1. Graphene Hysteresis Response
Edward Cazalas
12/21/12

2. Graphene Hysteresis Mechanism - Intro
Xu, H., et al. determine through experiment that hysteresis requires presence of water molecules AND oxygen.
Graphene is hydrophobic and free-floating graphene does not display hysteresis.
Hysteresis “dopants” cling onto SiO2 substrate.
Electron transfer to/from graphene/dopant causes change in carrier density of graphene.
Electrochemical doping
Graphene
SiO2
SiC or Si
Backgate
Xu, H., et al., “Investigating the Mechanism of Hysteresis Effect in Graphene Electrical Field Device Fabricated on SiO2 Substrates using Raman Spectroscopy”, Small, 8, No. 18, , 2012.

3. Graphene Hysteresis Mechanism – Redox Reaction
Absorption and desorption of electrons from/to graphene occurs through a redox reaction.
Reaction dynamics can be described by Marcus-Gerischer (MC) theory, which describes charge-transfer electrochemistry between metals and redox systems[1,2].
MC theory expresses probability of charge transfer though concepts of density of states (DOS) of energy level potentials[3].
Reduction (gain e-)
Redox Pair
(relative to graphene)
O2 + 2H2O +4e- (graphene) = 4OH- [4]
Oxidation (lose e-)
[1] J. O. M. Bockris, S. U. M. Khan, Surface Electrochemistry: A Molecular Level Approach.New York, 1993, p 496–500. [2] A. J. Bard, L. R. Faulkner, Electrochemical Methods: Fundamentals and Applications, 2nd ed. John Wiley & Sons: New York, 2001,
p833–836.
[3] P. L. Levesque, S. S. Sabri, C. M. Aguirre, J. Guillemette, M. Siaj, P. Desjardins, T. Szkopek, R. Martel, Nano Lett. 2011, 132.
[4] Xu, H., et al., “Investigating the Mechanism of Hysteresis Effect in Graphene Electrical Field Device Fabricated on SiO2 Substrates using Raman Spectroscopy”, Small, 8, No. 18, , 2012.

4. Graphene Hysteresis Mechanism – Fermi Levels
DOS is maintained for a reduction and oxidation potential (Ere and Eox, respectively)
A Fermi level (Eredox) is obtained where DOS is in equilibrium for reduction and oxidation species.
The graphene/dopant system will want to reach equilibrium.
Since the Fermi level of graphene (Ef = -4.6) is higher than the redox Fermi level (Eredox = -5.3), electrons will naturally want to go to the lower potential and engage in the redox reaction with the dopants.
O2 + 2H2O +4e- (graphene) = 4OH-
The absorption of electrons by the dopants causes the graphene to be naturally h+ doped with no applied backgate bias.

5. Graphene Hysteresis Mechanism – Dirac Peak
The Dirac peak is the result of minimizing the number of carriers on graphene
Graphene
SiO
SiC or Si
Graphene
SiO
SiC or Si
Bg = -20 V
Bg = +20 V
To the left of the Dirac peak, resistance is lower for the equivalent voltage (to positive bias) due to the greater electric field caused by combined dopant and backgate electrons
Due to capacitive shielding of the backgate by dopant electrons (ovals), resistance of graphene is expected to be higher to the right of the Dirac peak.
As expected, electron carriers exists to the right of the Dirac peak, and hole carriers to the left of the Dirac peak.
Graphene
SiO
SiC or Si
Bg = 0 V
Graphene
SiO
SiC or Si
Backgate = +6V
Dirac taken with bg = 0 V for long time prior.

6. Graphene Hysteresis Mechanism – Breaking Equilibrium
After some time, Ef = Eredox, and equilibrium is reached (stabilized).
Changing the backgate bias immediately breaks the equilibrium.
Applying positive backgate voltage causes the graphene to initially gain a greater population of e- carriers with backgate charges.
Applying negative backgate voltage causes the graphene to initially gain a greater population of +h carriers with backgate charges.
Graphene
SiO
SiC or Si
Bg = -V
Graphene
Graphene
SiO
SiO
SiC or Si
SiC or Si
Bg = +V
Backgate = 0V
Graphene resistance immediately and always goes down.
Graphene resistance may go up or down depending on magnitude of +V.
With no applied bias, graphene is +h doped.

7. Graphene Hysteresis Mechanism – Equalizing
Since Ef has shifted relative to Eredox, electrons will undergo redox to equalize the DOS and the Fermi potentials. If the density of both dopant species (H2O and O2) does not change, the DOS and, thus, Eredox remains unchanged.
Dirac peak location shifts in direction of applied backgate voltage.
Since DOS is probabilistic, it is unlikely to fully oxidize or reduce the dopants.
At Bg = +V, equilibrium is achieved through oxidation, transferring electrons from graphene to dopants.
At Bg = -V, equilibrium is achieved through reduction, transferring electrons from dopants to graphene.
Dirac peak shifts toward -V
Dirac peak shifts toward +V
Graphene
Graphene
SiO
Graphene
SiO
SiO
SiC or Si
SiC or Si
SiC or Si
Bg = -V
Bg = +V
As e- are removed from dopants, graphene resistance should gradually increase.
Backgate = 0V
As e- are added to the dopants, graphene resistance should gradually increase.
With no applied bias, graphene is +h doped.

8. Graphene Hysteresis Mechanism – Equalizing
As times goes on and equilibrium continues…
At Bg = -V, dopant will continuously give up e- until it becomes positively charged. Graphene carrier type never switches.
At Bg = +V, graphene will experience minimum resistance as number of e- transferred to dopants. Graphene charge carrier switches from e- to h+.
When equilibrium is obtained, a higher resistance for a +V backgate will be observed than for the same magnitude –V backgate.
Graphene
Graphene
SiO
Graphene
SiO
SiO
SiC or Si
SiC or Si
SiC or Si
Bg = -V
Bg = +V
As e- are removed from dopants, graphene resistance should gradually increase.
Backgate = 0V
After carrier switch, graphene resistance should go down.
With no applied bias, graphene is +h doped.

9. Graphene Hysteresis Mechanism – Explaining Exp
The theory can be used to explain the location of the Dirac peaks after stabilization at positive and negative backgate voltages.
When equalization occurs, the capacitive shielding between the backgate and dopants causes the graphene to essentially see zero backgate bias, and the graphene becomes h+ doped again. This causes the Dirac peak to be located to the right of the positive stabilized voltage.
Again, the resistance for +BG, once stabilized, is much higher than for –BG of same magnitude, as is shown here.
Capacitive shielding is established once enough electrons have left dopant to leave it positively charged.
Graphene
SiO
SiC or Si
Bg = - 20 V
Graphene
SiO
SiC or Si
Backgate = 0V
Graphene
SiO
SiC or Si
Bg = + 20 V

10. Graphene Hysteresis Mechanism – Explaining Exp
The theory can also explain time dependent graphene response.
Graphene
SiO
SiC or Si
Bg = -20 V
Graphene
SiO
SiC or Si
Bg = - 20 V

11. Graphene Hysteresis Mechanism – Explaining Exp
The theory can also explain time dependent graphene response.
Graphene
SiO
SiC or Si
Bg = 0 V
Graphene
SiO
SiC or Si
Backgate = 0V

12. Graphene Hysteresis Mechanism – Explaining Exp
The theory can also explain time dependent graphene response.
Graphene
SiO
SiC or Si
Backgate = +20V
Graphene
SiO
SiC or Si
Bg = + 20 V

13. Graphene Hysteresis Mechanism – Response to Radiation
Radiation causes ionization and conductivity to increase in substrate.
Charges are then allowed to drift to and away from graphene and magnify field effect response.
In the absence of hysteretic dopants, field effect principle is straightforward.
Here we assume backgate/substrate forms an ohmic contact.
Graphene
SiO2
SiC or Si
Bg = - 20 V
Graphene
SiO2
SiC or Si
Backgate = 0V
Graphene
SiO2
SiC or Si
Bg = + 20 V - - - - - - - + + + + + + + + +
e-/h+ pairs formed by radiation are separated by an applied bias to the backgate.
In the case of no applied bias, charges should eventually recombine.
Graphene
SiO2
SiC or Si
Bg = - 20 V
Graphene
SiO2
SiC or Si
Backgate = 0V
Graphene
SiO2
SiC or Si
Bg = + 20 V - - - + + +
When backgate bias is applied, radiation should further increase the charge carriers already present in graphene.