nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
ISFET (IonSelective Field Effect Transistor): Electrolyte Gate AlGaN/GaN Field Effect Transistor as pH Sensor
Author:
Stefan Birner
If you want to obtain the input file that is used within this tutorial, please contact stefan.birner@nextnano.de.
> 1DGaN_electrolyte_sensor.in
This tutorial is based on the following paper
Theoretical
study of electrolyte gate AlGaN/GaN field effect transistors
M. Bayer, C. Uhl, P. Vogl
Journal of Applied Physics 97 (3), 033703, (2005)
as well as on the diploma theses of both Christian Uhl and Michael Bayer.
Note that this tutorial only briefly sketches the underlying physics. So
please check these references for more details.
Acknowledgement: The author  Stefan Birner  would like to thank
Christian Uhl and Michael Bayer for helping to include the electrolyte features into nextnano³.
Electrolyte Gate AlGaN/GaN Field Effect Transistor as pH Sensor
Here, we predict the sensitivity of electrolyte
gate AlGaN/GaN field effect transistors (FET) to pH values of the electrolyte
solution that covers the semiconductor structure. Particularly, we need to take
into account the piezo and pyroelectric polarization fields according to the
pseudomorphic growth of the nitride heterostructure on a sapphire substrate
(However, here we assume that the heterostructure is strained with respect to
GaN).
The charge density due to chemical reactions at the oxidic
semiconductorelectrolyte interface is described within the sitebinding model ($interfacestates ).
We calculate the spatial charge and potential distribution both in the
semiconductor and the electrolyte (PoissonBoltzmann equation)
selfconsistently.
The AlGaN/GaN FET that is exposed to an electrolyte solution has the
following schematic layout:
Fig. 1: AlGaN/GaN FET with electrolyte gate.
The polarization charges are included for the case of Gaface polarity.
The polarization fields lead to the formation of a 2DEG.
We simulate the structure along the z direction
and neglect source and drain so that the structure is effectively
onedimensional, i.e. laterally homogeneous.
The nitride heterostructure is assumed to be grown along the hexagonal [0001]
direction and is of Gaface polarity. The AlGaN layer is strained with respect
to GaN but the remaining layers are unstrained.
The piezo and pyroelectric polarization of wurtzite GaN and Al_{0.28}Ga_{0.72}N
result in huge polarization fields within the structure. The divergence of the
total polarization across the interface between adjacent layers causes a fixed
interfacial sheet charge density.
The following figure shows the influence of the interface charge densities:
Fig. 2: Schematic layout of the calculated AlGaN/GaN heterostructure
including the interface charge densities. The magnitudes are indicated by the
filled symbols.
The file
densities/interface_densitiesD.txt gives us information about the
relevant interface charge densities:
 Interface 1 (1 nm): sigma_{boundary} =  2.2 * 10^{13}
e / cm²
as defined in the input file:
$interfacestates
statenumber =
1
! between Metal / GaN at 1 nm
statetype =
fixedcharge !
sigma_boundary
interfacedensity = 2.2d13
! 2.2 x 10^13 [e/cm^2]
 Interface 2 (1500 nm): sigma_{polarization} (Interface 2) =
sigma_{piezo} + sigma_{pyro} = 1.38 * 10^{13} e / cm²
> sigma_{piezo} = 6.61 * 10^{12} e / cm²
> sigma_{pyro} = 7.14 * 10^{12} e / cm²
 Interface 3 (1535 nm): sigma_{polarization} (Interface 3)
=  sigma_{polarization } (Interface 2)
 Interface 4 (1538 nm): sigma_{boundary} as for Interface 1
(but with different sign) plus an additional charge sigma* = 1.0 * 10^{13}
e / cm² as defined in the input file:
$interfacestates
statenumber =
3
! between GaN / Oxide at 1538 nm
statetype =
fixedcharge ! sigma*
interfacedensity = 1.0d13
! 1 x 10^13 [e/cm^2]
 Interface 5 (1543 nm): sigma* as for Interface 4 (but with
different sign) plus an additional charge sigma_{adsorbed} that
results from the sitebinding model that describes chemical reactions at the
oxidic semiconductorelectrolyte interface. More details:
$interfacestates
$interfacestates
statenumber =
5
! between Oxide / Electrolyte at 1543 nm
statetype
= electrolyte !
sigma_adsorbed
interfacedensity = 9.0d14
! [cm^2]  total
density of surface sites, i.e. surface hydroxyl groups
adsorptionconstant = 1.0d8
! K_{1} = adsorption constant
dissociationconstant = 1.0d6
! K_{2} = dissociation constant
$electrolyte
...
pHvalue
= 5.3d0 !
pH = lg(concentration) = 5.3 > concentration in [M]=[mol/l]
(The point of zero charge for GaO is at pH = 6.8.)
For the origin of sigma_{boundary} and sigma* please refer to the
references that are given above.
The GaN region (1 nm  1500 nm) is homogeneously ntype doped with a
concentration of 1 * 10^{16} cm^{3}.
The electrolyte region (1543 nm  2999 nm) contains the following ions:
!!
! The electrolyte (NaCl, Hepes) contains four types of ions:
! 1) 100 mM singly charged cations (Na^{+})
! 2) 100 mM singly charged anions (Cl^{})
! 3) 10 mM doubly charged cations (Hepes^{2+} solution)
! 4) 20 mM singly charged anions (Hepes^{
} solution)
!!
$electrolyteioncontent
ionnumber =
1
!
100 mM singly charged cations
ionvalency =
1d0 !
charge of the ion: Na^{+}
ionconcentration = 0.100d0
! Input in units of: [M] = [mol/l] = 1d3 [mol/cm³]
ionregion =
1543d0 2999d0 ! refers to region where
the electrolyte has to be applied to
ionnumber =
2
! 100 mM singly charged anions
ionvalency =
1d0 !
charge of the ion: Cl^{}
ionconcentration = 0.100d0
! Input in units of: [M] = [mol/l] = 1d3 [mol/cm³]
ionregion =
1543d0 2999d0 ! refers to region where
the electrolyte has to be applied to
ionnumber =
3
! 10 mM doubly charged cations
ionvalency =
2d0 !
charge of the ion: Hepes^{2+}
ionconcentration = 0.010d0
! Input in units of: [M] = [mol/l] = 1d3 [mol/cm³]
ionregion =
1543d0 2999d0 ! refers to region where
the electrolyte has to be applied to
ionnumber =
4
! 20 mM singly charged anions
ionvalency =
1d0 !
charge of the ion: Hepes^{}
ionconcentration = 0.020d0
! Input in units of: [M] = [mol/l] = 1d3 [mol/cm³]
ionregion =
1543d0 2999d0 ! refers to region where
the electrolyte has to be applied to
In addition to these four types of ions, the pH value (as specified in $interfacestates )
automatically determines inside the code four further types of ions, namely the
concentration of H_{3}O^{+}, OH^{} and the
corresponding anions^{} (conjugate base: [anion^{} ]
= 10^{pH}  10^{pOH} = 10^{5.3}  10^{8.7} =
5.01 x 10^{6})
and cations^{+} (conjugate acid; zero in this tutorial because pH = 5.3
< 7). For
details, confer $electrolyteioncontent .
We have to solve the nonlinear Poisson equation over the whole device, i.e.
including the PoissonBoltzmann equation that governs the charge density in the
electrolyte region.
As for the boundary conditions we assume at the right boundary
(Electrolyte/Metal) a Dirichlet boundary condition where the electrostatic
potential phi is equal to U_{G} where U_{G} is the gate voltage
determined by an electrode in the electrolyte solution and which is constant
throughout the entire electrolyte region. In this example the applied gate
voltage is U_{G} = 0 V. Note that the reference potential U_{G}
enters the PoissonBoltzmann equation and also the equation for the sitebinding
model at the oxide/electrolyte interface. So the Dirichlet boundary condition is
phi = 0 V. This corresponds to the fact that at the right part of the
electrolyte, i.e. at 'infinity' (at 2999 nm) the ion concentration is the
'equilibrium' (default) concentration as defined in
$electrolyteioncontent .
At the left boundary (Metal/GaN) we use a generalized Neumann boundary
condition with a potential gradient that corresponds to the polarization charge
sigma_{boundary} =  2.2 * 10^{13} e / cm². Thus the electric
field E is given by
E = d phi / d z = sigma_{boundary} / (epsilon_{0}
* epsilon_{r}) * (1) =  3.977 * 10^{8} V/m.
epsilon_{r} = vacuumpermittivity (see $physicalconstants )
epsilon_{0}(GaN along [0001] axis) = 10.01
Note that the generalized Neumann boundary condition is automatically taken
into account as sigma_{boundary} is specified at the left contact. Thus
it is not necessary to specify the electric field of
3.977d8 V/m.
$poissonboundaryconditions
poissonclusternumber = 1
regionclusternumber = 1
boundaryconditiontype = Neumann
! Not necessary:
! electricfield =
3.977d8 ! 3.977 * 10^{8} [V/m] ,
corresponds to sigma_{boundary} =  2.2 * 10^{13}
[e/cm^{2}] at the left boundary
! Specify this instead:
electricfield
= 0d0 ! 0 [V/m]
poissonclusternumber = 2
regionclusternumber = 7
boundaryconditiontype = Dirichlet
potential
= 0d0 ! phi = 0 [V]
<=> U_{G} = 0 [V]
Oxide/electrolyte interface potential as a function of pH value
The GaN heterostructure acts as a sensor via the semiconductorelectrolyte
interface potential that reflects sigma_{adsorbed}, the pH value and the
spatial dependence of the electrostatic potential in the solution as described
by the PoissonBoltzmann theory.
Choosing flowscheme = 30 , several
calculations are performed while sweeping over the pH value from 0 to 12.
pHvalue
= (value is overwritten internally in the program)
The file InterfacePotential_vs_pH1D.dat gives us the
information about the electrostatic potential at the oxide/electrolyte interface
for different pH values.
We performed these calculations three times where we varied the adsorption
and dissociation constants.
adsorptionconstant = 1.0d8
! K_{1} = adsorption constant
(best fit to experiment)
dissociationconstant = 1.0d6
! K_{2} = dissociation constant
(best fit to experiment)
adsorptionconstant = 1.0d10
! K_{1} = adsorption constant
dissociationconstant = 1.0d10
! K_{2} = dissociation constant
adsorptionconstant = 1.0d10
! K_{1} = adsorption constant
dissociationconstant = 1.0d6
! K_{2} = dissociation constant
The total
density of surface sites, i.e. surface hydroxyl groups, was taken to be the
same in all three cases:
interfacedensity = 9.0d14
! [cm^2]
The surface potential is defined as the difference of the electrostatic
potential at the oxide/electrolyte interface and the reference potential U_{G}
(Here: U_{G} = 0 V).
Fig. 3: Calculated oxide/electrolyte interface
potential as a function of the pH value.
The solid line shows the result for K_{1} = 10^{8}, K_{2}
= 10^{6}.
Also included are the cases for K_{1} = 10^{10}, K_{2}
= 10^{10} (dashed line), K_{1} = 10^{10}, K_{2}
= 10^{6} (dotted line)
and the experimental data (G. Steinhoff et al., APL 83, 177 (2003).
Note that in Fig. 3 only the slope (d phi / d pH) is relevant but not the
absolute values of the potential. The experiment gives 56.0 +/ 0.5 mV/pH. Our
best fit parameters (solid line) yield 55.9 mV/pH and reproduce the constant
slope over the entire pH range.
Oxide/electrolyte interface charge density sigma_{adsorbed} as a
function of pH value
From the file InterfacePotentialDensity_vs_pH1D.dat we also obtain
information about the oxide/electrolyte interface sheet charge density sigma_{adsorbed}
(as a function of pH value) that is determined by the amphoteric reactions at
the oxide surface.
We plot in Fig. 4 for the following constants the oxide/electrolyte interface
charge density sigma_{adsorbed}:
adsorptionconstant = 1.0d8
! K_{1} = adsorption constant
(best fit to experiment)
dissociationconstant = 1.0d6
! K_{2} = dissociation constant (best
fit to experiment)
Fig. 4: Calculated variation of the
oxide/electrolyte interface charge density sigma_{adsorbed} of the
amphoteric oxide surface with the pH value of the electrolyte solution.
Note that there is a range of pH values where the net surface charge density is
close to zero.
The calculated point of zero charge for the GaO surface is reached for pH = 6.8.
Electrostatic potential for different electrolyte gate voltages U_{G}
Now we want to plot the electrostatic potential for different values of U_{G},
i.e. for applying a gate voltage to the electrolyte. Note that U_{G} is
both the Dirichlet boundary condition for the electrostatic potential at the
right contact as well as the reference potential that enters into the
PoissonBoltzmann equation (i.e. into the exponential term of the ion charge
density).
U_{G} is constant throughout the entire electrolyte region.
U_{G} = 0.5 V:
poissonclusternumber = 2
regionclusternumber = 7
boundaryconditiontype = Dirichlet
potential
= 0.5d0 ! phi = 0.5 [V]
<=> U_{G} = 0.5 [V]
U_{G} =  0.5 V:
poissonclusternumber = 2
regionclusternumber = 7
boundaryconditiontype = Dirichlet
potential
= 0.5d0 ! phi = 0.5 [V]
<=> U_{G} = 0.5 [V]
The pH value is set to 5.3.
Fig. 5: Spatial electrostatic potential distribution for pH = 5.3 in the
electrolyte.
Depicted are the cases U_{G} = 0.5 V (solid line) and U_{G} = 
0.5 V (dotted line).
The inset illustrates the effect of an applied voltage on the potential near the
position of the 2DEG.
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[at] nextnano.com .
