Electrical Conductivity of Porous Silver Made from Sintered Nanoparticles

Electron. Mater. Lett., Vol. 11, No. 2 (2015), pp. 315-321

Electrical Conductivity of Porous Silver Made from Sintered Nanoparticles

Abu Samah Zuruzi1,* and Kim S Siow

2

1Engineering Product Development Pillar, Singapore University of Technology and Design, 20 Dover Drive,

S(138682), Singapore2Institute of Microengineering and Nanoelectronics, Universiti Kebangsaan Malaysia, 43600 Bangi,

Selangor D.E., Malaysia

(received date: 21 October 2014 / accepted date: 23 November 2014 / published date: 10 March 2015)

1. INTRODUCTION

Silver (Ag) has been used as filler in electrically conductive

adhesives because of its high electrical conductivity,

resistance to oxidation and high melting temperature.[1] Such

adhesives are used to join silicon dies to a substrate or lead

frame to facilitate heat dissipation. Silver paste (without the

adhesive component) has been used to bond dies directly to

copper for higher power application such as in power

modules.[2] The microelectronic packaging industry sees

low-temperature sintered Ag pastes as a potential replacement

for PbSn solders used for die attach.[3-5] Similarly, the printed

electronics industry is developing low sintering temperature

Ag ink as electrical conductors compatible with low cost

plastic or paper substrates for flexible circuits.[6-8]

Silver pastes or inks with low-temperature sintering

capability utilize Ag in the form of nanorods,[9] nanowires,[10,11]

nanoparticles.[8,7,12] Sintering temperatures of these materials

are between 100 to 200°C; much lower than melting

temperature of bulk Ag, 961.8°C.[13] Prior studies suggest

that at these temperatures, surface diffusion and/or surface

melting occur on nanoscale Ag leading to metallurgical

bonding and, consequently, electrical conduction.[14,15] Ag

pastes and inks sintered at 150°C for up to about 30 minutes

are often porous. As sintering proceeds at a constant

temperature, pores in Ag pastes and inks shrink. Porosity in

metals suppresses electrical and thermal conductivities;

properties which are of paramount importance in electronic

applications.[16] In a recent study, degradation in the

performance of an optical device has been attributed to pores

in its sintered Ag die attach.[17] Pores act as hotspots and

degrade thermal dissipation resulting in a temperature

increase and, subsequently, an undesired wavelength drift.

Here we study the relationship between electrical con-

ductivity and porosity when Ag nanoparticles are sintered at

150°C up to 5 mins. We propose a model that relates

electrical conductivity to morphology. Although there are

existing physical models,[18-20] our model captures salient

features not considered in prior models; such as material

accumulation at vertices during sintering, due to amplified

surface tension effects at the nanoscale, and shape of

ligaments. There is close agreement between electrical

conductivity predicted by our model, which does not contain

any fitting parameter, and that determined experimentally.

Electrical conductivity of open cell porous silver (Ag) with sub-micrometer featureswas studied. Porous Ag was formed from annealing Ag nanoparticles at 150°C up to5 minutes. Porous Ag is a network of cylindrical ligaments joined at their ends tospherical vertices. Electrical conductivity of porous Ag was ~20% of bulk value after5 mins annealing. Kelvin cells (truncated octahedrons) with cylindrical ligaments andspherical vertices (CLSV) were used to compute electrical conductivity which is inagreement with experimental data, without any fitting parameter. Results of theCLSV model are also in agreement with the well-established Koh-Fortini empiricalrelation.

Keywords: porous silver, electrical conductivity, silver nanoparticle, sintering,interconnects

DOI: 10.1007/s13391-014-4357-2

*Corresponding author: [email protected]©KIM and Springer

316 A. S. Zuruzi et al.

Electron. Mater. Lett. Vol. 11, No. 2 (2015)

2. MATERIALS AND METHODS

2.1 Materials and methods

Silver (Ag) paste was printed on electrically insulating

plastic support sheet. Ag paste contains Ag nanoparticles

(average radius 25 nm) suspended in viscous organic phase.

Initial thickness of the Ag paste layer is 450 μm. The Ag

paste layer/plastic sheet was then cut to obtain 1.5 cm square

samples. Mass of Ag paste layer/plastic sheet samples was

determined using a high resolution analytical mass balance

with repeatability down to 0.01 mg (Mettler Toledo). Mass

of a 1.5 cm plastic support sheet was determined in the same

way. Mass of the Ag paste layer in each sample was

determined after off-setting the mass of the 1.5 cm plastic

support sheet. All samples were annealed at 150°C in air

with no applied load. After annealing for the required time,

their mass measured again. Hence, mass change of samples

after annealing was determined. Thickness of Ag paste layer,

τ, after annealing was determined, at a step made at an edge,

using a contact profilometer. Surface roughness of Ag paste

layer is less than 1% of its thickness and does not contribute

significantly to error in volume computation.

2.2 Characterization

Morphology of porous Ag layer formed was studied using

a scanning electron microscope (SEM) equipped with a field

emission gun (JEOL). Dimensions of microstructural features

were measured in-situ in the SEM chamber. Electrical

conductivity was computed from current-voltage measurements

collected at room temperature using a four point probe

system comprising of a Keithley source meter coupled to a

Keithlink prober. The probe head has Cu-Be alloy pins

positioned collinearly at 1.25 mm spacing. Voltage was

measured as current was sourced from 0 to 100 mA at

10 mA increments.

3. EXPERIMENTAL RESULTS

Scanning electron microscopy (SEM) shows porous

structure in annealed Ag paste layer, Fig. 1(a) to (d).

Nanoparticles in pristine Ag paste layer exist as distinct

particles held together by organic binders. After annealing at

150°C, Ag paste layers consist of a porous network of

cylindrical ligaments connected to spherical vertices at their

ends. Also, morphology of porous Ag coarsens with

increasing annealing time.

During annealing, porosity of the Ag paste layer decreased

with time, Fig. 2(a). There was no observable change in edge

length of samples during annealing, hence it plausible to

assume volume change in the Ag paste layer after annealing

is due to the reduction in thickness only. Mass and thickness

of Ag paste layer decreased during annealing. The rate of

decrease for mass and thickness of the Ag paste layer was

greatest at the start and decreased with annealing time. Mass

and thickness loss after 5 minutes annealing are 2.5 mg

(0.8% of initial mass) and 9 μm (2% of initial thickness),

respectively. This suggest that organic components had

largely vaporized after 5 mins annealing; this is supported by

Fig. 1. Morphology of silver paste: After (a) 1 min (b) 3 min and (c) 5 min annealing; (d) During annealing, sintering between Ag nanoparticlesoccur resulting in formation of porous morphology with cylindrical ligaments joined at their ends to spherical vertices.

A. S. Zuruzi et al. 317


Raman spectroscopy data not presented here. Hence, it is

reasonable to assume that mass of Ag in all samples to be

272.8 mg. Since residual organics exist as superficial films,

they do not add to volume of the porous Ag. Although mass

of Ag is 272.8 mg for all samples, thickness of the Ag paste

layer decreased with annealing time due to compaction;

hence the apparent density, ρ of porous Ag increased.

Porosity, ϕ, of porous Ag plotted in Fig. 2(b), was computed

using the equation ; where ρr is relative

density of porous Ag and is given by the ratio between

density of sample, ρ, to that of bulk Ag, ρo; we used

10.49 gcm−3 for ρo.[13] Apparent density, ρ, of the Ag paste

layer in the as-received state and after 5 mins annealing was

3.20 and 5.40 gcm−3 respectively. Porosity, ϕ, decreased

from 0.69 in the as-received condition to 0.48 after 5 mins

annealing.

Porous Ag formed after annealing has significantly higher

electrical conduction than pristine Ag paste. The relative

conductivity, σr, is defined as the ratio between conductivity

of sample, σ, to that of bulk Ag, σo; we assumed σo = 6.30 ×

107 Sm−1.[13] Using appropriate correction factors to com-

pensate for sample geometry, the relative electrical con-

ductivity, σr, of samples annealed at different annealing

times were computed, Fig. 2(b). Electrical conductivity of

porous Ag increased significantly upon annealing at 150°C.

Between 1 and 2 min annealing, σr increased from 0.10 to

0.17; σr was 0.19 after 5 mins annealing.

The increase in electrical conductivity in porous Ag is due

to conducting paths formed during annealing. Although bulk

Ag has a melting point of 961.8°C,[13] surface melting on Ag

nanostructures has been reported between 110 to 150°C.[21,22]

As organic components vaporized, metallic bonds are

formed at contact points between particles through surface

melting and/or diffusion. cAg paste changes from one

consisting of distinct nanoparticles held by organic binders

in the pristine condition, to an electrically conducting porous

structure with cylindrical ligaments connected at their ends

to spherical vertices after annealing.

Electrical conductivity in porous metals depends on

geometry of conduction paths. Accordingly, apparent electrical

conductivity of porous Ag changes as ligaments and joints

coarsen with annealing time. High resolution scanning

electron microscope images show that porous Ag consists of

cylindrical ligaments connected to spherical vertices, Fig.

1(d). Radii of vertices are noticeably larger than those of

cylindrical ligaments. During sintering surface tension draws

material away from ligaments resulting in accumulation at

vertices.[16]

The radius A and length l of ligaments and radius R of

vertices at various annealing times are listed in Table 1. For

each annealing time, 25 sets of data were collected for radius

A and length l of ligaments and radius R of vertices; the

average and one standard deviation for each parameter were

then computed. For any one set, one measurement each of A,

l and R were made at a ligament-vertex joint. In general, the

radius A and length l of ligaments and radius R of vertices

increased with annealing time; spread or variability of these

parameters increased with as well.

4. MORPHOLOGICAL MODEL FOR ELEC-

TRICAL CONDUCTIVITY

Physical models relating electrical conductivity to

morphology of porous metals exist.[18-20] These models were

developed using macroporous metals with microstructures

ϕ 1 ρr– 1 ρ/ρo–= =

Fig. 2. Densification of Ag paste layer: Porosity of Ag pastedecreases while electrical conductivity increases with annealingtime.

Table 1. Parameters of porous Ag after annealing at various times. As annealing time increased, radius and length of ligaments and vertex radiusall increased.

Annealing Time

(min)

Ligament radius, A (nm) Ligament length, L (nm) Noderadius, R (nm)

Average Standard Deviation Average Standard Deviation Average Standard Deviation

1 98 18 373 122 204 36

2 236 46 488 176 240 44

3 296 106 503 253 350 73

4 316 56 524 198 335 42

5 336 89 639 231 366 88



in the range from tens to hundreds of micrometers. However,

at those length scales surface tension, that leads to material

accumulation at vertices at nanoscale, are less pronounced.[16,23]

In the present study, material accumulation results in

formation of spherical vertices. Material accumulation at

vertices has a significant impact as it takes away material

from ligaments causing resistance to increase. The model we

develop here takes into consideration cylindrical ligaments

between spherical vertices.

A Kelvin cell is a truncated octahedron with slight

curvature to hexagonal faces and fills space with minimal

surface area. From an energy perspective and in the context

of morphological evolution during annealing, a model based

on Kelvin cell is attractive as it reduces surface area. Here,

we model porous Ag as an assembly of truncated octahedron

unit cells, Fig. 3(a) and (b). In our model, cell edges are

cylindrical ligaments which are joined at their ends to

spherical vertices. Cylindrical ligaments have radius A and

length l while spherical vertices have radius R, Fig. 3(c). The

height of spherical cap, the volume defined by the overlap

between ligament and vertex, is denoted by H, Fig. 3(d).

Each spherical vertex has 4 cylindrical ligaments attached to

it and forms a tetrahedron. Although our model does not

perfectly describe actual morphology of porous Ag, there is

reasonable agreement with observed structural features, Fig.

3(e). From this point on, we denote the proposed model as

truncated octahedron with cylindrical ligaments and spherical

vertex (CLSV).

The CLSV model was used to compute porosity and

electrical conductivity of porous Ag using average values

of radius A and length l of ligaments and radius R of

vertices measured experimentally. In a truncated octahedron

unit cell, there are 36 cylindrical ligaments and 24 spherical

vertices. Each ligament and vertex are shared by 3 and

4 contiguous unit cells, respectively. The volume of Ag, VAg,

associated with a unit cell is given by

; the negative term in the square bracket

accounts for the volume of overlap with the 4 connecting

ligaments. In Fig. 3(d), the volume of overlap between a

vertex and a ligament is a spherical cap and is denoted as

Vcap. From geometry, height of a spherical cap is H = R −

. Accordingly, length L of each ligament is

, Fig. 3(c). Volume of Ag, VAg,

can then be expressed as

.

Mass of Ag associated with 1 truncated octahedron is VAg

ρ0; where ρ0 is density of bulk Ag, taken as 10.49 gcm−3. The

volume, VUC, of a truncated octahedron unit cell with edge of

length L, is given by ; all

variables in expressions for VUC and VAG are listed in Table 1.

Density of porous Ag, which is the mass of Ag associated

VAg

36

3------πA

2l

24

4------

4

3---πR

3–+=

4

6---πH 3A

2H

2+{ }

R2

A2

–

L l 2 R H–( )+= l 2 R2

A2

–+=

VAg 12πA2l 8πR

3– 4π 4R

22A

2+[ ]+=

R2

A2

–

VUC 8 2L3

8 2 l 2 R2 A2–+[ ]3= =

Fig. 3. Porous silver modeled as truncated octahedrons with cylindrical ligaments and spherical vertices (CLSV). Schematic representation of(a) a single and (b) 3 truncated octahedron unit cells with cylindrical ligaments (edges) and spheres (vertices); (c) Schematic of a ligament joinedto spherical vertices at its ends; (d) Tetrahedral formed when 4 ligaments are joined to a spherical vertex; (e) Overlay of (d) on an SEM image.Note the multiple grains in ligaments and vertex in (e).



per unit volume of 1 truncated unit cell, was determined by

computing the ratio of VAg ρ0 and VUC. Hence, porosity can

be computed .

The octahedron unit cell can be viewed as a network of

discrete conductors. To extract the apparent electrical

conductivity, we treat each edge as a discrete conductor with

resistance RL. Each edge in the truncated octahedron

comprised of a cylinder and two truncated spheres, Fig.

3(c). Each cylinder and sphere is shared by 3 and 4

contiguous unit cells, respectively. Accordingly, resistance

of a single discrete conductor, RL, can be expressed as

the sum of resistances of a cylinder and 2 truncated spheres

; σ0 is conductivity of

bulk Ag and the factors 1/3 and 1/4 accounts for effective

areas for conduction. The expression can be further sim-

plified to give

; where .

In practice, pathways an electrical current take when

passing through a collection of unit cells depend on how

these cells are orientated with respect to the overall direction

of current. We adopt a commonly used analysis method that

assumes current direction normal through 2 parallel faces on

opposite sides of a unit cell.[16,18,19,24] In the truncated

octahedron unit cell, there are 2 distinct current directions.

One direction is between opposing hexagonal faces such as

from aeijfb to txwsmn; this direction is denoted as D6. The

other is between opposite square faces such as from abcd on

top of the unit cell to another square face uvwx at the bottom;

denoted as D4. These faces are shown in Fig. 3(a).

To determine the electrical resistance along D4 and D6, we

solve the respective circuit equivalents; Fig. 4 shows that for

D6. There are 6 possible current flow paths each starting at

spherical vertices. Consider the path from vertices e to t. At

vertex e, current flows through resistor ep to vertex p only as

vertices a, e and i are at the same potential; resistors between

vertices at the same potential are indicated in lighter tone in

Fig. 4. At vertex p, current flows through resistor po to

vertex o only. There is no current flow to vertex q because

vertices p and q are equipotential. At vertex o, by the same

reasoning, current flows to vertex t only. Hence the total

resistance for path from vertices e to t is 3RL. Similar

argument holds for the other 5 current flow paths. Since the

6 current paths are in parallel configuration, the equivalent

resistance of the unit cell for D6 is .

To extract apparent conductivity, σapp, using the model,

one has to consider a truncated octahedron unit cell

having equivalent resistance RUC and filled with ‘homo-

geneous’ porous Ag. By geometry, distance between any

pair of opposing hexagon faces such as aeijfb to txwsmn is

. The average cross-section area of the unit cell

along D6 is . Hence, for a unit cell

filled with porous Ag with apparent conductivity, σapp, one

can write the resistance encountered by current flowing

along D6 as .

The two expressions for RUC6 are equated

and from which conductivity along D6

ϕ 1 ρ/ρ0

–=( )

RL

1

1

3---πA

2

σ0

-----------------2

1

4---σ0

-------- xd

π R2

x2

–[ ]---------------------

0

R2

A2

–( )

∫+=

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

RL

3l

πA2σ0

--------------4

πR0

--------n R2

A2

– R+

R2

A2

– R–

---------------------------+3l

πA2σ0

--------------4C

πRσ0

------------+ = =

3Rl 4A2C+

πA2

Rσ0

-------------------------= C ln R2

A2

– R+

R2

A2

– R–

---------------------------=

RUC6

RL

2-----

3Rl 4A2C+

2πA2Rσ0

-------------------------= =

a6

= 6L

A6

VUC

a6

--------8 2L

3

6L---------------

8

3------L

2= = =

RUC6

18L

σapp8L2

------------------18

σapp8L----------------

18

σapp8 l 2 R2

A2

–+( )-----------------------------------------------= = =

1

2---

3Rl 4A2

C+

2πA2Rσ0

--------------------------⎝⎛

18

σapp8 l 2 R2

A2

–+( )---------------------------------------------

⎠⎟⎞

Fig. 4. Equivalent circuit for unit cell for D6 current direction: Each resistor represents a unit cell edge which is a ligament with electrical resis-tance RL. There is no current flow across resistors in lighter tone; these connect vertices at the same potential.



was obtained .

Hence, we are able to relate electrical conductivity to

morphological parameters; namely radius A and length l of

ligaments and radius R of vertices.

The preceding methodology was used to derive con-

ductivity along D4, σr, 4. By geometry, distance between

opposing square faces in a truncated octahedron is a4 =

. The average cross-section area along D4 is A4 =

4L2. The unit cell resistance as a function of morphological

parameters is then RUC4 = . Another expression for

RUC4 was obtained after solving the circuit equivalent resistor

network along .[18] Equating these expres-

sions for RUC4 and using appropriate substitutions for C, L

and RL, conductivity along D4 is given by =

.

Figure 5(a) compares electrical conductivity obtained

from experimental measurements and computed using

CLSV model. The well-known Hashin-Shtrikman bound, an

upper limit for isotropic composites, was computed using

volume fraction determined and is included for comparison.[25]

Significant dispersion in computed conductivity along D4

and D6 directions is attributed to variability of radius A and

length l of ligament as well as vertex radius. Nevertheless,

computed electrical conductivities have the same trend as

experimental values. The computed values are lower than

the Hashin-Shtrikman bound, as expected. In addition, our

results agree with those of Dharmasena and Wadley.[18]

Relative electrical conductivity along the D4 pathway from

their corrected truncated octahedron unit cell model, was

lower than those obtained experimentally which is similar to

results obtained in our study.

Electrical conductivity obtained experimentally fits a

power law dependence on annealing time with pre-exponent

and time exponent values of 0.37 and 0.11, respectively; R2

value is 0.86. The corresponding values for the Hashin-

Shtrikman bound are 0.22 and 0.28 with an R2 value of 0.99.

For the D4 and D6 pathways, the pre-exponent values are

0.07 and 0.08 while time exponent values are both 0.48,

respectively. However, the goodness to fit is poor; R2 values

for both are 0.41. The poor fit is due to significant variability

in radius A and length l of ligaments and radius R of vertices.

A few empirical relationships had been proposed to relate

electrical conductivity to porosity for porous metals. The

Koh-Fortini relationship provided good correlation between

these properties without consideration for initial com-

pactness.[26] Recently, Montes et. al. employ tap porosity to

relate these two properties.[27] Tap porosity of a powder

sample is porosity after vibration and is a measure of initial

compactness. We analyzed our experimental data and those

from CLSV model in light of these two empirical relationships.

Electrical conductivity data was fitted to the Koh-Fortini

relationship with a sensitivity factor as a fitting parameter.[28]

For the experimental data, the sensitivity factor was 5.9; the

R2 value is 0.74. For D4 and D6 pathways, the sensitivity

values are 3.8 and 3.2, respectively while R2 values are both

0.96, Fig. 5(b). These sensitivity factor values suggest

conductivity is sensitive to structural features through

σr 6,

σapp

σo

---------2 18πA

2

R

8 3Rl 4A2

C+[ ] l 2 R2

A2

–+[ ]----------------------------------------------------------------------= =

⎝ ⎠⎜ ⎟⎛ ⎞

2 2L

1

σapp 2L--------------------

D4

RUC4

3RL

4---------=⎝ ⎠

⎛ ⎞

σr 4,

σapp

σ0

----------=

2 2πA2R

8 3Rl 4A2C+[ ] l 2 R

2A2

–+[ ]--------------------------------------------------------------------

Fig. 5. Relative electrical conductivity of porous Ag: (a) Agreementto experimental data is better for D6 (flow between opposing hexago-nal faces) than D4 (flow between opposing square faces). Linesshown are power law fits. (b) Filled and dashed lines are fits of elec-trical conductivity data to Koh-Fortini empirical relationship using asensitivity factor as fit parameter. Dotted line is fit to Montes modelusing tap porosity of 0.48; which is experimental porosity after5 mins annealing.



porosity. Agreement to the Montes model was investigated

using tap porosity as a fitting parameter. While the

relationship proposed by Montes et al. gives a better fit, tap

porosity obtained from the fit was negative. Also, tap

porosities computed using conductivity from the CLSV

model were unrealistically low; 0.09 and 0.24 for the D4 and

D6 pathways, respectively. The curve of the Montes model

plotted in Fig. 5(b) is for a tap porosity of 0.48; which is

porosity of porous Ag after 5 min annealing. It can be

observed that the Koh-Fortini relationship provides better

correlation between electrical conductivity and porosity.

5. CONCLUSIONS

We investigated the electrical conductivity of porous Ag

having sub-micrometer structural features. Porous Ag were

formed by sintering nanoparticles at 150°C for up to 5 mins.

Electrical conductivity of porous Ag was about 20 % of bulk

values after 5 mins annealing. A model based on Kelvin

cells (truncated octahedrons) with cylindrical ligaments and

spherical vertices (CLSV) was used to compute electrical

conductivity normal to square (D4) and hexagonal (D6)

faces; better agreement was observed in the D6 direction.

Experimental and computed electrical conductivity in our

present study are correlated well to porosity through the

Koh-Fortini relationship.

ACKNOWLEDGMENT

A.S.Z and S. K. S. are grateful for the use of equipment

and other resources at the Singapore University of

Technology and Design as well as Universiti Kebangsaan

Malaysia, respectively, during the course of conducting this

work. We acknowledge support from Universiti Kebangsaan

Malaysia grant GGPM-2013-079 for this work.

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