Rs (Ω) RON (kΩ)
Bias amplitude (V)
RON/Rs ROFF/Rs
ROFF (kΩ)
a) b) c)
Rs (Ω)
0 200 400 0 200 400
1 2 3 4
0.1 0.2 0.3 0.4 0.5
0.2 0.3 0.4 0.5
2 4 6
2 4 6
RON = Rs
Figure 6.5: Average and standard deviation of the ON (a) and OFF (b) state resistances acquired at various temperatures at4.2 K<Tbath <300 Kand atV0drive= 0.6 V as a function of the series resistance RS. c) RON/RS (blue squares) and ROFF/RS (red dots) as a function of the bias voltage amplitude measured in a single junction at Tbath = 300 K and RS = 50 Ω. [O3]
On the other hand, the ON state resistance is proportional to RS as indicated by the blue squares in Figure 6.5.a. The latter behavior is attributed to the decreasing voltage drop on the junction as its resistance is decreasing toward RS upon a set process resulting in a rate limitation for a further resistance change as described in Section 6.2. This is also in agreement with the observed dependence of ROF F/RS and RON/RS on the bias voltage amplitude as displayed in Figure 6.5.c for a repre-sentative single junction measured at room temperature. RON saturates at the scale of RS as the driving amplitude is increased while ROF F remains unaltered.
Summary
The major conclusions of this Ph.D. work are summarized in the following thesis points.
1. I have developed a room temperature, 3D-tunable point contact measurement setup including the design of the sample holder and the implementation of measurement control programs. I have studied the resistance change behav-ior of metallic nanofilaments formed in Ag2S thin films situated between Ag and PtIr electrodes. I have shown that it is possible to induce reproducible resistance changes by bipolar voltage pulses of the width of 10 ns. Utilizing a unipolar, custom built avalanche pulse generator I have demonstrated that significant resistance changes also occur due to 500 ps long voltage pulses. [O1]
2. I have investigated the dynamics of the resistive switching process in Ag/Ag2S/PtIr nanojunctions. I demonstrated that at a fixed ROF F/RON ratio linearly increasing the driving amplitude results in an exponential ac-celeration of the switching process over six decades in the frequency domain.
I showed that the resistance change exhibits a strongly nonlinear time de-pendence upon a constant driving voltage acting on the memristive cell and a conventional serial resistor. High voltage drop on the cell induces a fast switch-ing while low voltage signals slowly modify the resistance state. This results in an elongated transition taking place over at least 11 orders of magnitude in the time domain. Consequently, it is not possible to attribute a well-defined characteristic time to the switching process. This elongated transition was accounted for by numerical simulations which revealed the direct connection between the bias dependence of theROF F/RON ratio and the temporal evolu-tion of the resistance change. All parameters of the simulaevolu-tions were deduced from experimental data. [O2]
3. I have studied the voltage polarity of set and reset processes in Ag/Ag2S/Ag cells. The generally accepted models attribute the polarity of the switching
83
to the specific sequence of inert and active electrode materials. In contrast, I found that the polarity can be solely determined by the local inhomogenity of the applied electric field at the active volume of the junction in agreement with molecular dynamical simulations. This inhomogenity arises from the geometrical asymmetry of the electrodes. Consequently, stable reproducible switchings with well-defined switching polarity were achieved in an STM setup using a silver tip and an Ag/Ag2S thin film sample. On the other hand, break-junction experiments utilizing silver wires and post-rupture sulfurization revealed a random initial switching polarity in agreement with the stochastic nature of the rupture process. [O4]
4. I have performed temperature dependent measurements to reveal the role of the ambient temperature and Joule heating in the fundamental properties of the hysteretic I(V) traces in Ag/Ag2S/PtIr cells. My measurements demon-strated that the switching threshold voltages increase with decreasing tem-perature while the metallic nature of the ON and OFF states is preserved down to cryogenic temperatures. This behavior was attributed to an excessive heat dissipation taking place in the active volume of the junction facilitating a structural transition of the Ag2S medium to its superionic argentite phase where rapid changes in the filament structure can occur. This implies that the dominating driving force of the non-isothermal resistance change is self-heating assisted electric field driven ionic transport. [O3]
Acknowledgments
I am grateful to my supervisor, Prof. Gy¨orgy Mih´aly who not only provided me the great opportunity to work in his team but guided through the course of my PhD and coordinated my work.
I would like to express my thanks to Prof. Andr´as Halbritter for his never ending support in building various measurement setups and for the useful scientific discussions during the interpretation of the results.
I highly acknowledge the support of Dr. Mikl´os Csontos who provided invaluable help in many aspects of my PhD work from the scientific background to the smallest technical details.
I really appreciated the help of Andr´as Magyarkuti in implementing measurement control programs and the assistance of Zolt´an Balogh in MCBJ sample preparation.
It was my pleasure to work with D´avid Zsolt Manrique and L´aszl´o P´osa. I would like to thank the members of the Solid State Physics Laboratory, the technical and administration staff for their cooperation. I am thankful to Dr. Ferenc Tanczik´o for silver thin film preparation and to ´Ad´am Sz¨ull˝o for his circuit designing contribution to the fast pulsing experiments.
85
List of publications
[O1] Attila Geresdi, Mikl´os Csontos, Agnes Gubicza, Andr´as Halbritter and Gy¨orgy Mih´aly. A fast operation of nanometer-scale metallic memristors:
highly transparent conductance channels in Ag2S devices. Nanoscale6, 2613 (2014).
[O2] Agnes Gubicza, Mikl´os Csontos, Andr´as Halbritter and Gy¨orgy Mih´aly.
Non-exponential resistive switching in Ag2S memristors: a key to nanometer-scale non-volatile memory devices. Nanoscale 7, 4394 (2015).
[O3] Agnes Gubicza, Mikl´os Csontos, Andr´as Halbritter and Gy¨orgy Mih´aly.
Resistive switching in metallic Ag2S memristors due to a local overheating induced phase transition. Nanoscale 7, 11248 (2015).
[O4] Agnes Gubicza, D´avid Zs. Manrique, L´aszl´o P´osa, Colin J. Lambert, Gy¨orgy Mih´aly, Mikl´os Csontos and Andr´as Halbritter. Asymmetry-induced resistive switching in Ag-Ag2S-Ag memristors enabling a simplified atomic-scale memory design. Scientific Reports 6, 30775 (2016).
87
Appendix
Numerical simulations of filament formation in Ag/Ag 2 S/Ag cells
The following simulations related to Chapter 5 were prepared by D´avid Zsolt Man-rique at the Physics Department of Lancaster University. [O4]
The simulations were carried out on a two-dimensional equilateral triangular lat-tice, where the lattice sites are either empty or occupied by a silver ion or atom.
This geometry mimics the distribution of silver ions in a polycrystalline Ag2S crys-tal [109] with the assumption that the silver atoms forming a conducting filament occupy crystal sites in the Ag2S matrix. The presence of sulphide ions is taken into account as a screening medium reducing the range of the silver ion interactions to nearest neighbor. The microscopic development is driven by room temperature ionic diffusion and redox processes, the latter taking place at the electrode surfaces. Sim-ilar filament growth simulations have been done before in Reference [110] utilizing different theoretical models. The calculations are done in a two-dimensional system for simplicity.
A typical equilateral triangular lattice used in the simulations can be recognized in Fig. A2 where the black dots denote silver atoms, the red dots represent silver ions. The lattice exhibits periodic boundary conditions along the horizontal direction while the vertical boundaries are terminated by two layers of silver atoms set toV and zero potentials on the top and bottom, respectively. The lattice constant is set to a=3.85 ˚A which approximately corresponds to one real atom per site. The electrostatic potential on each site is computed by solving the Poisson’s equation
∇(r∇u) = −ρ/0 on the lattice. The charges of the silver ions are considered to be screened, therefore they are excluded from the electrostatic calculation. r = 1 is set to zero outside, whereas inside the silverr = 1−i1.25×105 is applied. The surface charge density is computed from the potential as4u=−ρ/0.
89
The time development is performed either by moving some of the silver ions or atoms to their neighboring empty site or by simulating a redox reaction, in which silver ions and atoms located at an electrode surface are exchanged. First the elec-trostatic potential is computed in each time step. This is followed by the calculation of a transition probability for each possible change. Finally the changes are executed with the calculated probabilities. The transition probability of a silver ion at site k+ moving to its neighboring empty site k◦ is computed as [111]
wkdiff+,k◦ = min 1,∆t τ+
e−
∆Ediff k+,k◦
kB T
!
, (5)
where ∆Ekdiff
+,k◦ is the energy change of the move, 1/τ+ is the attempt frequency of the silver ion to jump and ∆tis the duration of the time steps. The wkdiff•,k◦ transition probability of a silver atom jumping from site k• to its neighboring empty sitek◦ is computed similarly using ∆Ekdiff•,k◦ and 1/τ•. The transition probability for a redox step, where a surface silver atom on site k• is oxidized and a surface silver ion on site k+is reduced, is computed as
wredoxk•,k+ = min 1, ∆t τredoxe−
∆Eox k•+∆Ered
k+ kB T
!
, (6)
where ∆Ekox• + ∆Ekred+ is the energy change due to the oxidation and reduction on site k• and site k+, respectively, and 1/τredox is the redox reaction rate. The ∆t duration of the time steps is chosen such that the typical transition probability is much smaller than 1. The energy changes are calculated as
∆Ekdiff+,k◦ = ˜µk◦(Ag+)−µ˜k+(Ag+) (7)
∆Ekdiff•,k◦ = ˜µk◦(Ag)−µ˜k•(Ag) (8)
∆Ekox• = ˜µk•(Ag+)−µ˜k•(Ag) + ˜µek0• (9)
∆Ekred+ = ˜µk+(Ag)−µ˜k+(Ag+)−µ˜ek0
+ (10)
where the electrochemical potential of a silver ion at site k is
˜
µk(Ag+) = γ++n+k +γ+•n•k+|e|uk, (11) where n+kand n•k are the numbers of silver ion and atom neighbors of sitek, respec-tively, γ++ and γ+• are the interaction energies, |e| is the charge of the silver ion
NUMERICAL SIMULATIONS OF FILAMENT FORMATION 91 and uk is the potential at site k. The electrochemical potential for the silver atom is
˜
µk(Ag) =γ+•n+k +γ••n•k, (12) where the γ•• is the interaction energy between silver atoms. The sites k0• and k0+ denote neighboring silver sites to k• and k+, respectively. They are chosen to provide the maximum probability for the given redox reaction. The electron’s electrochemical potential on a surface site is calculated as [112]
˜
µek =EF − |e|uk− ρk/|e|
g(EF), (13)
whereρkis the charge density at the surface sitek andg(EF) is the surface density of states in silver. The structural evolution of the contact is determined by the locations of the redox processes. The above formulae of the chemical potential imply that the typical locations of the redox reactions are determined by the average numbers of neighbors i.e., the ion concentration, and by the surface charge density. The numbers of neighbors are weighted with the interaction energies, therefore this contribution is coupled to the energetics of the diffusion process, whereas the surface charge density terms depend on the metal properties.
The ∆E energy change depends on the interaction of the participating atom or ion with its neighbors. In case of silver ions it also depends on the electrostatic potential. The ion-ion and ion-atom interactions are parameterized close to room temperature in order to keep the silver ions sufficiently mobile. The atom-atom interaction is set to be strongly attractive enabling the growth of stable metallic branches which resist to thermal diffusion.
Figure A1 shows a typical simulated evolution of the junction at negative and positive tip potentials (left and right panels in Figure A1, respectively). The struc-tural development can be followed from the top to bottom panels. The initial asym-metrical arrangement, representing an STM tip - flat surface setup, develops in time very differently at opposite bias polarities. Figures A1.a and A1.b demonstrate that at the initial phase of the filament formation the region of the most intensive structural changes are located at the apex of the tip, where the electric field is the highest. At a negative tip potential the silver atoms are being deposited on the apex and a filament starts growing towards the bottom surface. Figure A1.c illustrates a dendritic filament growth which is predominantly fueled by the oxidation of the silver atoms of the bottom electrode located under the filament. At a reversed po-larity the tip gradually loses its apex leading to an effective shrinking while opposite
Figure A1: Snapshots of initial silver filament formation within the Ag2S layer at an asymmetric initial tip versus flat surface arrangement of the Ag electrodes.
The semi-transparent color map indicates the electrostatic potential and the stream lines visualize the electric field direction and magnitude across the Ag2S layer. The left and right panels show the evolution of the junction for negative and positive tip potential (Vbias), respectively, at identical initial geometries. The time evolution of the filamentary structure can be followed from the top to bottom panels. [O4]
filament growth starting from the flat surface toward the tip is taking place (Fig-ure A1.d). Fig(Fig-ures A1.e and A1.f show the filament struct(Fig-ures after establishing a metallic contact between the electrodes. Figure A1.e demonstrates that the initial asymmetry of the contact is qualitatively preserved, while dendritic features in the filament and a shallow dip in the planar electrode also appear. These qualitative features are in good agreement with in-situ experimental studies on the dynamics of nanoscale metallic inclusions in dielectrics [61], where due to the high redox rates
NUMERICAL SIMULATIONS OF FILAMENT FORMATION 93
Figure A2: Snapshot of initial silver filament formation within the Ag2S layer at an asymmetric tip versus flat surface arrangement of the Ag electrodes. The semi-transparent color map indicates the electrostatic potential. The black (red) dots represent Ag atoms (Ag+ ions) The grey dots denote empty sites. [O4]
and ion mobility [1] in stoichiometric Ag2S, qualitatively similar filament growth was observed. A snapshot of the electrostatic potential corresponding to a dendritic filament formation is displayed along with the underlying lattice in high resolution in Figure A2. The time evolution of complete filaments were also studied as it is presented in Section 5.1.2 and plotted in Figure 5.2.
The simulated resistive states upon repeated biasing cycles are shown in Fig-ure A3. The two-terminal resistance is calculated from the bias voltage and the current through the contact which is I ∼ −R
∇u·dˆy where the integration is over the horizontal cross-section of the two-dimensional plane. Due to its reduced di-mensionality, this simple model is not expected to provide a quantitative agreement with the experimentally observed resistance values. Nevertheless, calculating the two-terminal resistance by taking, as a rude estimate, the bulk conductivity of sil-ver and the narrowest (one-dimensional) cross-section of the junction into account,
1 15 1 10 1 05 1 00 0 95
Figure A3: Simulated resistive switching upon a time-dependent bias voltage. a) Time dependence of the bias voltage. b) The corresponding resistance across the nanojunction. R0 is a normalization factor accounting for the two-dimensional as-pects of the simulation. [O4]
the basic qualitative features of the hysteretic I(V) traces can be reproduced.
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