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Space-Charge-Limited Conduction Mechanism II Space-charge-limited conduction mechanism II Yifan Yuan 2019-10-10 D. R. Lamb, Electric Conduction Mechanisms in Thin Insulating Films (Methuen, London, 1967) 1. Space-charge-limited flow • One-carrier space-charge-limited flow without traps. (electrons) • One-carrier space-charge-limited flow with traps. • Two-carrier space-charge-limited flow without traps or recombination centers. (cathode electrons, anode holes) • Two-carrier space-charge-limited flow with recombination centers 2. One-carrier space-charge-limited Definition: if an electron injecting contact is applied to an insulator, electrons will travel from the metal into the conduction band of the insulator and form a space-charge similar to that of a vacuum diode. Accumulation of charges in a particular region is referred to as space charge. In vacuum (Child's law): 3. Theory • At low voltages where the injected carrier density is less than 푛0, which is the thermally generated free carrier density, Ohm’s law will be obeyed: s: film thickness 푉 퐽 = 푒푛 휇 (1) 휇: mobility 0 푠 V: voltage k: dielectric constant • At transition voltage, 푉푡푟, the transition from Ohm’s law to Mott and Gurney law takes place: n: free electron density 9 푉2 D: diffusion coefficient 퐽 = 푘휇 (2) 8 푠3 The theory is based on purely field driven currents and diffusion current: 푑푛 퐽 = 푛푒휇퐸 − 퐷푒( ) (3) 푑푥 3. Shallow and deep trapping Shallow Trap ∆휀 Deep Trap 휖퐹−휖푐 Free electron density: 푛 푥 = 푁푐exp[ ] 푘퐵푇 Increase voltage 푁푐: effective density of states in the conduction band 휖퐹: Fermi level 휖푡: trap level 휖푐: bottom of conduction band 3. Shallow and deep trapping Trapped charge density (occupancy of trap): 푁 푛 = 푡 푡 1 Fermi–Dirac distribution 1 + 푔 exp 휖푡 − 휖퐹 /푘퐵푇 푁푡 For shallow trap: 푛푡 ≈ 1 휖 Shallow Trap exp 휖 −휖 /푘 푇 푡 ∆휀 Deep Trap 푔 푡 퐹 퐵 휖퐹 For deep trap: 푛푡 = 푁푡 Increase voltage 푁푡: trap density 휖퐹: Fermi level 휖푡: trap level 휖푐: bottom of conduction band 3. Shallow and deep trapping Thus the ratio of free to trapped charge is 휖 − 휖 푁 퐹 푐 푛 푥 푐exp[ 푘 푇 ] 휃 = = 퐵 푛푡 푁푡 1 1 + 푔 exp 휖푡 − 휖퐹 /푘퐵푇 Shallow Trap 휖푡 ∆휀 Deep Trap −7 휃 can be as low as 10 , very large effect. 휖퐹 휖푡−휖푐 푁푐exp[ ] For shallow trap:휃 ≈ 푘퐵푇 푔푁푡 휖퐹−휖푐 For deep trap: 휃 = 푁푐exp[ ]/푁푡 푘퐵푇 Increase voltage Therefore, 2 9 푉2 9 푉 퐽 = 푘휇 퐽 = 푘휇휃 8 푠3 8 푠3 3. Shallow trapping The charge which has been injected into the insulator can be distributed in three parts: Shallow Trap (1) Free charge in the conduction band 휖푡 ∆휀 Deep Trap (2) Trapped charge above the Fermi level 휖퐹 (3) Trapped charge in the states between the initial Fermi level and the final Fermi level. Assumption that all injected charge will in fact be trapped in (3). 푄 ∆휀 = ≈ 푉퐶/(푒푁푡푠) 푒푁푡푠 푄 푉퐶 푛 = = Increase voltage 푡 푒푠 푒푠 The free carrier density is given by Q: injected charge 휖 − 휖 ∆휀 퐹 푐 휖퐹: Fermi level 푛 = 푁푐exp[ ]exp[ ] 푘퐵푇 푘퐵푇 휖푡: trap level ∆휀 푉퐶 푡푉 휖푐: bottom of conduction band = 푛0exp[ ] =푛0exp[ ]=푛0푒 푘퐵푇 푒푁푡푠푘퐵푇 푛 푛 푒푡푉 푛 푒푠 9 푉2 9 푉 푛 푒 Hence 휃 = = 0 = 0 푒푡푉 Therefore, 퐽 = 푘휇휃 = 푘휇 0 푒푡푉 푛푡 푉퐶/푒푠 푉퐶 8 푠3 8 푠3 퐶 4. Experiments Fig. S2 Typical J-E characteristics of a Au/BFO/Au structure (BFO1) at 300 K, (b) SCLC. Choi, T., et al. "Switchable ferroelectric diode and photovoltaic effect in BiFeO3." Science 324.5923 (2009): 63-66. (Supporting information) 4. Experiments Amorphous selenium (20 u)/ tin oxide / glass substrate For film 2, the dependence of current on voltage was between V and V2 at lower voltages. 퐼 = 2.2 × 10−11푉푒푉/31.1 퐼 = 1.3 × 10−11푉푒푉/57.0 For voltages less than 10 v the current was probably a mixture of ohmic and SCLC. This suggests that the thermal equilibrium Fermi level was less than kT above a uniform distribution of hole capture levels Hartke, Jerome Luther. "Drift mobilities of electrons and holes and space-charge-limited currents in amorphous selenium films." Physical Review 125.4 (1962): 1177. Derivation of Mott and Gurney law The theory is based on purely field driven currents and diffusion current: 푑푛 휇: mobility 퐽 = 푛휇퐸 − 퐷( ) (3) 푑푥 n: charge carrier density Using Gauss’s law and D: diffusion coefficient 푛 푑퐸 k: dielectric constant = 푘 푑푥 푑퐸 푑2퐸 퐽 = 푘휇퐸 − 푘퐷 푑푥 푑푥2 V Using Einsteins’ relation 휇 퐷 = 푘 푇 푒 퐵 푑퐸 푑2퐸 퐽 = 푘휇퐸 − 푘휇푘 푇 J 푑푥 퐵 푑푥2 푑퐸 퐸 Taking ≈ , if 푘 푇 ≪ 푒퐸푠, one can neglect the diffusion term s 푑푥 푠 퐵 푑퐸 퐽 = 푘휇퐸 푑푥 And integrating 2퐽 퐸 = (푥 + 푥 ) 푘휇 0 where x0 is a constant Derivation of Mott and Gurney law 푠 푠 2퐽 휇: mobility 푉 = න 퐸푑푥 = න (푥 + 푥0) 푑푥 0 0 푘휇 n: charge carrier density D: diffusion coefficient 2 2퐽 3/2 k: dielectric constant = 푠 + 푥 3/2 − 푥 3 푘휇 0 0 Thus, for x0 << s, neglecting x0 ퟗ 푽ퟐ 푱 = 풌흁 ퟖ 풔ퟑ The theory in fact cannot give an accurate description of the physical situation near the injecting cathode where the field will be zero and the current must be a pure diffusion current..
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