ST AN2386 APPLICATION NOTE
AN2386
Application note
How to achieve the threshold voltage thermal coefficient
of the MOSFET acting on design parameters
Introduction
Today, the MOSFET devices are used mainly as switches in electronic circuits. In such
operational conditions, the MOSFET device works in switch on and switch off modes.
However, in some applications, as in audio amplifiers or air conditioning, the MOSFET
works in a linear zone. The MOSFET works in a linear zone when either it is subject to a
high voltage, or a high current passes through the device. As it is well known in literature,
during the linear zone operation mode the MOSFET could fail if a thermal run-away occurs.
The failure conditions depend on either of the internal structure of MOSFET or of the
package used. The threshold voltage thermal coefficient (TVTC) is one of the big elements
that could bring the MOSFET to fail. TVTC is achieved deriving the MOSFET threshold
voltage against the temperature. TVTC is a negative coefficient because of when the
temperature increases the threshold voltage decreases. When TVTC increases in absolute
value, the MOSFET becomes thermally instable and a failure could occur. Therefore, in
order to understand if a MOSFET device can be used in an application working in linear
zone in safety conditions, a device with a low TVTC value must be considered and, thus, it is
important to achieve a theoretical expression for it.
June 2006 Rev 1 1/30
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Contents AN2386
Contents
1 MOS structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Some considerations on VTH and TVTC equations and real examples .
14
3 Case of DEVICE3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Revision history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2/30
AN2386 List of figures
List of figures
Figure 1. Cross section view of a MOS capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Figure 2. Energy band diagram of an ideal MOS capacitor under thermal equilibrium.. . . . . . . . . . . . 5
Figure 3.
Figure 4.
Figure 5.
Figure 6. DEVICE1 V
Figure 7. DEVICE1 TVTC - comparison between simulated and measured data . . . . . . . . . . . . . . . 16
Figure 8. Weight of single term of (Equation 42) on TVTC at different temperatures . . . . . . . . . . . . 17
Figure 9. DEVICE2 V
Figure 10. DEVICE2 TVTC - comparison between simulated and measured data . . . . . . . . . . . . . . . 19
Figure 11. DEVICE1 V
Figure 12. DEVICE1 TVTC simulated data - comparison between different N
Figure 13. DEVICE1 V
Figure 14. DEVICE1 TVTC simulated data - comparison between different N
Figure 15. DEVICE1 V
Figure 16. DEVICE1 TVTC simulated data - comparison between different tox . . . . . . . . . . . . . . . . . 21
Figure 17. DEVICE1 V
Figure 18. DEVICE1 TVTC simulated data - fixing T and acting on N
Figure 19. DEVICE1 V
Figure 20. DEVICE1 TVTC simulated data - fixing T and acting on N
Figure 21. DEVICE1 V
Figure 22. DEVICE1 TVTC simulated data - fixing T and acting on t
Figure 23. V
Figure 24. TVTC simulated data - comparison between DEVICE1 and the new device . . . . . . . . . . . 23
Figure 25. V
Figure 26. TVTC simulated data considering a p-gate doped MOS . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 27. Energy band diagram at low and high doping concentration . . . . . . . . . . . . . . . . . . . . . . . 25
Figure 28. DEVICE3 V
Figure 29. DEVICE3 TVTC - comparison between simulated and measured data . . . . . . . . . . . . . . . 26
Energy band diagram and charge distribution in an ideal MOS capacitor in accumulation condition.. 6
Energy band diagram and charge distribution in an ideal MOS capacitor in accumulation condition. . 6
Energy band diagram and charge distribution in an ideal MOS capacitor in inversion condition . . . . 7
- comparison between simulated and measured data . . . . . . . . . . . . . . . . 16
TH
- comparison between simulated and measured data . . . . . . . . . . . . . . . . 18
TH
simulated data - comparison between different Ng. . . . . . . . . . . . . . . . . . . 19
TH
simulated data - comparison between different Na. . . . . . . . . . . . . . . . . . . 20
TH
simulated data - comparison between different tox. . . . . . . . . . . . . . . . . . . 20
TH
simulated data - fixing T and acting on Ng . . . . . . . . . . . . . . . . . . . . . . . . . 21
TH
simulated data - fixing T and acting on Na . . . . . . . . . . . . . . . . . . . . . . . . . 22
TH
simulated data - fixing T and acting on tox . . . . . . . . . . . . . . . . . . . . . . . . . 22
TH
simulated data - comparison between DEVICE1 and the new device . . . . . . . . . . . . 23
TH
simulated data considering a p-gate doped MOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
TH
- comparison between simulated and measured data . . . . . . . . . . . . . . . . 26
TH
. . . . . . . . . . . . . . . . . . . . . . . . 21
g
. . . . . . . . . . . . . . . . . . . . . . . . 22
a
. . . . . . . . . . . . . . . . . . . . . . . . 23
ox
. . . . . . . . . . . . . . . . . 19
g
. . . . . . . . . . . . . . . . . 20
a
3/30
List of tables AN2386
List of tables
Table 1. Main electrical parameter simulated by DEVICE1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Table 2. Main electrical parameter simulated by DEVICE2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Table 3. Revision history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4/30
AN2386 MOS structure
1 MOS structure
As it is well known, a MOS structure is composed by three layers: the first one is metal or
heavily doped polycrystalline silicon, the second one is an insulator of SiO
one is the semiconductor (see Figure 1.).
Figure 1. Cross section view of a MOS capacitor
and the third
2
Considering an ideal MOS system with a p-doped semiconductor, the energy band diagram
can be illustrated as in Figure 2.
Figure 2. Energy band diagram of an ideal MOS capacitor under thermal
equilibrium.
q Φ
is the work function (energy that needs to extract an electron from the metal); q ΦB is
m
the energy difference between the oxide conduction band and the metal Fermi energy level
(metal-to-oxide barrier energy); q Φ
is the work function of the semiconductor; q χ is the
sc
energy difference between the vacuum level and the conduction band edge.
When a negative voltage (V
the Fermi level of the metal raises of qV
) is applied on the gate terminal respect to the semiconductor,
g
compared to the semiconductor side. In moving
g
from the semiconductor to the metal, the vacuum level must bend up gradually to
5/30
MOS structure AN2386
accommodate the gate voltage applied. Part of this bending occurs in the semiconductor
and the rest in the oxide. The metal and the semiconductor affinity remain the same (see
Figure 3.).
Figure 3. Energy band diagram and charge distribution in an ideal MOS capacitor
in accumulation condition.
The negative charge on the gate creates an opposite charge on the semiconductor
(enhanced concentration of holes near the oxide interface). When a small positive bias is
applied to the gate, holes are pushed away from the oxide interface and create a depletion
layer in the semiconductor, consisting on the negative charges due to the acceptor ions. The
energy band diagram and the charge distribution are shown in Figure 4.
Figure 4. Energy band diagram and charge distribution in an ideal MOS capacitor
in accumulation condition.
6/30
AN2386 MOS structure
The charge Qs in the semiconductor side near the oxide interface is equal to Qg and it can
be written as:
Equation 1
QsqNaW–=
N
is the acceptor concentration and W is the width of the surface depletion layer. Increasing
a
the gate voltage the bands continue to bend downward until E
energy between the conduction and the valence bands) equals E
(the intermediate level
i
(the Fermi energy level)
F
at the surface. In this condition, at the interface near the oxide, the semiconductor becomes
intrinsic. Increasing the gate voltage again, E
crosses EF and, thus, the minority carriers, in
i
this case the electrons, are attracted to the oxide-semiconductor interface. In this new
condition, the surface layer contains more electrons than holes and it becomes n-type
(inversion layer) (see Figure 5.).
Figure 5. Energy band diagram and charge distribution in an ideal MOS capacitor
in inversion condition
Now, Q
can be written as:
s
Equation 2
Q
is the inversion layer charge.
n
Q
Q–
s
qNaW–=
n
The relationships between the band bending, the electron and hole concentrations in the
interface oxide-semiconductor may be obtained assuming that the semiconductor is non
degenerate and that the doping is uniform. The electron and hole concentrations on the bulk
semiconductor can be written respectively as:
Equation 3
φ
F
------ -
–
•
q
kT
7/30
n
0ni
EFEi–
----------------- -
kT
e
==
nie
MOS structure AN2386
Equation 4
EFEi–
----------------- -
p0nie
is the intrinsic electron concentration; Ei is the intrinsic Fermi level; EF is the Fermi level; k
n
i
kT
==
is the Boltzmann constant and it is equal to 1.38x10
Considering that the semiconductor is doped of N
φ
F
------ -
•
q
kT
nie
-23
J/K; ΦF is the Fermi potential.
acceptor, (Equation 4) becomes:
a
Equation 5
•
q
nie
------ -
kT
φ
F
Now, it is possible achieve
Φ
F
as:
N
ani
EFEi–
------------------
–
kT
e
==
Equation 6
N
kT
------ -
Φ
F
q
------ -
ln=
a
n
i
For an intrinsic semiconductor ni can be written in two different ways as:
Equation 7
EcEF–
-------------------
–
n
=
iNc
kT
e
Equation 8
EcEF–
-------------------
–
niNVe
=
kT
Multiplying (Equation 7) and (Equation 8) it is possible obtain:
Equation 9
EcEv–
------------------
2
n
NcNVe
i
N
and NV are the effective density of states at the conduction and valence edges
C
respectively and E
is the energy band-gap. NC and NV can be achieved by the electron and
g
kT
NcNve
==
EgT()
---------------
kT
hole distribution functions and they are equal to:
Equation 10
3
-- -
2
2πm
Nc2
--------------------------- -
=
kT•
e
2
h
Equation 11
3
-- -
2
N
m
* and mh* are the electron and hole density of the states effective masses and h is the
e
v
2πm
--------------------------- -
2
=
kT•
h
2
h
Planck constant. The energy band gap depends on the temperature and it can be written as:
Equation 12
E
gEg0b1
T–=
8/30
AN2386 MOS structure
where b1 is equal to 4.76 x 10
E
is equal to 1.21eV and it is the extrapolate value of the band-gap at 0°K while b1 is the
g0
-23
J/K.
rate of energy gap shift with temperature. Now, (Equation 9) can be rewritten as:
Equation 13
Eg0b1T–
-------------------------
2
n
NcNVe
i
kT
NcNve
==
b
1
-----
k
E
g0
---------
kT
e
Considering that for the silicon:
Equation 14
b
1
-----
cNV
k
e
3.88 1016• T
N
3
-- -
2
cm3–[]=
and that:
Equation 15
E
g0
---------
14047 K[]=
k
n
becomes:
i
Equation 16
ni3.88 1016• T
3
-- -
–
2
e
E
g0
----------
2kT
3.88 10
• T
16–
3
–
-- -
2
e
7023
-------------
T
cm3–[]==
In the interface layer oxide-semiconductor, the electron and hole concentrations depend on
the gate voltage and can be written as:
Equation 17
n
sn0
EisEi–
------------------
–
kT
e
==
q
•
n0e
------ -
kT
Φ
s
Equation 18
sn0
e
When
n
Φ
is equal to ΦF the semiconductor surface becomes intrinsic. Instead, increasing Φs
s
again, the layer becomes n-type. When n
EisEi–
------------------
–
kT
==
equals to p0 a "strong inversion" occurs and the
s
q
•
n0e
------ -
kT
Φ
s
minority carriers at the surface become the same of the majority carriers in the bulk. The
potential condition for the strong inversion occurs when:
Equation 19
2Φ
=
Φ
s
F
As explained above, when V
is applied on the MOS structure, VOX (a part of VG) will
G
appear across the oxide and the rest on the silicon surface:
Equation 20
V
GVOXΦs
–=
9/30
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