Supplementary material for “On the validity of MIS-CELIV for mobility determination in
organic thin-film devices” by
Oskar J. Sandberg, Mathias Nyman, Staffan Dahlström, Simon Sandén, Björn Törngren, JanHenrik Smått, and Ronald Österbacka
Theory of SCL extraction current transients in metal-insulator-semiconductor structures
In the following the theory of space-charge-limited (SCL) extraction current transients for a MIS
structure, with a finite insulator capacitance, is derived. Averaging the total transient current
density,S1
𝑗(𝑡) = 𝐽𝑐 (𝑥, 𝑡 ) + 𝜖𝜖0
𝜕𝐸(𝑥,𝑡)
(S1)
𝜕𝑡
over the semiconductor region (0 ≤ 𝑥 ≤ 𝑑𝑠 ), we find
𝑑
1
𝑑
1
𝑗(𝑡) = 𝑑 ∫0 𝑠 𝑗(𝑡)𝑑𝑥 = 𝑑 ∫0 𝑠 𝐽𝑐 (𝑥, 𝑡)𝑑𝑥 +
𝑠
𝑠
𝜖𝑠 𝜖0 𝜕𝑉𝑠 (𝑡)
𝑑𝑠
𝜕𝑡
(S2)
𝑑
where 𝑉𝑠 (𝑡) = ∫0 𝑠 𝐸(𝑥, 𝑡)𝑑𝑥 is the potential difference across the semiconductor. Noting that the
conduction current is zero in the insulator, a similar averaging over the insulator region yields
0
1
𝑗(𝑡) = 𝑑 ∫−𝑑 𝑗(𝑡)𝑑𝑥 =
𝑖
𝑖
since 𝐴 =
𝜕𝑉(𝑡)
𝜕𝑡
=
𝜕𝑉𝑠 (𝑡)
𝜕𝑡
+
𝜕𝑉𝑖 (𝑡)
the insulator and 𝑉(𝑡) =
Solving Eq. (S3) for
𝜕𝑡
𝜖𝑖 𝜖0 𝜕𝑉𝑖 (𝑡)
𝑑𝑖
𝜕𝑡
𝜖𝑖 𝜖0
𝑑𝑖
[𝐴 −
𝜕𝑉𝑠 (𝑡)
𝜕𝑡
]
(S3)
0
, with 𝑉𝑖 (𝑡) = ∫−𝑑 𝐸(𝑥, 𝑡)𝑑𝑥 being the potential difference across
𝑖
𝑑𝑠
∫−𝑑 𝐸(𝑥, 𝑡)𝑑𝑥
𝑖
𝜕𝑉𝑠 (𝑡)
𝜕𝑡
=
we find
= −𝑈𝑜𝑓𝑓 + 𝐴𝑡 the total potential across the device.
𝜕𝑉𝑠 (𝑡)
𝜕𝑡
= 𝐴 − 𝑗(𝑡)𝑑𝑖 /𝜖𝑖 𝜖0 , and after re-substitution into Eq.
(S2), we obtain
1
𝑑
𝑗(𝑡) = 𝑗0 + 𝑑 ∫0 𝑠
𝑠
with 𝑗0 ≡
𝜖𝑠 𝜖0 𝐴
𝑑𝑠
𝜖 𝑑
−1
(1 + 𝜖𝑠𝑑 𝑖 ) .
𝑖 𝑠
𝐽𝑐 (𝑥,𝑡)
𝜖 𝑑 𝑑𝑥
(1+ 𝑠 𝑖 )
𝜖𝑖 𝑑𝑠
(S4)
Now, assuming that all carriers are initially concentrated at 𝑥 = 0, the current density 𝑗(𝑡) can be
solved analytically within the time interval 0 < 𝑡 < 𝑡𝑠𝑐 , where tsc is the time at which the leading
front of charge carriers reach the anode. Since no carriers arrive at the anode before 𝑡 = 𝑡𝑠𝑐 , the
conduction current at the anode is zero within this time interval, 𝐽𝑐 (𝑑𝑠 , 𝑡) = 0. The total current
transient evaluated at the anode then reads
𝑗(𝑡) = 𝜖𝑠 𝜖0
𝜕𝐸𝑎𝑛 (𝑡)
(S5)
𝜕𝑡
where 𝐸𝑎𝑛 (𝑡) = 𝐸(𝑑𝑠 , 𝑡). Moreover, assuming a drift-only transport model, 𝐽𝑐 (𝑥, 𝑡 ) =
𝑒𝜇𝑝(𝑥, 𝑡)𝐸(𝑥, 𝑡) =
by 𝑝(𝑥, 𝑡) =
𝜇𝜖𝑠 𝜖0 𝑑
2
𝜖𝑠 𝜖0 𝑑𝐸(𝑥,𝑡)
𝑒
𝑑𝑥
𝑑𝑥
[𝐸 2 (𝑥, 𝑡)] with 𝐸(0, 𝑡) = 0, where the injected hole density is given
by Gauss law, the integral in Eq. (S4) can be evaluated:
𝜇𝜖 𝜖 𝐸 2
0 𝑎𝑛
𝑗(𝑡) = 𝑗0 + 2𝑑𝑠 (1+𝑓)
= 𝑗0 [1 +
𝑠
2
𝜇𝐸𝑎𝑛
2𝐴
]
(S6)
𝜖 𝑑
where 𝑓 ≡ 𝜖𝑠𝑑 𝑖 and 𝐸𝑎𝑛 (0) = 0 was used. Upon equating Eq. (S5) and (S6) we obtain
𝑖 𝑠
𝜖𝑠 𝜖0
𝜕𝐸𝑎𝑛 (𝑡)
𝜕𝑡
𝜇
2 (𝑡)]
= 𝑗0 [1 + 2𝐴 𝐸𝑎𝑛
2𝐴
𝑡
1
𝐸𝑎𝑛 (𝑡) = √ 𝜇 tan (𝑡
0 √1+𝑓
(S7)
)
(S8)
2𝑑2
where 𝑡𝑡𝑟 = √ 𝜇𝐴𝑠 (1 + 𝑓). Hence, the corresponding extraction current transient reads
𝑗(𝑡) = 𝑗0 [1 + tan2 (
𝑡
𝑡𝑡𝑟 √1+𝑓
)]
(S9)
This expression is valid for 0 < 𝑡 < 𝑡𝑠𝑐 , where 𝑡𝑠𝑐 is obtained from
𝑡𝑠𝑐
𝑡𝑠𝑐
𝑑𝑠 = ∫ 𝜇𝐸𝑎𝑛 (𝑡)𝑑𝑡 = −2𝑑𝑠 (1 + 𝑓) ln [cos (
)]
𝑡𝑡𝑟 √1 + 𝑓
0
or equivalently,
1
𝑡𝑠𝑐 = 𝑡𝑡𝑟 √1 + 𝑓 cos−1 [𝑒 −2
(1+𝑓)−1
]
(S10)
which, depending on the value of 𝑓 is close to 𝑡𝑠𝑐 ≈ 0.92𝑡𝑡𝑟 .
For 𝑡 > 𝑡𝑠𝑐 , on the other hand, the electric field at the anode has time to redistribute and maintain
3𝑉
conditions similar to quasi-equilibrium, 𝐸𝑎𝑛 → 2𝑑𝑠 , in accordance with the theory of SCL
𝑠
injection current transients.S1 Inserting into Eq. (S3): 𝑗(𝑡) =
2𝑑𝑠 𝜕𝐸𝑎𝑛 (𝑡)
3
𝜕𝑡
𝜖𝑖 𝜖0
𝑑𝑖
[𝐴 −
𝜕𝑉𝑠 (𝑡)
𝜕𝑡
]=
𝜖𝑖 𝜖0
𝑑𝑖
[𝐴 −
] and equating with Eq. (S6) we find
[1 −
2𝑑𝑠 𝜕𝐸𝑎𝑛 (𝑡)
𝑗0
𝜇 2
]=
[1 +
𝐸 (𝑡)]
3𝐴
𝜕𝑡
𝑗𝑠𝑎𝑡
2𝐴 𝑎𝑛
and hence,
3𝑡
𝑗
𝑗(𝑡) = 𝑗0 + (𝑗𝑠𝑎𝑡 − 𝑗0 ) tanh2 (2𝑡 √𝑗 0 )
𝑡𝑟
9𝑡 2
𝑠𝑎𝑡
(S11)
Note that the result 𝑗(𝑡) = 𝑗0 [1 + 4𝑡 2 ] by Juška et al.S2 is reobtained in the limit of
𝑡𝑟
𝑗𝑠𝑎𝑡
𝑗0
→ ∞, as
expected.
The effect of diffusion under equilibrium conditions
In Fig. S1 and S2, the effect of diffusion under equilibrium dc conditions prior to the application
of the transient voltage pulse (𝑡 < 0) and on the extracted MIS-CELIV mobility, is demonstrated
with numerical drift-diffusion simulations.
Fig. S1. (a) Simulated equilibrium hole density (prior to the pulse) is simulated at different dc offset voltages for the
case with an ohmic contact at the anode (𝑥 = 𝑑𝑠 ). (b) The corresponding simulated MIS-CELIV mobilities [Fig. 3(a)
in the main text] are plotted on log-log scale over a wide range of 𝐴∗ 𝑡1 .
Fig. S2. The energy level diagrams for the HOMO level of the semiconductor under dc conditions (with large offset
voltage, prior to the extraction pulse) in the case of different energy level offsets at the anode (𝑥 = 180 nm). The
energy level offset between the semiconductor and the anode, typically referred to as the injection barrier, is given by
𝑘𝑇 ln[𝑁𝑣 /𝑝𝑎𝑛 ], where 𝑝𝑎𝑛 is the hole density at 𝑥 = 𝑑𝑠 , here given in units of 𝑝0 ≡ 𝜖𝑠 𝜖0 𝑘𝑇/𝑒 2 𝑑𝑠2 . It can be seen that
a (bias-induced) built-in voltage 𝑉 ∗ is formed over the semiconductor layer in the case when an injection barrier is
present at the anode. The device is assumed to have reached equilibrium with the current being exactly zero across
the device under dc conditions. In reality this condition is not necessarily met in case of a large injection barrier
and/or imperfect insulator.
Experimental MIS-CELIV mobilities on hole-only P3HT devices
MIS-CELIV measurements on sample structure ITO/SiO2/P3HT/Ag were carried out. Samples
were prepared as follows; ITO covered borosilicate glass (from Präzisions Glas & Optik GmbH)
was used as substrate. Half of the substrate was etched with HCl for 40 minutes. The etched
substrates were cleaned in a 1:1:5 blend of H2O2, NH3 and water in an ultrasonicator for 30
minutes at 70 °C and blown dry with nitrogen. The SiO2 layers were made by dip coating the
ITO-substrates in a silica precursor solution. The silica precursor solution was made by first
dissolving 0.0152 g of the block copolymer Pluronic F-127 in 17.38 ml of EtOH and adding 0.26
ml of 12.5 vol% aqueous HNO3 solution and 1.93 mL of tetrahydrofuran under stirring.
Thereafter, 1.0 g of tetraethyl ortosilicate was added dropwise to the solution under stirring. The
solution was left stirring at room temperature for 2h in order to reach equilibrium before
proceeding with dip coating. The dip coating was carried out in room temperature with a constant
humidity of 15%. After dip coating, the SiO2 films were sintered at 400 °C and cleaned in an
ultrasonicator in water, acetone and IPA at 60 °C for 5 minutes each and blown dry with
nitrogen. The cleaned SiO2 samples were transferred into a nitrogen glove box where the rest of
the fabrication took place. P3HT films (450 nm) were spin-cast at 700 rpm from 30 mg/ml
solution in chlorobenzene and annealed at 120 °C for 10 minutes. Subsequently a 50 nm layer of
Ag was thermally evaporated as a top contact for hole injection and extraction. The active layer
thickness was determined using atomic force microscopy. MIS-CELIV measurements were
performed in a vacuum cryostat at room temperature. The complete device was kept in air 16
hours prior to measuring in order to obtain a hole-ohmic injecting contact by oxidation of the Ag
top contact leading to an increased work function of the metal. No air-doping of the P3HT layer
could be detected in the MIS-CELIV measurements, see Fig. S3.
In Fig. S3, experimental MIS-CELIV mobilities are shown as a function of 𝐴∗ 𝑡1 . The mobilities
are extracted from the current transients either using Eq. (9) or Eq. (11). The voltage amplitude of
the CELIV pulse has been varied between 3 V and 7 V and the pulse length between 2.5 μs and
10 ms. The mobility is extracted from the current transients under conditions when a large hole
reservoir is accumulated at the insulating SiO2 layer. When using Eq. (9), an overestimation of
the mobility at lower 𝐴∗ 𝑡1 values can be seen in Fig. S3, as predicted by simulations in the case
of an ohmic injecting contact. Using Eq. (11), where also diffusion has been taken into account,
however, a mobility of roughly 2 × 10−4 cm2/Vs is obtained at larger 𝐴∗ 𝑡1 values, in agreement
with the literature.S3 We note that for P3HT at room temperature a small increase in mobility
towards lower electric field has been previously observed using CELIV and time-of-flight
measurementsS3. This field dependence might partly explain the increasing mobility towards
lower 𝐴∗ 𝑡1 values seen in Fig. S3 using Eq. (11).
Fig. S3. Above: experimental MIS-CELIV transients with varying offset voltage, 𝑈𝑜𝑓𝑓 , are shown. Below:
experimental MIS-CELIV mobilities calculated either using Eq. (9) or Eq. (11) are plotted as a function of 𝐴∗ 𝑡1 .
[S1]
M. A. Lampert and P. Mark, Current Injection in Solids (Academic Press, New York, 1970).
[S2]
G. Juška, N. Nekrašas, and K. Genevičius, J. Non-Cryst. Sol. 358, 748 (2012).
[S3]
A. J. Mozer, N. S. Sariciftci, A. Pivrikas, R. Österbacka, G. Juška, L. Brassat, and H. Bässler,
Phys. Rev. B 71, 035214 (2005).