thermal gradient can be bigger than electron wind forces. If they are in opposite
direction, it can negate the electron wind forces and if they are in the same direction
can hasten the failure process significantly.
The temperature gradient-induced driving force is usually represented by
(Huntington 1972).
F tm ¼
Q
Ã
T
∇T
ð8:9Þ
where Q
* is the heat of transport, which is the isothermal heat transmitted by the
moving atom in the act of jumping a lattice site less the intrinsic enthalpy and T is
temperature in Kelvin.
The most accepted thermomigration theory is very much based on the same
concepts as electromigration. In electromigration, the force acting on the diffusing
atom is the momentum exchange due to the collisions of diffusing atoms and
scattered electrons in the conduction band. However, in thermomigration, in the
absence of electrical current, the driving force in the direction of the temperature
gradient results from the fact that the energy, therefore the momentum of the atoms
(phonons) at higher temperatures, is greater than that of the lower ones. This gradient
in the momentum produces a driving force for mass transport.
Based on this argument, Q
* for a metal with high Z
* value should also be high.
Similar to Z
* , the sign of Q
* can either be positive or negative depending on the
effective mass of the charge carriers.
According to Huntington (1972), thermomigration driving force can be divided
into an intrinsic part and a part coupled to carriers, i.e. electrons, and phonons:
Q
Ã
¼ Q
Ã
in þ Q
Ã
el þ Q
Ã
ph
ð8:10Þ
The intrinsic considerations include the energy transported by the moving atom,
the energy required to prepare a place [a lattice site] to receive it, and, for vacancy
diffusion mechanism, the formation energy for the counter moving vacancy, which
is given by.
Q
Ã
¼ βE m À E f
ð8:11Þ
Energy, E
E f
l
Momentum, k
m* ∝
d 2 E
dk 2
Fig. 8.4 Energymomentum relationship that
determines the value of
effective mass (after Lloyd
et al. 2004)
8.2 Physics of Electromigration Process
403
direction, it can negate the electron wind forces and if they are in the same direction
can hasten the failure process significantly.
The temperature gradient-induced driving force is usually represented by
(Huntington 1972).
F tm ¼
Q
Ã
T
∇T
ð8:9Þ
where Q
* is the heat of transport, which is the isothermal heat transmitted by the
moving atom in the act of jumping a lattice site less the intrinsic enthalpy and T is
temperature in Kelvin.
The most accepted thermomigration theory is very much based on the same
concepts as electromigration. In electromigration, the force acting on the diffusing
atom is the momentum exchange due to the collisions of diffusing atoms and
scattered electrons in the conduction band. However, in thermomigration, in the
absence of electrical current, the driving force in the direction of the temperature
gradient results from the fact that the energy, therefore the momentum of the atoms
(phonons) at higher temperatures, is greater than that of the lower ones. This gradient
in the momentum produces a driving force for mass transport.
Based on this argument, Q
* for a metal with high Z
* value should also be high.
Similar to Z
* , the sign of Q
* can either be positive or negative depending on the
effective mass of the charge carriers.
According to Huntington (1972), thermomigration driving force can be divided
into an intrinsic part and a part coupled to carriers, i.e. electrons, and phonons:
Q
Ã
¼ Q
Ã
in þ Q
Ã
el þ Q
Ã
ph
ð8:10Þ
The intrinsic considerations include the energy transported by the moving atom,
the energy required to prepare a place [a lattice site] to receive it, and, for vacancy
diffusion mechanism, the formation energy for the counter moving vacancy, which
is given by.
Q
Ã
¼ βE m À E f
ð8:11Þ
Energy, E
E f
l
Momentum, k
m* ∝
d 2 E
dk 2
Fig. 8.4 Energymomentum relationship that
determines the value of
effective mass (after Lloyd
et al. 2004)
8.2 Physics of Electromigration Process
403