Valleytronics : Another Degree of Freedom

Yes. You are reading it properly – ‘Another Degree of Freedom’. In addition to Charge and Spin degree of  freedom, Bloch electrons in 2D materials are endowed with ‘Valley’ degree of freedom, which leads to ‘VALLEYTRONICS’, a new branch of material science. I’ll try to skip technical details as much as possible for readers who are not expertise.


I’ll talk about hexagonal 2D materials – Graphene (monolayer of Graphite) and Group-VI Transition Metal Dichalcogenides ( MoS2, MoSe2, WS2, WSe2).



If we see band structure of materials, a local minimum in conduction band and a local maximum in valence band is known as VALLEY.  Generally, we can see several degenerate valleys which are labelled by different ‘Valley Index’ to distinguish them in Band Structure.



There are several motivations behind valleytronics.

  1. Quantum Computing:- In QC, we use Spin Up & Spin Down as 0 & 1 to store information. Similarly, can we use degenerate valleys (for 2) as qubit?
  2. Optoelectronics:- We can use valley polarization to make source, detect and control lights. So, Valleytronics is connected with photonics also.


Lattice structure of Graphene


2D Hexagonal Systems:-

In massive Graphene (i.e with Band Gap) and Group- VI TMD odd layers, there are two valleys k+ & k- in momentum space who are degenerate in energy. But due to Inversion Symmetry Breaking , they have Berry Curvatures of opposite sign.

Berry Curvature is analogous to magnetic field for Spintronics.

Berry Curvature is defined as,

\Omega _{{n,\mu \nu }}({\mathbf  R})={\partial  \over \partial R^{\mu }}{\mathcal  {A}}_{{n,\nu }}({\mathbf  R})-{\partial  \over \partial R^{\nu }}{\mathcal  {A}}_{{n,\mu }}({\mathbf  R}).

{\mathbf  \Omega }_{n}({\mathbf  R})=\nabla _{{{\mathbf  R}}}\times {\mathcal  {A}}_{n}({\mathbf  R}).


{\mathcal  {A}}_{n}({\mathbf  R})=i\langle n({\mathbf  R})|\nabla _{{{\mathbf  R}}}|n({\mathbf  R})\rangle

n(R) eigenstate of 2D system in parameter space.

These valleys have different optical selection rules due to Valley Polarization. If we use different circular polarized light, we can excite electrons in different valleys as shown in fig. Below. This ‘VALLEY SWITCHING’ may serve as building block for future Valleytronics.Page_0

We can’t say whether Valleytronics will lead to devices that complement or outperform Semiconductor Technology. But, ‘Valley Degree of Freedom’ has ignited a new direction of material science that needs deep exploration.



  1. Nature Review Materials1 , Article No:- 16055(2016)
  2. Topological Insulators and Topological Superconductors, B.A. Bernevig, T.L. Hughes
  3. Valley Polarization in MoS2 monolayers by Optical Pumping, Nature Nanotechnology, DOI: 10.1038/NNANO.2012.95
  4. Quora


Dipankar Sen


IISER Bhopal

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