APL Supplementary Material
Carrier dynamics
InGaAs/InAlAs
in
Beryllium
doped
low-temperature-grown
B. Globisch, R. J. B. Dietz, D. Stanze, T. Göbel, M. Schell
Fraunhofer Institute for Telecommunications, Heinrich Hertz Institute, Einsteinufer 37, 10587 Berlin, Germany
This supplementary material contains the derivation of the analytical equations for the carrier population in the
CB in the cases of trap saturation and partial trap filling, i.e. the derivation of Eq. (6) and Eq. (7) of the main
article are described.
First, we account for carrier conservation in Eqs. (3). Electrons in the CB and electrons trapped by ionized
arsenic antisites equal holes in the VB and trapped holes in ionized Be dopants. Hence, the following equation
holds for all times t:
nt   nT t   nBe t   ht 
(A1)
Additionally, we introduce dimensionless variables in order to simplify the subsequent analysis:

N  n / N As
;

NT  nT / N As
;

N Be  nBe / N As
;
t   t / e
(A2)
Here, carrier densities are normalized to the number of ionized antisites NAs+ and time is measured in units of
the electron trapping time τe. Inserting Eq. (A1) and Eq. (A2) into Eqs. (3) of the main article yields:
dN ~
 G(t ,N)  N 1  NT 
dt 

dNT
~ N
 N 1  N T   BR  As
 1  N T  N Be



dt 
 N As


dN Be  e
~ N
 N Be .
 N  N T  N Be   BR  As

1

N
T
 N

dt 
τh
As


(A3a)
(A3b)
(A3c)
Here, Gt , N  and BR  BR c N As denote the normalized generation rate and the normalized recombination
rate. The dynamic equation for the valence band holes has been eliminated with Eq. (A1). Subsequently,
Eqs. (A3a)-(A3c) are expanded under trap saturation conditions and partial trap filling, respectively.
~
~

Trap saturation
The condition of complete electron trap saturation means that the each arsenic antisite defect is occupied by an
electron from the CB. Saturation occurs when the density of excited electrons outnumbers the density of
trapping states. After the initial trap filling process NT is clamped to its maximum value NT  1 since electron
trapping is much faster than electron recombination. Taking this situation as the starting point of the dynamic
development each recombined electron is directly replaced by another free electron from the CB (detailed
balance). Hence, the two terms on the RHS of Eq. (A3b) are equal as long as the saturation condition holds true.
The same assumption holds true for holes captured by negatively charged Be dopants in Eq. (A3c). In order to
simplify the terms in Eqs. (A3) we consider small deviations from this temporary equilibrium and write:
NT t   1  ut  ,
N Be t   1  vt  .
(A4a)
(A4b)
Here, ε is a small parameter and u t  , vt  denote the time dependent deviations from the saturation condition.
Inserting Eq. (A4a) and Eq. (A4b) into Eqs. (A3) and neglecting terms in ε2 yields:
dN ~
 G(t ,N)  Nu ,
dt 
 ~ N
du
~ N

 Nu  BR  As
v  u   BR As
,




dt 
N As
 N As

 ~ N
dv  e
~ N

 Nv  BR  As
v  u   BR As
.


N

dt   h
N As
 As

(A5a)
(A5b)
(A5c)
Now, Eq. (A5b) and Eq. (A5c) can be solved by the detailed balance relation mentioned above:
 ~ N
~ N
0  Nu  BR  As
v  u   BR As
,


N

N As
 As

0
(A6a)
 ~ N As
e
~ N
  BR
Nv  BR  As
v

u
.

 N

h
N As
 As

(A6b)
Solving for u and v and inserting the results in Eq. (A5a) leads to the expression:
dN
 N
dt 
~ N e
B R As
 
N As
h


~ N
N e  B R  As
 1

N

h
 As

.
(A7)
In order to simplify Eq. (A7) we solve Eq. (A3b) for NT  N Be  1 . In this case one neglects the refilling of
traps with conduction band electrons and simply accounts for carrier recombination.
~ N
N Tsat t   1  B R As
t .

N As
(A8)
~

Eq, (A8) describes a linear decay of the trapping centers with BR N As / N As . Since carrier capture is assumed to
be much faster than carrier recombination the condition:
~ N
B R As
 1

N As
(A9)
holds true. Using this relation in Eq. (A7) we obtain:
dN
~ N
  B R As
.

dt 
N As
(A10)
Hence, the dynamics of the population in CB is completely determined by the recombination process. Solving
Eq. (A10) and transforming back to real quantities leads to:
~

n sat t   n0sat  BR N As N As
t,
sat
(A11)
Here, n0 is the conduction band population when trap saturation occurs. The discussion of Eq. (A11) is done in
the main article.
Partial trap filling
For partial trap filling the number density of electrons excited to the CB is still small compared to the total
number of trapping centers. Nevertheless, it is assumed to be high enough to decrease the probability for carrier
trapping due to partial trap filling. Since carrier recombination is much slower than carrier capture we neglect
the second term on the RHS of Eq. (A3b). For simplicity, we account for the generation term in Eq. (A3a) by an
appropriate initial condition. Hence, the dynamic equations reduce to:
dN
  N 1  NT  ;
dt 
dNT
 N 1  N T  ;
dt 
N t  0  N ex ,
(A12)
NT t  0  0 .
(A13)
These above equations can be solved analytically and the result for the conduction band population reads:
N t  
N ex  1
,
1  1 / N ex exp  N ex  1 t 
(A14)
Note that N is normalized to NAs+. Hence the term N ex  1 describes the difference between excited electrons in
CB and the number of available trapping centers NAs+. Next, we analyze Eq. (A14) with respect to partial trap
filling, i.e. 1  N ex  1  0 .
N t  
1  N ex 
 N ex 1  N ex  exp  1  N ex  t  ,
1 / N ex exp 1  N ex  t   1
(A15)
with 1 / N ex  1 , exp 1  N ex  t   1 for all t   0 . Since the number of excited carriers has assumed to be
smaller than NAs+ the quadratic term in Nex can be neglected. Transforming back to real quantities yields Eq. (7)
of the main article:
 
n
n(t )  nex exp  1  ex
N As
 

with nex  N ex N As .
 t 
 ,
e 
 
(A16)