Supplementary Information
High Efficiency Silicon 1310 nm Detector without Defect States or
Heteroepitaxy
Yu-Hsin Liu,1),a) Yuchun Zhou,2) and Yu-Hwa Lo,2)
1
Materials Science and Engineering Program, University of California at San Diego, La Jolla,
California 92093-0418, United States.
2
Department of Electrical and Computer Engineering, Jacobs School of Engineering, University
of California at San Diego, La Jolla, California 92093-0407, United States.
I. Absorption Coefficient Calculation
From the Fermi Golden Rule and the time-dependent perturbation theory1-2, the absorption coefficient for
an indirect bandgap semiconductor that is heavily doped to produce an impurity band can be written as,

 Ecv   

c
vi
2
 e* (r , t ) H vi (r , t )

2
 f v  f c    2
ci
v

 ei* (r , t ) H h (r , t )
2
 f v  f c  /
I 

   
(1)
In (1),  e* ( r , t ) and  h ( r , t ) are the wave functions of electrons and holes in the conduction and valance
bands.  ei* and  vi are the wave functions of electrons and holes in the impurity states. H is the
interaction Hamiltonian. For indirect bandgap semiconductor such as silicon, absorption of subbandgap
photons by electrons and holes in the conduction and valence bands has little contribution due to the
constraint of k-selection rule. Therefore, this term is neglected in (1). The Fermi functions f c and f v
represent the probability for a state to be occupied by an electron or hole, and I is the optical power
intensity upon the device.
In the following, we describe how we calculate subbandgap light absorption due to quasi 2D electron
states and spatially confined acceptor states. All the electron states contributing to absorption of
subbandgap energy photons need to satisfy the following condition required by the law of energy
conservation:
    e bi ( x0 )  V ( x0 )  Ee,no  Eg
(2)
___________________________________________
a)
[email protected]
1
Supplementary Information
Here  is the energy of the incident photons,  is the ionization energy of acceptor, assumed to be 45
meV3 above the valence band edge for boron, e bi ( x0 )  V ( x0 ) is the sum of the built-in potential and
the applied reverse bias voltage at position x0 , Ee , no is the electron energy of each subband in a confined
p/n junction, and E g is the bandgap energy. The relations among different energies in (2) are shown
schematically in Fig. S1.
Figure. S1 | Absorption due to all available quasi 2D electron states and acceptor states.
The wave function overlap integral for electrons at each sub-band and the acceptor impurity will be
calculated next. Since silicon has six conduction band minima, we define the axis in parallel with the Efield as x-axis. Then the electron wave functions can be represented as 
*
e
 e* 
ei ( k e ) r uc (r )e, n ( x)eik x 0 x
V
ei ( k e  k 0 ) r uc (r )e, n ( x)
V
and
for states with 4-fold degeneracy and 2-fold degeneracy, where kxo
represents the position of conduction band minimum in parallel with the x-axis.  e , n ( x ) is the envelope
function for electrons in the nth sub-band.
The localized acceptor state wave function can be approximated as  h   (r , x0 )uh (r ) , where u h (r ) is
the usual atomic wave function and  (r , x0 ) represents the spatial distribution of an impurity state at
location x0 . For simplicity, we approximate  ( x, y, z , x0 )  (1 /( 2a* )3 / 2 ) if x  x0  a* , y  a* , z  a* ,
and  ( x, y, z, x0 )  0 elsewhere, with a * being the Bohr radius calculated from the equation
a*  (4 2 / mh*e 2 ) .
By substituting these wave functions into (1), we can obtain the expressions of absorption coefficient
from the 4-fold and 2-fold degeneracy states of quasi 2D electrons:
2
 4 fold ( E ) 
e2h M b 1 1
2A
( 2a * ) 3
2 0 m02cn E V L
(2 ) 2
 P( x )  
1/ 3
x0
0
n
2
*
e ,n


2
sin 4 ke a* 2me
( x0 ) ( f v  f c ) P( x0 )
4

2
k e a *


(3)
Supplementary Information
 2 fold ( E ) 
e2h M b
2
1 1
2 A 2 sin( k0 x a* )
(2a * ) 4
2
2 0 m0 cn E V L
(2 ) 2 k0 (2a* ) 3 / 2
2
 P( x ) 
1 / 3
x0
0
n
2
 e*,n ( x0 ) ( f v  f c ) P( x0 )


sin 4 ke a * 2me (4)
4
2
k e a *


In Eqs. 3 and 4, L is the thickness of the epitaxial silicon p/n junction, and A and V are the area (y-z plane)
and volume of the device. The value of M b
2
2
can be determined approximately by M b  1.33m0 Eg in
a quasi-quantum confined system4. P( x0 ) is the doping concentration profile of acceptor and
k  (2me*E )1 / 2 /  , where E is the kinetic energy of an electron in a specific sub-band. Other
parameters in Eqs. 3 and 4 have their usual meanings. Eqs. (3) and (4) produce the theoretical results in
Fig. 3(b).
The theory and experiment show good agreements in their general characteristics. The differences in the
numerical values could be due to the approximate wave function for impurity states, the aproximation of
the impurity band into a single energy level, and the difference between the real and designed impurity
profiles.
Reference
1.
H.C. Casey Jr, and M.B. Panish, Heterostructure Lasers -Part A. (Academic, New York , 1978) p.
110-136.
2.
S. L. Chuang, Physics of Optoelectronic Devices. (Juhn Wilry & Sons, Inc., New York, 1995) p.
337-345.
3.
M. Lax, and E. Burstein, Phys. Rev., 100, 592 (1995)
4.
Y. Arakawa, and A. Yariv, IEEE J. Quantum Electron. 22, 1887 (1986)
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Supplementary Information
II. Internal Quantum Efficiency Measurement
To obtain the internal quantum efficiency (QE) without the uncertainties of extrinsic factors such as
insertion loss and SiN waveguide loss, we took the following approach to obtain the internal QE at 1310
nm wavelength for the ring/disk detector. Both 635 nm light and 1310 nm light were coupled into the
same single-mode optical fiber and then into the same device. The ratio of the bias-dependent
photocurrent is represented in the following relation:
I1310 ( L,V )  P1310  1310  1310  QE1310 ( L,V )  P1310  1310  1310 





QE1310 ( L,V )

 
I 635 ( L,V )  P635  635  635  QE635
 P635  635  635 
(5)
where I1310 and I 635 are the measured photocurrents, P1310 and P635 are the amounts of optical power
coupled into the SiN waveguide,  635 and 1310 , are the coupling efficiency between the SiN waveguide
and the Si detector at both wavelengths. The quantities of P1310 , P635 , 1310 , and  635 are hard to measure
precisely. The internal quantum efficiency QE635 is verified to be very close to unity (i.e. within 10 %
from unity) and independent of the device length while the internal quantum efficiency QE1310 at 1310
nm wavelength depends on both the bias voltage, V, and device dimension, L. Here L can be considered
as the “effective length” of the resonator detector and its value can vary significantly for wavelengths that
are on-resonance or off-resonance. To precisely determine the internal quantum efficiency QE1310 ( L,V ) ,
 P1310 
 
 and  1310  . This value can be obtained from the photocurrent
 P635 
  635 
we need to find the product of 
ratio of a very long (7 mm), straight waveguide detector since the internal QE at both 630nm and 1310
nm wavelengths is nearly identical for such a long device:
I1310 (,V )  P1310  1310  1310 




I 635 (,V )  P635  635  635 
(6)
As a result, we can obtain the internal QE of the ring/disk resonator detector at 1310 nm
wavelength under different bias voltage from the following relation:
I1310 ( L,V ) I1310 (,V )

QE1310 ( L,V )
I 635 ( L,V ) I 635 (,V )
(7)
We applied the same method to measure the internal QE at other wavelengths in the 1310 nm wavelength
regime.
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