1. Toward Quasi-Band Gap from SPR of Gold NP

Supporting Information

1. Toward quasi-band gap from SPR of gold NP

The motion of the free electron in metal is with damping. The velocity is reduced by collisions of electrons with atoms of a nonideal lattice. The equation of motion, Drude model, is written as the following.

  d 2 r dr  m  m  (e)E dt 2 dt where  is the damping factor, m is the mass of an electron, and E is the applied external field. For a sphere particle, the polarization, P, is proportional to the E. And P is the product of the dipole moment of one times the number of all dipoles, n.

 4  4  E  P  ner 3 3

Then by combining the above equation, the equation of the free electron motion with damping, Drude model, is represented.

  d 2 r dr 4   4ne 2  m  m  (e) ner  r dt 2 dt 3 3

  d 2 r dr 4ne 2  m  m  r  0 dt 2 dt 3

Solving the above equation, the stationary solution for weak damping is

  4ne 2  2 4ne 2  2 r(t)  exp( t)[A exp(i  t)  A exp(i  t)] 2 1 3m 4 2 3m 4 .  4ne2  2  Aexp( t)cos(  t  ) 2 3m 4  4ne 2  2 exp( t) is the damping wave. cos(  t  ) is the undamping wave. 2 3m 4

For no damping system (  0 ), the plasmon modes of a sphere is

4ne 2   . Otherwise, for a weak damping system (  0 ), the plasmon mode of 0 3m

4ne 2  2 a sphere is    , which is also called the angular frequency of the 0 3m 4 damped oscillator. The frequency at the angular frequency of the damped oscillator is a result of the electrons vibrating freely without an external force. In addition, the drift

 dr    velocity is  ( r   Aexp( t)sin( t  )) . dt 2 0 2 0

From classical dynamics, we can obtain the total energy of the damped oscillator as follows. The kinetic energy,

 1 dr 1    T  m( ) 2  m[( r   Aexp( t)sin( t  ))]2 2 dt 2 2 0 2 0 2 . 1     2  m( r 2   Aexp( t)sin( t  )r   A2 exp(t)sin 2 ( t  )) 2 4 0 2 0 0 0

Moment, p, and wave vector, k, of electron are given from the velocity.

 dr    p(t)  m  m( r   Aexp( t)sin( t  ))  k(t) . dt 2 0 2 0

m    k(t)   ( r   Aexp( t)sin( t  )) .  2 0 2 0

Then k() is transformed from time space to frequency space by Fourier-sin transformation.

1 2  1 2 The potential energy is U  m r 2  m A2 exp(t) cos 2 (  ) . The total 2 0 2 0 0 energy, E, is the combination of kinetic energy and potential energy. In addition, the system will radiate after being vibrated by an external electromagnetic wave, which causes the energy loss phenomenon of the system. Combining the expressions for T,

U, and loss energy to find the total energy,

 E(t)  T U   S nˆd 2 2 4 1  4c k 2  mA2 exp(t)( cos 2 ( t  )   sin( t  ) cos( t  )  2 )  P 2 4 0 0 0 0 0 3 2 2 4 1  2 4c k   mA2 exp(t)( cos 2 ( t  )   sin( t  ) cos( t  )  )  (ne r ) 2 2 4 0 0 0 0 0 3 1  2 4c 2 k 4  mA2 exp(t)( cos 2 ( t  )   sin( t  ) cos( t  )  2 )  n 2 e 2 A2 exp(t) cos 2 ( t  ) 2 4 0 0 0 0 0 3 0

Then E() is transformed from time space to frequency space by Fourier-sin transformation.

The number of allowed valued of k is under Fermi-energy, E f . Then we set the

Fermi-energy, E f is zero. The number of allowed valued of k is between  and d , which is the energy fluctuation of the electrons with damping under the Drude model.

L L dN  2( )3 d 3k  2( )3 (4  k() 2 )dk 2 shell 2

The general expression for D() , the number of states per unit frequency range, is presented as the following. L is the lattice constant of metal. dN dk dN dN dk d D()    dk d dE dk d dE dE d L 2( ) 3 (4  k() 2 )dk dk() 2   dk d dE() d

D() , the number of states per unit frequency range, is plotted in Fig. S1.

(Parameters: L=4.068 Angstrom, A=1, m=9.1*10-28 g,   6.7 1013 s -1 , and

16 1 0  1.208310 s ) s e t a t S

y t i s n e D

200 400 600 800 1000 1200 1400 1600 1800 Wavelength (nm)

Fig. S1 Density of states of surface plasmon resonance. The range of peak is ranged from 510 nm to 580 nm. Surface plasmon resonance produces exiton to separate charge after the visible light radiation.

2. The chemical potential of electrolyte applied an external electrical field is rewritten as the following. 2 2 ' X Ce4 GCe4 'X Ce3 GCe3 '2X Ce4 X Ce3 2 2  X 4 [G 4  (4q)V ]  X 3 [G 3  (3q)V ]  2X 4 X 3 Ce Ce Ce Ce Ce Ce 2 2 2 2  X 4 G 4  X 3 G 3  (4q)VX 4  (3q)VX 3  2X 4 X 3 Ce Ce Ce Ce Ce Ce Ce Ce 2 2    (4q)VX Ce4  (3q)VX Ce3 2 2  q(4X Ce4  3X Ce3 )V  

The conductive level of a semiconductor is assumed to be lack of change in an

applied electric field. The open-circuit voltage (Voc) is represented as the following.

2 2 Voc '(V )  C.B. ' C.B. (q(4X Ce4  3X Ce3 )V  ) 2 2  C.B.   q(4X Ce4  3X Ce3 )V 2 2  V  q(4X 4  3X 3 )V oc Ce Ce

The open-circuit voltage (Voc) is proportional to the external electrical field.

3. SPR effect is applied in solar conversion efficiency (')

2 2 When SPR is applied in a solar cell, Voc ' Voc  q(4X Ce4  3X Ce3 )V and

' ' ' J sc  V are put into the above equations. And Vmp , J mp , and FF are represented in the SPR solar cell case.

2 2 k T eV e q(4X 4   3X 3 )V V ' (V ) V B ln(1 oc ) ln( Ce Ce ) q(4X 2 3X 2 )V mp  oc    Ce4   Ce3 e k BT k BT

2 2 a  V  q(4X 4   3X 3 )V  V  V mp Ce Ce mp e

2 2 ' eVmp  e  q(4X Ce 4  3X Ce3 ) ' J mp (V )  J s (exp( 1)  J sc k BT ' eVmp ' (J mp  exp(0)  J sc ) ' J mp ' exp(0)  J s exp( )  J sc   J sc   J sc (1  ) k BT J s J s J s 2 2 2 2 eV / k T  e  q(4X 4  3X 3 )V / k T  ln(1 eV / k T  e  q(4X 4  3X 3 )V / k T ) ' oc B Ce  Ce  B oc B Ce  Ce  B FF (V )  2 2 1 eV / k T  e  q(4X 4  3X 3 )V / k T oc B Ce  Ce  B (b 1)  aV / k T  ln(b  aV / k T )  B B b  aV / k BT

2 2 b 1 eV / k T a e q(4X 4 3X 3 ) where oc B and   Ce   Ce  .

' (b 1)  aV / kBT and  ln(b  aV / kBT) act as the antagonism for the FF , which makes the relationship between the SPR intensity and the fill factor irregular as shown in Table 1. The solar conversion efficiency (') is given by

a J mp ' exp(0) (Vmp  V )(  J sc (1 )) , P Vmp J mp e J J .  (V )  max   s s Ps Ps Ps