | 0.05 |

Pt | 7.8 | 0.023 |

InAs | 0.354 | 0.05 |

(Bi_{0.5}Sb_{0.5})_{2}Te_{3} |
0.43 | 0.71 |

Bi_{2}Se_{3} |
1.12 | 0.63 |

Sb_{2}Te_{3} |
0.7 | 0.24 |

Bi_{2}Te_{2}Se |
1 | 0.78 |

To illustrate the simulation method using the transmission line model, we consider a three contact based structure shown below. We discretize the structure into *N* small sections and represent each of the small sections with the circuit model shown above, with (*G*_{0} ≠ 0 indicated by block 2) or without (*G*_{0} = 0 indicated by block 1) external contact. We connect the charge (*c*) and spin (*s*) terminals of the blocks for all the small sections in a modular fashion using standard circuit rules to construct the simulation setup of the corresponding structure. We assumed point contacts for contacts 1, 2, and 3.

We apply the charge open and spin ground boundary conditions at the two boundaries:

We perform a SPICE simulation using the full transmission line model on the three contact based structure shown above. We observe the charge-spin interconversion in a SML channel, namely, (i) charge current induced spin voltage and (ii) spin current induced charge voltage.

(i) *Charge current induced spin voltage*: We inject a charge current *I _{c}* between the charge terminals of contacts 1 and 2 and observe the open circuit spin voltage

which has been used by a number of experimental groups [2] to quantify their experiments.

(ii) *Spin current induced charge voltage*: The reciprocal effect of the above effect is known as the spin current *i _{s}* induced charge voltage Δ

The reciprocal relation between these two effects have been discussed in details in Ref. [7].

*Spin current induced charge current*: The Δ*V _{c }*described above is the open circuit charge voltage in the channel induced by injected spin current

We can derive a very simple expression for a parameter widely used to quantify IREE in 2D channels as (see Refs. [1], [7])

which we have compared with SPICE simulations as well as available experiments on diverse materials like Cu|Bi, Ag|Bi, and LaAlO3|SrTiO3 (see Refs. [8]) shown below. For SPICE simulation we have injected a current *i _{s}* through the spin terminal of contact 3 and observed the induced short circuit charge current

Module versions available:

- v1.0 (Release: April 2018): Contains two modules for with and without external contact cases.

Any update / modification on the module will be posted to this page.

The module, as a part of the library, can be downloaded here.

To download the SPICE simulation files for the example shown above, click here.

Please contact Shehrin Sayed ( *ssayed AT purdue DOT edu* ) for questions or comments regarding this page.

[1] S. Sayed, S. Hong, and S. Datta, “Transmission Line Model for Charge and Spin Transport in Channels with Spin-Momentum Locking”, *arXiv:1707.04051 [cond-mat.mes-hall]*, 2017.

[2] C. H. Li *et al.*, Nature Nanotechnol. 9, 20325, 2014; J. Tang *et al.*, Nano Letters 14, 5423, 2014; A. Dankert *et al.*, Nano Letters 15, 7976, 2015; L. Liu *et al.*, Phys. Rev. B 91, 235437, 2015; J. Tian *et al.*, Sci. Rep. 5, 14293, 2015; J. S. Lee *et al.*, Phys. Rev. B 92, 155312, 2015; F. Yang *et al.*, Phys. Rev. B 94, 075304, 2016.

[3] J.-H. Lee, H.-J. Kim, J. Chang, S. H. Han, H.-C. Koo, S.Sayed, S. Hong and S. Datta, “Multi-terminal spin valve in a strong Rashba channel exhibiting three resistance states”, *Scientific Reports* 8, 3397, 2018.

[4] V. T. Pham *et al.*, Nano Lett. 16, 6755, 2016; V. T. Pham *et al.*, Appl. Phys. Lett. 109, 192401, 2016; Li and Appelbaum, Phys. Rev. B 93, 220404, 2016.

[5] S. Sayed, "Transport Theory for Materials with Spin-Orbit Coupling: Physics to Devices," Ph.D. dissertation, Purdue University, 2018.

[6] S. Hong *et al.* "Modeling potentiometric measurements in topological insulators including parallel channels," Phys. Rev. B 86, 085131, 2012; "Spin Circuit Model for 2D Channels with Spin-Orbit Coupling," Sci. Rep. 6, 20325, 2016.

[7] S. Sayed *et al. *"Multi-Terminal Spin Valve on Channels with Spin-Momentum Locking," Sci. Rep. 6, 35658, 2016.

[8] Sanchez *et al.*, Nat. Commun. 4, 2944, 2016; Isasa *et al.*, Phys. Rev. B 93, 014420, 2016; Lesne *et al.*, Nat. Materials 15, 1261, 2016.