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June 16th, 2019

To confirm the ATLAS results and also to simulate the mobility behaviour of our samples, we have developed an analytical model to explain the sheet resistance and mobility characteristics of our bilayers. As far as we know, this model has not been proposed in the literature and for this reason we will explain it from a general point of view below.

General Model for the Sheet Resistance of a Bilayer

We will assume a bilayer composed by layer 1 and layer 2 as in Fig. 13.11. Both layers are connected through resistors Rt/2, which model the current limitation for the current flux from one layer to the other.

Currents injected on layer 1 and 2,I1 and I2 could be obtained as:

where V is the voltage drop at the current source terminals. Obviously, I — I1 +12. RCi and RC2 are the resistances between two consecutive electrodes (a and b on Fig. 13.12) for layer 1 and (a’ and b’) for layer 2.

As we are interested in sheet resistance RS and not in the resistance between two contacts RC, it will be mandatory to obtain the relation between both magnitudes. We will call a — RC/RS. This figure is solely dependent on the relative size of the electrical contact respective to the sample size. We will assume triangular contacts.

The resistance RC has to be calculated numerically as there is not analytical model for it. We can use a code as ATLAS to determine the relation between this resistance and the parameter a quoted above but it is easy and more instructive to use the popular PSPICE code [24]. We will simulate the layer as an array of resistors with 19 rows and 19 columns. Resistor at the limits should have twice value because

Fig. 13.11 Exploded view of the bilayer showing the current limiting resistances at the four corners, currents on each layer introduced at corners a and b and the voltages developed at the opposite corners c and d through a RC resistance |

Fig. 13.12 Exploded view of the bilayer showing the diodes at the four corners that simulate the rectifying behaviour of the TIL-substrate interface |

there is not current path outside these limits [25]. For easy calculation, we will use 1 ohm resistor and we will inject a 1 ampere current between two consecutive contacts and we will register the potential on the current source (V) and the potential difference at the opposite corners (AV) for different contact size. In the Table 13.1, we give the results of this simulation.

Column 1 is the contact size, i. e. the number of nodes that have been short circuited at the corners to simulate the equipotential triangular contact. The relative

Cont. size |
AV |
Rs |
a factor |
Error(%) |

1 X 1 |
0.222 |
1.006 |
5.174 |
0.610 |

2 X 2 |
0.222 |
1.004 |
2.390 |
0.429 |

3 x 3 |
0.220 |
0.999 |
1.901 |
-0.115 |

4 x 4 |
0.217 |
0.983 |
1.542 |
-1.746 |

5 x 5 |
0.210 |
0.951 |
1.255 |
-4.919 |

Table 13.1 Potential at measurement comers (AV), sheet resistance (Rs), a factor (Rc/Rs) and sheet resistance error for different contact sizes |

Table 13.2 a factor for Hall configuration

Cont. size |
a factor |

1 x 1 |
5.394 |

2 x 2 |
2.610 |

3 x 3 |
2.140 |

4 x 4 |
1.711 |

size could be obtained having in mind that the mesh size is 19 x 19. The second one is the differential voltage at potential measurement corners, the third one is the sheet resistance obtained with the classical van der Pauw formula RS = n/ln(2) AV/I. It is important to realize that the theoretical sheet resistance is just the resistance of each one of the differential elements we used to represent the complete sample, that is to say 1 ohm. The a factor is simply the voltage developed at the current source because the current is 1 Ampere and the sheet resistance is 1 ohm. For our sample which has approximately 1.5 mm side triangular contacts the a factor is very close to 2. The error in the sheet resistance measurement is as low as 0.1%. Table 13.2 gives the same factor a when the electrodes are not consecutive but opposite as is the case for Hall measurements.

Rewriting expressions 13.1, h = V/aRS and I2 = V/(Rt C aRS2/, where RS1 and RS2 are the sheet resistances of the layer 1 and 2, respectively. Due to the presence of the Rt, the developed voltage on the opposite corners (c and d for layer 1 and c’ and d’ for layer 2 on Fig. 13.11) are not the same for both layers (it is the same only if the injected current on each layer is in a inverse proportion to the sheet resistance). To obtain the measured voltage we have to bear in mind the internal impedance across which the voltage is developed [16]. Of course, the resistance between electrodes c and d is the same than the one observed between electrodes a and b, that is to say RC1 . The same happen for electrodes c’ and d’, which have a resistance RC2.

Each one of the voltages AV could be expressed according van der Pauw as:

AVi = Ii • Rs1 • ln(2)/n AV2 = 12 • Rs2 • ln(2)/n (13.2)

and the equivalent voltage measured on the electrodes of the top layer (electrodes d and c on Fig. 13.11) with a infinite impedance voltmeter will be:

where GC1 = 1/RC1 and GC2 = 1/RC2 are the conductances of layer 1 and layer 2, respectively, and Gt is 1/Rt.

Equation 13.3 could be written in a more compact way if we introduce a function F = Gt/(Gt C Gc2):

AV1GC1 C AV2GC2F

AV =——- 1 C———— 2 C2 . (13.4)

Gc1 C Gc2F

The equivalent sheet resistance, Rsheet, i. e., the one measured on top of the layer 1 is:

Introducing the function F in the sheet resistance formula, we can write:

In (13.5) and (13.6), Rsheet! RS1 when Gt or F! 0 and to RS1 // RS2 when Gt! і and F! 1

General Model for the Hall Mobility of a Bilayer

Same procedure can be used to find the Hall mobility obtaining an effective mobility as:

_ M1GC1 C MGt/(Gt C Gc2))2 M1GC1 C M2GC2F2 (13 7)

Gc1 C Gc2(G,/(G, C Gc2))2 GC1 C GC2F2

where Д4 and д2 are layer 1 and layer 2 mobilities. These mobilities have to be considered with its corresponding signs according if they are related to p type carriers or n type ones. Now the a coefficient has to be changed from 2 to 2.2, which is the correct value for RC/RS when the current is injected at the opposed electrodes in a square sample with relative electrode size as the one we have, as it is obtained from Table 13.2.

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