# nextnano^{3} - Tutorial
## next generation 3D nano device simulator
## 1D Tutorial
## ISFET (Ion-Selective 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
submit a support ticket.
`-> 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 next***nano**³.
## 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
semiconductor-electrolyte interface is described within the site-binding model (`$interface-states` ).
We calculate the spatial charge and potential distribution both in the
semiconductor and the electrolyte (Poisson-Boltzmann equation)
self-consistently.
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 Ga-face 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
one-dimensional, i.e. laterally homogeneous.
The nitride heterostructure is assumed to be grown along the hexagonal [0001]
direction and is of Ga-face 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:
` $interface-states`
state-number =
1
! between Metal / GaN at 1 nm
state-type =
fixed-charge !
-sigma_boundary
interface-density = -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:
` $interface-states`
state-number =
3
! between GaN / Oxide at 1538 nm
state-type =
fixed-charge ! sigma*
interface-density = 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 site-binding model that describes chemical reactions at the
oxidic semiconductor-electrolyte interface. More details: ```
$interface-states
```
` $interface-states`
state-number =
5
! between Oxide / Electrolyte at 1543 nm
state-type
= electrolyte !
sigma_adsorbed
interface-density = 9.0d14
! [cm^-2] - total
density of surface sites, i.e. surface hydroxyl groups
adsorption-constant = 1.0d-8
! K_{1} = adsorption constant
dissociation-constant = 1.0d-6
! K_{2} = dissociation constant
$electrolyte
...
pH-value
= 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 n-type 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)
!---------------------------------------------------------------------------!
$electrolyte-ion-content
ion-number =
1
!
100 mM singly charged cations
ion-valency =
1d0 !
charge of the ion:` Na`^{+}
ion-concentration = 0.100d0
! Input in units of:` [M] = [mol/l] = 1d-3 [mol/cm³]`
ion-region =
1543d0 2999d0 ! refers to region where
the electrolyte has to be applied to
ion-number =
2
! 100 mM singly charged anions
ion-valency =
-1d0 !
charge of the ion:` Cl`^{-}
ion-concentration = 0.100d0
! Input in units of:` [M] = [mol/l] = 1d-3 [mol/cm³]`
ion-region =
1543d0 2999d0 ! refers to region where
the electrolyte has to be applied to
ion-number =
3
! 10 mM doubly charged cations
ion-valency =
2d0 !
charge of the ion:` Hepes`^{2+}
ion-concentration = 0.010d0
! Input in units of:` [M] = [mol/l] = 1d-3 [mol/cm³]`
ion-region =
1543d0 2999d0 ! refers to region where
the electrolyte has to be applied to
ion-number =
4
! 20 mM singly charged anions
ion-valency =
-1d0 !
charge of the ion:` Hepes`^{-}
ion-concentration = 0.020d0
! Input in units of:` [M] = [mol/l] = 1d-3 [mol/cm³]`
ion-region =
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` $interface-states` )
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` $electrolyte-ion-content` .
We have to solve the nonlinear Poisson equation over the whole device, i.e.
including the Poisson-Boltzmann equation that governs the charge density in the
electrolyte region.
As for the boundary conditions we assume at the right boundary
(Electrolyte/Metal) a Dirichlet boundary condition where the electrostatic
potential phi is equal to U_{G} where U_{G} is the gate voltage
determined by an electrode in the electrolyte solution and which is constant
throughout the entire electrolyte region. In this example the applied gate
voltage is U_{G} = 0 V. Note that the reference potential U_{G}
enters the Poisson-Boltzmann equation and also the equation for the site-binding
model at the oxide/electrolyte interface. So the Dirichlet boundary condition is
phi = 0 V. This corresponds to the fact that at the right part of the
electrolyte, i.e. at 'infinity' (at 2999 nm) the ion concentration is the
'equilibrium' (default) concentration as defined in```
``` `$electrolyte-ion-content` .
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} = `vacuum-permittivity ` (see` $physical-constants` )
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.
```
$poisson-boundary-conditions
```
poisson-cluster-number = 1
region-cluster-number = 1
boundary-condition-type = Neumann
** ! Not necessary:**
! electric-field =
-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:**
electric-field
= 0d0 ! 0 [V/m]
poisson-cluster-number = 2
region-cluster-number = 7
boundary-condition-type = 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 semiconductor-electrolyte
interface potential that reflects sigma_{adsorbed}, the pH value and the
spatial dependence of the electrostatic potential in the solution as described
by the Poisson-Boltzmann theory.
Choosing` flow-scheme = 30` , several
calculations are performed while sweeping over the pH value from 0 to 12.
```
pH-value
=
``` *(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.
```
adsorption-constant = 1.0d-8
! K
```_{1} = adsorption constant
(best fit to experiment)
dissociation-constant = 1.0d-6
! K_{2} = dissociation constant
(best fit to experiment)
```
adsorption-constant = 1.0d-10
! K
```_{1} = adsorption constant
dissociation-constant = 1.0d-10
! K_{2} = dissociation constant
```
adsorption-constant = 1.0d-10
! K
```_{1} = adsorption constant
dissociation-constant = 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:
interface-density = 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}:
```
adsorption-constant = 1.0d-8
! K
```_{1} = adsorption constant
(best fit to experiment)
dissociation-constant = 1.0d-6
! 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
Poisson-Boltzmann 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:
` poisson-cluster-number = 2`
region-cluster-number = 7
boundary-condition-type = Dirichlet
potential
= 0.5d0 ! phi = 0.5 [V]
<=> U_{G} = 0.5 [V]
U_{G} = - 0.5 V:
` poisson-cluster-number = 2`
region-cluster-number = 7
boundary-condition-type = 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. |