Imaginary Band Structure and Its Role in Calculating Transmission Probability in Semiconductors

Imaginary Band Structure and Its Role in Calculating Transmission Probability in Semiconductors Jamie Teherani Collaborators: Paul Solomon (IBM), Mathieu Luisier(Purdue) Advisors: Judy Hoyt, DimitriAntoniadis Acknowledgements: Steve Laux(IBM) 1

Outline Introduction Quantum mechanics refresher Imaginary k-vector and WKB approximation Imaginary band structure Applications of Imaginary Band Structure Directional dependence of GIDL in MOSFETs Strain dependence of tunneling current in Si tunnel diodes 2

Outline Introduction Quantum mechanics refresher Imaginary k-vector and WKB approximation Imaginary band structure Applications of Imaginary Band Structure Directional dependence of GIDL in MOSFETs Strain dependence of tunneling current in Si tunnel diodes 3

QM Transmission Through a Square Barrier Kinetic Energy Classical Analogy Potential Energy Trial Solution Boundary Conditions 4

Imaginary Momentum Vector Plane Wave Solution = V(x) V 0 I II III E potential barrier k is real k is imag. k is real x Momentum in the tunneling direction = 2 ħ In the basic example of a plane wave tunneling through a square potential barrier, the momentum in the tunneling direction is imaginary inside the barrier which results in a decaying exponential. 5

Imaginary Momentum Vector Plane Wave Solution = V(x) V 0 I II III E potential barrier k is real k is imag. k is real x Momentum in the tunneling direction = 2 ħ = In the basic example of a plane wave tunneling through a square potential barrier, the momentum in the tunneling direction is imaginary inside the barrier which results in a decaying exponential. 6

Energy Tunnel Probability Start with the WKB approximation for tunneling probability ~ 2 Action Integral a b x 7

E Transmission Coefficient for PN Junction Decaying Exponential conduction band potential barrier Initial Wave Transmitted Wave E c Energy valence band tunneling of electrons a b E v x a b x Start with the WKB approximation for transmission probability ~ 2 Action Integral 8

Constant Field Approximation For a constant field across the junction, action integral can be rewritten as = Where F is the field, q is the charge of an electron, and E is energy. Since: = 9

Constant Field Approximation For a constant field across the junction, action integral can be rewritten as = Where F is the field, q is the charge of an electron, and E is energy. Since: = Assumes parabolic band = 2 ħ 10

Constant Field Approximation For a constant field across the junction, action integral can be rewritten as = Where F is the field, q is the charge of an electron, and E is energy. Since: = Assumes parabolic band Can be calculated from imaginary band structure = 2 ħ 11

Outline Introduction Quantum mechanics refresher Imaginary k-vector and WKB approximation Imaginary band structure Applications of Imaginary Band Structure Directional dependence of GIDL in MOSFETs Strain dependence of tunneling current in Si tunnel diodes 12

Full Band Structure of Silicon Abhijeet Paul et al., Band Structure Lab, May-2006. [Online]. Available: http://nanohub.org/resources/1308. 13

Full Band Structure of Silicon Abhijeet Paul et al., Band Structure Lab, May-2006. [Online]. Available: http://nanohub.org/resources/1308. 14

Full Band Structure of Silicon The electric field provides energy for tunneling from valence band to conduction band. Valence Bands Conduction Bands Abhijeet Paul et al., Band Structure Lab, May-2006. [Online]. Available: http://nanohub.org/resources/1308. 15

Imaginary Band Structure [100] Tunneling in Silicon Imaginary k-vector, Real k-vector k 0 k =0 k i i (X-valley) k k S. Laux, Computation of Complex Band Structures in Bulk and Confined Structures, in Computational Electronics, 2009. IWCE '09. 13th International Workshop on, pp. 1-2, 2009. 16

Imaginary Band Structure [100] Tunneling in Silicon Imaginary k-vector, Real k-vector k 0 k =0 k i i (X-valley) k k S. Laux, Computation of Complex Band Structures in Bulk and Confined Structures, in Computational Electronics, 2009. IWCE '09. 13th International Workshop on, pp. 1-2, 2009. 17

Imaginary Band Structure [100] Tunneling in Silicon Imaginary k-vector, Real k-vector k 0 k =0 Action Integral k i i (X-valley) k k S. Laux, Computation of Complex Band Structures in Bulk and Confined Structures, in Computational Electronics, 2009. IWCE '09. 13th International Workshop on, pp. 1-2, 2009. 18

Outline Introduction Quantum mechanics refresher Imaginary k-vector and WKB approximation Imaginary band structure Applications of Imaginary Band Structure Directional dependence of GIDL in MOSFETs Strain dependence of tunneling current in Si tunnel diodes 19

GIDL in MOSFETs (Gate Induced Drain Leakage) (1a) Source Gate Oxide n+ Si p Si n+ Si n-mosfet Drain x (1b) Energy n-mosfet (On State) thermionic emission of electrons n-mosfet n-mosfet (Off State) (1c) Energy tunneling of electrons n-tfet n-tfet (On State) conduction band (2a) Source Gate Oxide p+ Si or SiGe n Si n+ Si n-tfet Drain x (2b) Energy tunneling of electrons valence band 20

Question: What is the effect of channel orientation on GIDL? (100) Wafer Surface S D <100> S D <110> Notch 21

GIDL Experiment General expression: Simplified Tunneling Equation For a triangular barrier: Tunneling Direction 2 4 3 Lobes with transverse electron mass [100] 4 [110] 2 [111] 0. 2 [100] Tunneling 4 lobes with transverse electron mass z [110] Tunneling z x [100] [110] y 2 lobes with transverse electron mass x [100] [110] y P. M. Solomon, S. E. Laux, L. Shi, J. Cai, and W. E. Haensch, Experimental and theoretical explanation for the orientation dependence gate-induced drain leakage in scaled MOSFETs, Device Research Conference, 2009. 22

GIDL Experiment Con t <110> channel MOSFETs have about 10x GIDL current compared to <100> channel MOSFETs P. Gilbert et al., Technology Elements of a Common Platform Bulk Foundry Offering (Invited), in Electron Devices Meeting, 2007. IEDM 2007. IEEE International, pp. 259-262, 2007. 23

GIDL Experiment Con t Tunneling Direction [100] Lobes with transverse electron mass 4 (Most Current) [110] 2 [111] 0 (Least current) <110> channel MOSFETs have about 10x GIDL current compared to <100> channel MOSFETs P. Gilbert et al., Technology Elements of a Common Platform Bulk Foundry Offering (Invited), in Electron Devices Meeting, 2007. IEDM 2007. IEEE International, pp. 259-262, 2007. 24

GIDL Experiment Con t Tunneling Current J (A/cm 2 ) 10000 1000 100 10 1 0.1 [001] [011] [111] 001-2 011-2 111-2 001-3 011-3 111-3 001-4 011-4 111-4 001-5 011-5 111-5 Vertical tunnel diodes were fabricated on (100), (110), and (111) Si substrates to study tunneling along these directions. 0.01 Increasing V 1E-3 3 4 5 6 7 8 9 10 11 W T (nm) Tunneling width determined by CV doping profile extraction and simulation. P. M. Solomon, S. E. Laux, L. Shi, J. Cai, and W. E. Haensch, Experimental and theoretical explanation for the orientation dependence gate-induced drain leakage in scaled MOSFETs, Device Research Conference, 2009. 25

GIDL Experiment Con t 2 Dashed lines are curves calculated using a band edge effective mass Band edge mass is insufficient in accurately calculating action integral Solid lines are calculated from the imaginary band structure Imaginary band structure is especially needed in analyzing tunneling in the [110] and [111] direction S. Laux, Computation of Complex Band Structures in Bulk and Confined Structures, in Computational Electronics, 2009. IWCE '09. 13th International Workshop on, pp. 1-2, 2009. 26

Outline Introduction Quantum mechanics refresher Imaginary k-vector and WKB approximation Imaginary band structure Applications of Imaginary Band Structure Directional dependence of GIDL in MOSFETs Strain dependence of tunneling current in Si tunnel diodes 27

Strain Dependence of Tunneling in Vertical Silicon Tunnel Diodes Objective: To determine the relative change in tunneling current due to applied uniaxial stress. Procedure:Experimentally measure the uniaxiallystressed vertical Si tunnel diodes and compare results to theoretical calculations. Tunneling Current Mechanically Applied Uniaxial Stress Vertical tunnel diodes were fabricated on (100), (110), and (111) Si substrates to study tunneling along these directions. The uniaxial bending apparatus was designed by the S. Thompson group at University of Florida. 28

Current (A) 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 1.E-11 1.E-12 1.E-13 1.E-14 IV Plot for (001) Wafer with [100] Stress Thermal Regime (63mV/decade) Trap Limited Regime Tunneling Regime -1-0.5 0 0.5 1 1.5 Voltage on N-Region (Volts) Forward Bias z Tunneling Current y x 3.E-08 2.E-08 1.E-08 [100] [110] Uniaxial Stress Compressive Strain Tensile Strain 0.7 0.75 0.8 Reverse Bias 29

20 Relative Change in Current Compared to 0% Strain (001) Wafer with [100] Stress 15 Percent Change in Current Compared to 0% Strain Case 10 5 0-5 -10-15 Increasing Compressive Strain Increasing Tensile Strain -0.13% Strain -0.065% Strain 0% Strain 0% Strain (remeasured) 0% Strain (remeasured 2) 0.065% Strain 0.13% Strain 0.195% Strain 0.195% Strain (remeasured) 0.26% Strain 0.325% Strain 0.39% Strain -20 0.2 0.4 0.6 0.8 1 1.2 1.4 Voltage on N-Region (Volts) 30

Imaginary Band Structure Calculation for Silicon (001) Wafer with [100] stress Analysis of the action integral calculated using imaginary band structure shows that the stress dependence is most sensitive to changes in band edge energies and occupancy of the bands 3 Imaginary Band Structure of Silicon with [100] Uniaxial Strain 2.5 Imaginary k-vector (1/nm) 2 1.5 1 0.5 SOH VB2 VB1 y-lobe Tunneling Action Integral x-lobe z-lobe T exp b a k x dx = exp b a k x de qf 0 0 0.2 0.4 0.6 0.8 1 1.2 Energy (ev) 31

Theoretical Analysis Imaginary band structure calculations for (001) wafer with [100] stress showed a large sensitivity of the transmission probability to energy band edge movements with stress. Little change in tunneling mass Strained imaginary band structure data was not available for all wafer/strain configurations Ultimately, the theoretical calculations were made without imaginary band structure calculations, taking into account the changes in band edge energy and occupation of the different lobes. This produced a reasonable fit, since we compared changes in current with strain for a set tunneling direction. Different tunneling directions were not compared to each other. 32

Relative Change in Current with Applied Strain 33

Summary of the Tunneling Analysis 34

Summary Imaginary k-vector,, represents a decaying exponential wavefunction Imaginary band structure calculations allow for accurate calculation of the action integral Band edge effective mass is insufficient to capture tunneling in different directions Band structure calculations that incorporated strain matched the trending behavior of relative change in tunneling current vsstrain 35

Acknowledgements DARPA Steep program IBM Network for Computational Nanotechnology (NCN) Jamie Teherani Collaborators: Paul Solomon (IBM), Mathieu Luisier(Purdue) Advisors: Judy Hoyt, DimitriAntoniadis Acknowledgements: Steve Laux(IBM) 36