Outline High-frequency (optical) conductivity of graphene; Optical - - PowerPoint PPT Presentation

Д.А. Свинцов1,2, В.И. Рыжий3, T. Otsuji3

• 1. Физико-технологический институт РАН
• 2. Московский физико-технический институт
• 3. Research institute of electrical communication, Tohoku university

Outline

 High-frequency (optical) conductivity of graphene;  Optical conductivity under population inversion

 Direct interband transitions  Carrier-carrier scattering and intraband absorption  Indirect interband transitions

Graphene-based THz electronics?

THz lasing in optically or electrically pumped graphene? GOOD IDEA! But what about

• Intraband Drude

absorption (strong at low frequencies)?

• F. Bonaccorso, Z. Sun, T. Hasan, and A.C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics 4, (2010).
• A. Tredicucci and M. S. Vitiello, “Device concepts for graphene-based terahertz photonics,” IEEE J. Sel. Top. Quant. 20 (2014).
• A. Dubinov, V.Ya. Aleshkin, M. Ryzhii, T. Otsuji, and V. Ryzhii, “Terahertz laser with optically pumped graphene layers and Fabri-

Perot resonator,” Appl. Phys. Express 2 (2009).

• V. Ryzhii, M. Ryzhii, V. Mitin, and T. Otsuji, “Toward the creation of terahertz graphene injection laser,” J. Appl. Phys. 110 (2011).

Optical conductivity of graphene

 

   

4 1 / 2 2 1

4 l 4 Re n 1

F T

e T e     

 

   

   

4

/ 2 / 2 4

V C

e f f         

• L. A. Falkovsky and A. A.

Varlamov, Eur. Phys. J. B 56, 281

Optical conductivity of graphene

 

   

4 1 / 2 2 1

8 l 4 Re n 1

F T

e T e     

 

   

4

/ 2 tanh 4 2

F

e T          

• V. Ryzhii, M. Ryzhii, and T. Otsuji,
• J. Appl. Phys. 101, 083114 (2007).

Drude conductivity of clean graphene under population inversion

Electron-hole, electron-electron and hole-hole scattering govern the intraband conductivity Velocity-momentum decoupling 1 2 3 4

   p p p p

0 / p



p

v p v

Does not necessarily lead to 1 2 3 4

   v v v v

 

1 2 3 4

1 ,

fi

e V c       A v v v v

• F. T. Vasko and V. Ryzhii, Phys. Rev. B 77, 195433 (2008).
• D. Sun et. al. New J. Phys. 14, 105012 (2012).
• D. Svintsov et. al., Optics Express 22, 19873 (2014).

Drude conductivity of clean graphene under population inversion

 

 

2 3 2 2 / 3

1 Re 1 / , / 4

T ee ee F

e e T e I T T



     



              v

 

 

2 3 2 2 / 3

2 Re 1 / , / 4

T eh eh F

e e T e I T T



     



              v    

2 2 2 1 2 , 1 2 1 2 1 2 2 2

( ) cos / 2 cos / 2 [ / ] ( )

ee ee

d d d I k k T k k Q Q



    

     

      



Q k k n

  

1 2 1 2

( ) ( ) 1 ( ) 1 ( ) f k f k f k f k

   

  

       

2 2 2 2 2 1 2 , 1 2 1 2 2 2

( ) cos / 2 cos / 2 sin / 2 sin / 2

eh eh

d d d I Q



    

   

       



Q k k n

  

1 2 1 2 1 2 1 2

[ / ] ( ) ( ) 1 ( ) 1 ( ) k k T k k f k f k f k f k  

       

     

Fermi golden rule + occupation numbers of initial and final states:

Interband amplification vs. carrier- carrier absorption

• Negative conductivity is possible in suspended graphene above ~6 THz;
• In high-k background above ~3 THz;
• Negative dynamic conductivity threshold slowly moves to lower values as k0

increases.

Real parts of net dynamic conductivity Re(σintra +σinter) normalized by σq=e2/4ђ at different quasi-Fermi energies εF in graphene structures with different background dielectric constants κ0 (T = 300 K).

Dependence on background dielectric constant

 

2 2 /

1 , 8 ln 1

F T

TF F q T

e e T q e q q v v V



     

Thomas-Fermi screening leads to weak dependence on k0

Threshold frequencies (solutions of Re(σintra+σinter) = 0) vs. background dielectric constant at different temperatures T (εF = 75 meV)

Raise interband amplification above 2.3%

• D. Svintsov, V. Ryzhii, T. Otsuji “Negative dynamic Drude conductivity in

pumped graphene” arXiv:1408.7023

Conductivity due to indirect inter- and intraband transitions

( , ) ˆ | | 2

cc S

e V c V c c 





    

p p pp

v v A p p ( , ) ˆ | | 2

cv S

e V c V v c 





    

p p pp

v v A p p

 

2 2 intra 3 , ,

2 ˆ Re ( ) ( ) | |

D q S

g f f V

    

           

    

                  



p p p p p p p p

p p v v

Energy conservation requires q=|p-p’|>w/v0!

 

2 2 inter 3 ,

2 ˆ Re ( ) ( ) | | .

D q v c S

g f f c V v          

   

                   



p p p p p p p p

p p v v

Energy conservation requires q=|p-p’|<w/v0! Need scattering potentials with either singular at or quickly decreasing as !

2

ˆ | |

S

V    p q p

q  q  

Conductivity due to indirect inter- and intraband transitions

 Scattering by Gaussian correlated disorder

2 2 2

( ) ( ) exp | | / c V V V l            r r r r

2

2 ( /2) 2 2

1 cos ˆ | | 2

c

ql S c

V l V e     





      

pp

p p

Intraband Drude absorption is suppressed due to requirement q=|p-p’|>w/v0

• F. T. Vasko and V. Ryzhii, Phys. Rev. B 76, 233404 (2007).
• G. M. Rutter, J.N. Crain, N.P. Guisinger, T. Li, P.N. First, J.A. Stroscio,

Science 317, 219 (2007)

Conductivity due to indirect inter- and intraband transitions

Calculated frequency dependencies of the interband, intraband and net Drude conductivity (normalized by sq) in pumped graphene with quasi-Fermi energy eF = 50 meV. The distribution of impurities is Gaussian. The dashed line indicates the region w<n, where our calculations are not rigorous. Real part of the net Drude conductivity (normalized by sq) vs. frequency at different correlation lengths lc.

Net dynamic conductivity of graphene with population inversion

Conclusions

Graphene with population inversion

exhibits negative dynamic conductivity (optical gain) in THz and IR range;

Carrier-carrier scattering sets the threshold

• f negative dynamic conductivity in clean

graphene;

Indirect interband transitions can improve

the optical gain above 2.3%.

Conductivity of graphene

   

 

         

     

               



2 2 2 /2 + /2 2 1 /2 + /2 + / , 2 /2 x

d v f f ie i

p k p k p k p k p k p k

p

   

 

 

 

2 2 12 21 /2 + /2 2 2 2 1 + /2 /2 + /2 /2

2

x x

d v v f f ie i          

   

              



p k p k p k p k p k p k

p

Intraband term Interband term

     

2

2 e / 2 / 4 R

k V C

e f f    



      

Uniform field k=0

• L. A. Falkovsky and A. A. Varlamov, Eur. Phys. J. B 56, 281

Optical conductivity of graphene: experiment

 Optical / IR range  Far IR and THz range

inter 2 inter

4 Re 2.3% e k c c      

intr int a 2 ra 2

4 Re

e e

k c        

• K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, Phys. Rev. Lett. 101, 196405 (2008).
• L. Ren, Q. Zhang et.al., Nano Lett. 12, 3711 (2012).

Population inversion in graphene

 Optical pumping + fast relaxation + slow

recombination

• S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, Phys. Rev. B 85, 035443 (2012).
• T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J.Wang, Phys. Rev. Lett. 108, 167401 (2012).

Negative conductivity and optical gain under population inversion

• S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, Phys. Rev. B 85, 035443 (2012).
• T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J.Wang, Phys. Rev. Lett. 108, 167401 (2012).

Conductivity under population inversion

   

 

4 / 2 2

/ 2 8 ln 1 tan R h 4 2 e

F T

e F e

T e e T



                         

• V. Ryzhii, M. Ryzhii, and T. Otsuji, J. Appl. Phys. 101, 083114 (2007).

Interband amplification vs. carrier- carrier absorption

Color map of real part of net dynamic conductivity Re(σintra +σinter)/σq vs. frequency and quasi-Fermi energy for κ0 = 5: at T = 300 K and at T = 200 K. The area Re(σintra+σinter)/σq < 0.75 is filled in solid color

Sub conclusions

 Carrier-carrier scattering in pumped graphene sets the

lower frequency threshold of negative dynamic conductivity ~5 THz;

 Increasing the background dielectric constant slightly

decreases the intraband conductivity;

 Intraband conductivity slowly (almost linearly) grows

with increasing the quasi-Fermi energy of pumped carriers.

Dependence on quasi-Fermi energy of pumped carriers

 

2 2 /

1 , 8 ln 1

F T

TF F q T

e e T q e q q v v V



     

Thomas-Fermi screening leads to weak dependence on eF

Real parts of intraband dynamic conductivity Reσintra (solid lines) and interband conductivity −Reσinter (dashed line) as functions of quasi-Fermi energy εF at fixed frequency ω/(2π) = 6 THz and different values of background dielectric constant κ0.

Drude conductivity of clean graphene under population inversion

Matching with variational solution of kinetic equation

1 ,

1 exp

p F

C f T

 

 



               

p p

Ev

2 2 2

2 Re ln 1 exp ,

cc B F intra B cc

k T e k T                     

2 ,0 ,0 13 1 2 /

10 s at 1 4 ln 1

F

ee eh c B cc T

I I k T e    



        

Construct a functional such that kinetic equation represents its minimum condition; Calculate the functional with trial function + minimize with respect to Cw

• D. Svintsov et. al., Optics Express 22, 19873 (2014).

A.B. Kashuba, Phys. Rev. B 78, 085415 (2008), L. Fritz et. al. Phys. Rev. B 78, 085416 (2008).

Net dynamic conductivity of graphene with population inversion