Contact resistivity
and suppression of Fermi level pinning
in side-contacted germanium nanowires
Maria M. Koleśnik-Gray1,2, Tarek Lutz1,2, Gillian Collins3,2, Subhajit Biswas3,2, Justin D. Holmes3,2, and
Vojislav Krstić1,2*
SUPPLEMENTARY MATERIAL
1
School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
Centre for Research on Adaptive Nanostructures and Nanodevices (CRANN), and AMBER at CRANN, Trinity College
Dublin, College Green, Dublin 2, Ireland
3
Department of Chemistry, University College Cork, Cork, Ireland
*
krsticv@tcd.ie
2
1
1) Nanowire synthesis details
Ge NW synthesis was carried out by thermal decomposition of an organogermane precursor,
diphenylgermane (DPG) in supercritical toluene. For ‘type A’ NWs, 17 ml of toluene was added to a 25
ml reaction cell containing an Au sputtered AAO membrane and heated to 400 oC. 40 μl of DPG
dissolved in 20 ml toluene was added to an injection cell which was pressurized to 24.1 MPa. The flow
rate was set to 0.05 ml min-1 and a reaction time of 2 h was used. ‘Type B’ NWs were synthesized on Au
sputtered Si wafers in a 5 ml reaction cell with 3 ml toluene. 20 μl of DPG dissolved in 20 ml toluene
was used. The reaction was carried out at 21 MPa for 3 hours. After the reaction, the NWs were washed
with chloroform, hexane and isopropanol and dried under N2. Type A NWs exhibited a diameter
distribution of 20 to 180 nm, whilst type B NWs displayed a diameter distribution of 15 to 70 nm.
2) Electrical measurements
Electrical characterization was carried out on a probe station at ambient conditions. Two- and four-point
current-voltage characteristics (Id/Vsd and Id/V4T, respectively) were taken along with transfer curves
(Id/Vg), where the degenerately doped Si backside was used as gate electrode.
3) Electrode/nanowire interface: quality analysis
Elemental composition of FIB-prepared contact-interface cross-sectional cuts was studied via EDX.
Spatial resolution of EDX scan is limited due to beam broadening as it passes through the sample. For a
material of thickness t, density ρ, and atomic weight A, the extent of beam-broadening, b, (in nm), is
approximated byS1:
𝜌 1/2
𝑏 = 0.198 (𝐴)
𝑍
(𝐸 ) 𝑡 3/2 ,
0
(S1)
where E0 is the incident beam energy (keV) and Z the atomic number. Values of the parameter b for Ge
and the metals used in this work are gathered in Table I.
Table I. Beam broadening b due to high-energy electrons in EDX. Values are calculated for 100 keV beam and 50 nm thin
lamellas.
Element
b (nm)
================
Ge
6.06
Au
17.31
Pd
10.82
Ag
18.11
Pt
10.26
2
The intermixing regions (corrected for relevant beam broadening) were found to be about 3.3 ± 2.1 nm,
2.1 ± 0.8 nm, 18.4 ± 4.7 nm and 12.2 ± 1.4 nm, for Au, Ag, Pd and Pt, respectively. We did not observe
any correlation between the contact-interface quality and the measured contact-resistivities, in
accordance with earlier studies on metal/semiconductor bulk junctionsS12.
Fig. S1 shows TEM images of the cross-sections. Fig. S2 shows results of elemental composition
analysis.
Fig. S1. Cross-sectional TEM images for (a) Au, (b) Pd, (c) Pt and (d) Ag – contacted Ge nanowires. The crystal-lattice
planes of the Ge (grey areas) are observed indicating that the images are not out of focus.
3
Fig. S2. EDX scans across contact-interfaces for different electrode materials. The intermixing regions (corrected for
relevant beam broadening) were found to be about 3.3 ± 2.1 nm, 2.1 ± 0.8 nm, 18.4 ± 4.7 nm and 12.2 ± 1.4 nm, for Au, Ag,
Pd and Pt, respectively. We did not observe any correlation between the contact-interface quality and the measured contactresistivities, in accordance with earlier studies on metal/semiconductor bulk junctions S12.
4) Examples of current-voltage and transfer characteristics
Exemplary two-point (𝐼𝑑 /𝑉𝑠𝑑 ), four-point (𝐼𝑑 /𝑉4𝑇 ) and transfer (𝐼𝑑 /𝑉𝑔 ) curves measured on different Ge
nanowire devices under ambient conditions are plotted in Fig. S3.
Fig. S3. Exemplary current-voltage characteristics of Ge nanowire devices. a, Two-point characteristic for a 52 nm
nanowire with Pd electrodes. b, Four-point measurement for a 22 nm Au-contacted nanowire. c, Transfer characteristics of
Ge NWs contacted with different metals.
5) Evaluation of transport parameters
The nanowire resistivity, 𝝆𝑵𝑾 , is extracted from the nanowire resistance RNW measured with the correct
electrically active conduction-channel-length, 𝑳𝑪 :
𝝆𝑵𝑾 =
𝝅 𝑹𝑵𝑾 𝒅𝟐
𝟒
𝑳𝑪
.
(S2)
From transfer measurements, the extrinsic transconductance gm (i.e. device-response) can be extracted
within the linear regime of the gate-dependence characteristic at given bias 𝑽𝒔𝒅 :
𝒅𝑰
𝒈𝒆𝒙 = |𝒅𝑽𝒅 |
𝒈
.
(S3)
𝑽𝒔𝒅
The intrinsic transconductance of the nanowire (i.e. material specific), 𝒈𝒊𝒏 , is defined as:
𝒈𝒊𝒏 = 𝒈𝒆𝒙
(𝑺) (𝑫)
𝑹 +𝑹𝑪
𝟏+ 𝑪
𝑹𝑵𝑾
(𝑺)
𝟏−𝒈𝒆𝒙 𝑹𝑪
,
(S4)
4
(𝑺)
(𝑫)
where 𝑹𝑪 and 𝑹𝑪 are the source- and drain-electrode resistances, respectively (cf. following parts
below).
Based on 𝒈𝒊𝒏 , the intrinsic field-effect mobility, 𝝁𝑵𝑾 , of the carriers in the nanowire is given by:
𝒈
𝝁𝑵𝑾 = 𝑽 𝒊𝒏
𝒔𝒅
𝑳𝟐𝑮
𝑪
,
(S5)
where the capacitance to the gate is calculated from a cylinder-on-a-plate modelS2:
𝑪 = 𝟐𝝅𝜺𝜺𝟎
𝑳𝑮
,
𝟐𝒕
𝐥𝐧( 𝒐𝒙 )⁡
(S6)
𝒅
with ε = 3.9 the dielectric constant for SiO2 (gate dielectric), ε0 the vacuum permittivity, and the SiO2
dielectric thickness tox = 300 nm.
The charge carrier concentration Nd is calculated using the expression:
𝑵𝒅 = (𝝆𝑵𝑾 𝒒𝝁𝑵𝑾 )−𝟏,
(S7)
where q is the elementary charge.
6) Total device resistance and contact resistance
Semiconductor devices consist of two Schottky diodes formed at the electrode/semiconductor interface,
connected back to back and separated by the semiconducting channel, as schematically depicted in Fig.
S4 for a p-type semiconductor with depletion type (negative barrier) contact-interface.
Fig. S4. Schematic band diagram of a metal-semiconductor (p-type)-metal device with depletion-type contacts. Green
dashed line represents the acceptor impurity level close to the valence band with states occupied by electrons (filled green
circles) along with the holes (open green circles) within the valence band (VB).
The total voltage drop V across the two contacts (1) and (2) is given by:
𝑽 = 𝑽𝒔𝒅 − 𝑽𝑵𝑾 = 𝑽𝟏 + 𝑽𝟐 ,
(S8)
5
The current Id through the contact interfaces (1) and (2) with cross-sections A1 and A2, respectively, is
constant according to Kirchoff’s law:
𝑰𝒅 = 𝑱𝟏 𝑨𝟏 = 𝑱𝟐 𝑨𝟐 ,
(S9)
where J1 and J2 are the current densities through the interfaces.
For 𝑽 < 𝟎, contact (1) is reverse biased and contact (2) is forward biased, and most of voltage drop
occurs across the depletion region of contact (1), such that 𝑽⁡→ 𝑽𝟏 . For 𝑽 > 𝟎, (1) is forward biased
and (2) is reverse biased and the predominant voltage drop occurs across depletion layer of contact (2),
and 𝑽⁡→ 𝑽𝟐 . The total contact resistance, excluding all parasitic contributions (lead, current crowding),
is defined as:
𝑽
𝑽𝒔𝒅 −𝑽𝑵𝑾
𝒅
𝑰𝒅
𝑹𝑪 = ⁡ 𝑰 =
For 𝑉 < 0, ⁡𝑅𝐶 = ⁡ 𝐽
𝑉1
1 𝐴1
.
(1)
(S10)
= 𝑅𝐶 , where for 𝑉 > 0, 𝑅𝐶 = ⁡ 𝐽
𝑉2
2 𝐴2
(2)
= 𝑅𝐶
(1)
(2)
, with 𝑅𝐶 and 𝑅𝐶
being the
contact resistances at the drain- and source-electrode, respectively. This holds for all Schottky-barrier
systems even in the case of very low 𝛷𝑆𝐵 .
7) Schottky barrier height and depletion region
Within the simple Schottky-Mott modelS3, the Schottky-barrier potential 𝛷𝑆𝐵 ⁡formed at the interface
between a p-type semiconductor with bandgap 𝐸𝑔 and electron affinity 𝜒𝑠 , and a metal of work function
𝛷𝑚 is given by:
𝛷𝑆𝐵 = 𝐸𝑔 + 𝜒𝑠 − 𝛷𝑚 .
(S11)
In the presence of Fermi-level pinning, generally due to the presence of high number of surface states on
the semiconductor, the Schottky-barrier height is defined asS4:
⁡⁡𝛷𝑆𝐵 = 𝛾(𝐸𝑔 + 𝜒𝑠 − 𝛷𝑚 ) + (1 − γ)𝑉𝐹𝐿𝑃 ,
(S12)
𝜕𝛷
where 𝑉𝐹𝐿𝑃 is the pinning potential, and the extent of pinning is defined by the parameter 𝛾 = | 𝜕𝛷𝑆𝐵 |,
𝑚
which takes values from 0 for completely pinned, and 1 for unpinned interfaces. In Table II, work
functions of the metals used in our study are gathered together with the 𝛷𝑆𝐵 ⁡ values predicted by the
Schottky-Mott model without Fermi-level pinning. Clearly, for Au, Pd and Pt, the contacts are expected
to be of the Schottky type whereas in the case of unpinned Ag ohmic behavior is expected.
6
Table II. Metal-electrode work functionsS5 and corresponding Schottky-barrier heights for p-type germanium according to
the Schottky-Mott model. For comparison, the Schottky-barrier height in the case of pinned metal/p-type Ge contacts is
given, equaling -1/3 𝐸𝑔 (with 𝐸𝑔 = 0.66 eV the bandgap of GeS6).
Metal
 m (eV)
 SB (eV)
=======================================================
Pinned metal/p-Ge interface
-
~0.22 eV (1/3 Eg)
Ag
4.52 – 4.74
(-0.08) – (+0.14)
Au
5.32 – 5.47
(-0.81) – (-0.65)
Pd
5.22 – 5.6
(-0.94) – (-0.56)
Pt
5.12 – 5.93
(-1.27) – (-0.46)
The alignment of Fermi levels at the metal/semiconductor interface in the case of Schottky contacts
leads to the formation of a depletion region - a layer within the semiconductor which, in the case of a
full depletion, is absent of mobile charge carriers. This region extends to a certain depth into the
nanowire. From our experimental results, the qualitative properties of the depletion region below
laterally contacted Ge NW devices can be drawn: Firstly, since the charge transport across the contactinterface was found to be realized by Thermionic Emission (see further below), the depletion layer, if
fully depleted, cannot be heavily extended; or, the depletion layer is not fully depleted and a sizable
charge-carrier concentration is present. Secondly, the result that the active conduction channel length
extends between the outer electrode edges suggests the very same aforementioned conclusions on the
depletion region irrespectively of the contact-material used.
8) Character of charge transfer across the contact-interface
Charge transport across a Schottky barrier can be realized via Thermionic Emission, Field Emission and
electron-hole recombination processes, depending on the doping level of the channel material and the
magnitude of the barrier potentialS8. Since the nanowires are low to moderately doped, Thermionic
Emission (TE) would be the first choice of dominating transport in Schottky-type devices. The current
density in TE approach is defined asS6:
𝛷
𝐽=
− 𝑆𝐵
𝐴∗ 𝑇 2 𝑒 𝑘𝐵 𝑇
𝑞𝑉
(𝑒 𝑛𝑘𝐵 𝑇 − 1),
(S13)
where 𝑉 = 𝑉𝑠𝑑 − 𝑉𝑁𝑊 and n is the ideality factor of the Schottky-device.
Numerical fitting of the terminal 𝐼 − 𝑉 curves for Au, Pd and Pt contacted devices yields n values in the
range of 1 to 1.15 (Fig. S5) which is expected for nearly ideal Schottky contacts.
7
Fig. S5. Ideality factor n for Schottky-barriers in the case of Pd,
Au and Pt Schottky contacts. For all contact types, the value of n is
below 1.17. Closest to ‘ideal’ Schottky-characteristics were observed
for Au-contacted devices (n ≈ 1).
The measured current density J in was found to range from 103 to 106 A m-2. Tunneling effects as
dominating transport mechanism were excluded since the expected current density J for tunneling
contacts under these conditionsS9 does not exceed 102 A m-2 as calculated in Fig. S6. Electron-hole
recombination processes, on the other hand, require the Schottky barrier ideality factor n to be equal
2S10, which is not the case in our devices.
Fig. S6. Maximum current density, Jmax , theoretically predicted for
Field EmissionS7 across Au, Pd and Pt Schottky barriers on p-type
germanium. The expected current density is significantly lower than 10 3
– 106 Am-2 measured in the devices, which proves that Field Emission is
not the dominant transport mechanism in the nanowire devices.
9) Parasitic resistances associated with contact geometry
Current crowding effects at the electrode/nanowire interface are expected to contribute to the total
(1)
nanodevice resistance. For side contacts to nanowiresS11, Rgeom has two components: 𝑅𝑔𝑒𝑜𝑚 associated
(2)
with the current crowding underneath the contact-interface, and 𝑅𝑔𝑒𝑜𝑚 , connected with the crowding at
the edges of the contacts. In our approach, we adapted the original modelS11 to the case of cylindrical
contacts to the nanowires to account for the transition from planar to nanodevice geometry. The first
parameter depends on the nanowire properties and the diffusion width, 𝜆𝑚 , and for nanowire with
fraction of the surface β covered by electrode material:
(1)
𝑅𝐿𝐶𝐶 = 𝜉
where 𝜉 =
(2𝑡+𝑑)2 −𝑑2
4𝑡𝑑
𝜌𝑁𝑊 𝜆𝑚
𝜋𝑑2 𝛽
,
(14)
accounts for the ‘bending’ of the contact interface (t: electrode material height).
8
In the original approach, the diffusion width, 𝜆𝑚 , was defined as the diffusion length within the
semiconductor in planar geometry perpendicular to the contact-electrode. For cylindrical side-contacted
nanowires, the maximum diffusion length 𝜆𝑚 ≅ 𝑑 due to the axial confinement of charge carriers. The
second geometrical contribution is defined as:
(2)
𝑅𝐿𝐶𝐶 =
𝜌𝑁𝑊 1
𝑘+1
[ 𝑙𝑛 𝑘−1 + 𝑙𝑛
𝜋𝑑𝛽 𝑘
𝑘 2 −1
𝑘2
],
(15)
1
with 𝑘 = 1−𝛽.
Based on the measurement data, the upper limit of Rgeom was found to be several orders of magnitude
lower than the RNW and RC for both types of nanowire devices (Fig. S7).
Fig. S7. Current crowding contributions to the total device resistance. For both type A (left) and type B (right) nanowires,
Rgeom is two orders of magnitude and more lower than the nanowire- and contact-resistances.
10) Transfer length
On the basis of contact resistivity values, we extracted the transfer length, LT ,
π ρ
LT = √4 ρ C d ,
(16)
NW
which is a measure for the maximum contact-width over which current injection can occur for a given
electrode/nanowire interface. For all devices investigated, we found LT > Δ as shown in Fig. S8, which
reconfirms the four-point measurement results (cf. Fig. 2 in manuscript) that the flow of charge carriers
occurs also over the nanowire length between and underneath the contacts.
Fig. S8. Transfer length, 𝐿𝑇 , in Ge nanowire devices compared to the
electrode width, Δ. In all cases, 𝐿 𝑇 > Δ.
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11) Channel and gate length – graphs for nanowires contacted with different metals
Fig. S9 shows the verification of the active conduction channel and gate length for different
electrode/NW diameter combinations (cf. Fig. 2 in manuscript).
Fig. S9. Verification of the active conduction channel and gate length (cf. Fig. 2 in manuscript). Nanowire resistivity and
mobility values calculated for according lengths – data shown for a range of NW diameter sizes and metal electrodes.
REFERENCES:
S1.
S2.
S3.
S4.
S5.
S6.
S7.
S8.
S9.
S10.
S11.
S12.
Ed. A. I. Kirkland and J. L. Hutchinson, Nanocharacterization, Ch. 4., p. 104, RSC Publishing, Great
Britain, 2007.
Wunnicke, O. Appl. Phys. Lett. 2006, 89, 083102.
Tersoff, J. Phys. Rev. B. 1985, 32, 6968-6971.
Malagu, C.; Guidi, V.; Carotta, M. C.; Martinelli, G. Appl. Phys. Lett. 2002, 91, 808-814.
CRC Handbook of Chemistry and Physics. 85th ed. CRC Press: Boca Raton, FL, 2004-2005.
Nishimura, T.; Kita, K.; Toriumi, A. Appl. Phys. Lett. 2007, 92, 123123(3).
Chang, C. Y.; Fang, Y.K.; Sze, S.M. Solid State Electron. 1971, 14, 541-550.
Yu, A. Y. C. Solid State Electron. 1970, 13, 239-247.
Zhang, Z.; et al. Adv. Funct. Mater. 2007, 17, 2478-2489.
Leonard, F.; Talin, A. A. Nature Nanotech. 2011, 6, 773-783.
Koleśnik, M. M.; et al. Small 2011, 7, 2873-2877.
Thompson, M. J., Niemanich, R. J., Tsai, C. C. Surf, Sci. 1983, 132, 250-263.
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