Supplementary information
Straintronics-Based Magnetic Tunneling Junction: Dynamic and
Static Behavior Analysis and Material Investigation
The supplementary information provides the mathematical derivation of the dynamic equations
of the Straintronics-based MTJ models. The susceptibility model is further investigated.
Developing the dynamic model based on the LLG equation is explained in detail. Finally,
theoretical calculations to derive the dynamic approximation model are explained.
i)
Critical stress for magnetic susceptibility model
In order to obtain the critical stress based on the variable susceptibility model, we need to
realize when the magnetic energy barrier between the major and the monir axis disappears. The
main sources of the intrinsic magnetic energy are shape anisotropy, 𝐸𝑠ℎ , and uniaxial anisotropy,
𝐸𝑢 . They are given by the following equations:
𝐸𝑠ℎ =
𝜇0 2
𝑀 𝑁 𝑉
2 𝑆 𝑠ℎ
𝐸𝑢 = 𝐾𝑢 𝑉 sin2 𝜃𝑀
(S1)
(S2)
Here, 𝐾𝑢 is the uniaxial anisotropy coefficient and 𝜃𝑀 is the angle between the the magnetization
vector of the free layer and the major axis. 𝑁𝑠ℎ is the demagnetization factor defined as 𝑁𝑠ℎ =
𝑁𝑧𝑧 cos2 𝜃 + 𝑁𝑦𝑦 sin2 𝜃 sin2 𝜑 + 𝑁𝑥𝑥 sin2 𝜃 cos2 𝜑. The parameters 𝑁𝑥𝑥 , 𝑁𝑦𝑦 , and 𝑁𝑧𝑧 are shape
dependent parameters. Typically, for a thin layer, we have: 𝑁𝑥𝑥 >> 𝑁𝑦𝑦 , 𝑁𝑧𝑧 . When the device
is a cylindrical ellipse, these parameters are defined by the following expressions, in which 𝑎, 𝑏,
and 𝑙 are the magnet’s major axis, minor axis, and thickness:
𝜋𝑙
1 𝑎−𝑏
3 𝑎−𝑏 2
𝑁𝑧𝑧 =
(1 − (
)− (
) )
4𝑎
4
𝑎
16
𝑎
(S3)
𝑁𝑦𝑦
𝜋𝑙
5 𝑎−𝑏
21 𝑎 − 𝑏 2
=
(1 + (
)+ (
) )
4𝑎
4
𝑎
16
𝑎
𝑁𝑥𝑥 = 1 − (𝑁𝑦𝑦 + 𝑁𝑧𝑧 )
(S4)
(S5)
The intrinsic magnetic energy has a net value of: 𝐸𝑡𝑜𝑡 = 𝐸𝑠ℎ + 𝐸𝑢 . In y-z plane, This total
energy, has a maximum along the y-axis, and a minimum along the z-axis. Therefore, the
magnetization vector tends to align itself along the major axis in the absence of any stress. When
a stress is applied across the magnet, the value of the shape anisotropy starts to decrease along
the y-axis and starts to increase along the z-axis. This is because 𝐸 = −𝑚. 𝐵, 𝐵 = 𝜇0 (1 + 𝜒𝑣 )𝐻,
and 𝐸𝑠ℎ =
𝜇0
2
𝑀. 𝐻𝑑 , with 𝑚 beign the magnetic depole moment, 𝐵 being the magnetic flux
density, and 𝐻𝑑 being the the demagnetization filed. As a result we will have:
𝜇0 1 + 𝜒 ⊥ 2
(
)𝑀 𝑁
2 1 + 𝜒0 𝑆 𝑧
(S6a)
𝜇0 1 + 𝜒 || 2
= (
)𝑀 𝑁
2 1 + 𝜒0 𝑆 𝑦
(S6b)
𝐸𝑠ℎ,𝑧 =
𝐸𝑠ℎ,𝑦
𝛥𝐸𝑠ℎ = 𝐸𝑠ℎ,𝑦 − 𝐸𝑠ℎ,𝑧 ≈
𝜇0 2
𝜒⊥
𝜒 ||
𝑀𝑆 (𝑁𝑦
− 𝑁𝑧 )
2
𝜒0
𝜒0
(S6a)
The last equality stands since 𝜒𝑣 , 𝜒 || , 𝜒 ⊥ ≫ 1. The energy barrier vanishes when 𝐸𝑡𝑜𝑡 reaches
zero. Since 𝐸𝑢,𝑧 = 𝐾𝑢 and 𝐸𝑢,𝑦 = 0, we will have:
𝜒|| (𝜎)
1
𝜒⊥ (𝜎)
{ 𝜇0 𝑀𝑆2 (𝑁𝑧
− 𝑁𝑦
)} ≈ 𝐾𝑢
2
𝜒0
𝜒0
(S7)
This, along with (2) and (3) can numerically predict the critical stress, for which the intrinsic
energy barrier disappears.
ii)
LLG modeling approach
The basis of the dynamic behavior of the magnet is the famous LLG equation. It can be given
in the Gilbert form as:
𝑑𝑀
𝛾
𝛾
(𝑀 × 𝐻) −
=−
(𝑀 × (𝑀 × 𝐻))
2
1
𝑑𝑡
(1 + 𝛼 )
𝑀𝑆 × (𝛼 + 𝛼 )
(S8)
Simplifying (7) in terms of spherical coordinates with 𝜃 being the angle of the free layer’s
magnetization vector with the z-axis, and 𝜑 being the angle of the vector’s x-y plane projection
with the x-axis, we have:
𝑑𝜃
𝛾0
=
(𝐻 + 𝛼𝐻𝜃 )
𝑑𝑡 1 + 𝛼 2 𝜑
(S9)
𝑑𝜑
𝛾0
1
=
(𝛼𝐻𝜑 − 𝐻𝜃 )
2
𝑑𝑡 1 + 𝛼 𝑠𝑖𝑛𝜃
(S10)
where, the two factors, 𝐻𝜑 and 𝐻𝜃 , will be expressed as:
𝐻𝜑 = −
1
1 𝜕𝐸
𝜇0 𝑉𝑀𝑆 𝑠𝑖𝑛𝜃 𝜕𝜑
(S11)
1 𝜕𝐸
𝜇0 𝑉𝑀𝑆 𝜕𝜃
(S12)
𝐻𝜃 = −
where, 𝐸 = 𝐸𝑠ℎ + 𝐸𝑢 + 𝐸𝜎 , with 𝐸𝜎 being the stress anisotropy energy, which is given by the
following equation:
3
𝐸𝜎 = 𝜆𝑆 𝜎𝑉 sin2 𝜃𝜎
2
(S13)
where, 𝜃𝜎 is the angle between the magnetization vector and the minor axis. If 𝜆𝑆 𝜎 > 0 the
stress to the magnet is tensile, while 𝜆𝑆 𝜎 < 0 leads to a compressive stress. In this work, the
direction of the applied voltage is chosen such that stress type is compressive, and therefore the
magnetization vector is forced to rotate towards the minor axis under an applied stress.
𝜃𝜎 is the angle between the magnetization vector and the minor axis. Since we choose z-axis as
the major axis in Fig 1a, we have:
𝑐𝑜𝑠𝜃𝜎 = 𝑠𝑖𝑛𝜃 × 𝑠𝑖𝑛𝜑
(S14a)
𝐸𝑠ℎ =
3
𝜆 𝜎𝑉(1 − sin2 𝜃 sin2 𝜑)
2 𝑠
(S14b)
By combining the energies and incorporating the effective fields together, we have:
𝐻𝜑 = −
𝐻𝜃 = −
1
𝜇0
3
( 𝑀𝑆2 𝑉(𝑁𝑥 − 𝑁𝑦 ) + 𝜆𝑠 𝜎𝑉) 𝑠𝑖𝑛𝜃𝑠𝑖𝑛2𝜑
𝜇0 𝑉𝑀𝑆 2
2
(S15)
1
𝜇0
3
( 𝑀𝑆2 𝑉(𝑁𝑦 sin2 𝜑 + 𝑁𝑥 cos 2 𝜑 − 𝑁𝑧 ) − 𝜆𝑠 𝜎𝑉 sin2 𝜑
𝜇0 𝑉𝑀𝑆 2
2
(S16)
+ 𝐾𝑢 𝑉) 𝑠𝑖𝑛2𝜃
By using the (S15) and (S16) in (S9) and (S10), θ and φ can be obtained at any time. This is
the basis of our LLG dynamic modeling.
iii)
Analytical model for magnetization’s damping behavior
When a voltage across the magnet exceeds the critical flipping voltage, the magnetization
vector starts rotating away from the major axis towards the minor axis. When the values of 𝛼 is
low (exp: Cobalt and Nickel), the magnetization vector shows large oscillations while damping
to the minor axis. This oscillation can be approximated using the 2nd order damping equation for
control systems:
𝑑2𝜃
𝑑𝜃
+ 2𝜁𝜔0
+ 𝜔0 2 𝜃 = 0
2
𝑑𝑡
𝑑𝑡
(S17)
where, 𝜁 is the general damping factor and 𝜔0 is natural oscillation frequency. Since 𝜃 and 𝜑
𝜋
tend to damp around 2 , we will use a change of variables in order to use the Taylor
𝜋
𝜋
approximations: 𝛽 = 𝜃 − 2 , and 𝜂 = 𝜑 − 2 . Now since at high stress values 𝛽 → 0 and 𝜂 → 0,
we can simplify (S15) and (S16) to:
𝐻𝜑 = 𝐾1 𝑠𝑖𝑛𝜃𝑠𝑖𝑛2𝜑 = −2𝐾1 𝜂
(S18)
𝐻𝜃 = 𝐾2 sin2 𝜑 𝑠𝑖𝑛2𝜃 + 𝐾3 cos 2 𝜑 𝑠𝑖𝑛2𝜃 + 𝐾4 𝑠𝑖𝑛2𝜃
= −2𝐾2 𝛽 − 2𝐾3 𝜂2 𝛽 − 2𝐾4 𝛽
(S19)
where, 𝐾1 … 𝐾4, driven from (S15) and (S16), are used for simplicity and are given by:
𝐾1 = −
1
𝜇0
3
( 𝑀𝑆2 𝑉(𝑁𝑥 − 𝑁𝑦 ) + 𝜆𝑠 𝜎𝑉)
𝜇0 𝑉𝑀𝑆 2
2
(S20a)
1
𝜇0
3
( 𝑀𝑆2 𝑉𝑁𝑦 − 𝜆𝑠 𝜎𝑉)
𝜇0 𝑉𝑀𝑆 2
2
(S20b)
1
𝜇0
( 𝑀𝑆2 𝑉𝑁𝑥 )
𝜇0 𝑉𝑀𝑆 2
(S20c)
1
𝜇0
( 𝑀𝑆2 𝑉(−𝑁𝑧 ) + 𝐾𝑢 𝑉)
𝜇0 𝑉𝑀𝑆 2
(S20d)
𝐾2 = −
𝐾3 = −
𝐾4 = −
Incorporating (S18) and (S19) into (S9) and (S10), and since 𝐾1 ≈ 𝐾3 due to our device
geometry, and by using a change of variable we can obtain the following coupled equations of 𝛽
and 𝜂:
𝑑𝛽
𝛾0
(−2𝐾1 𝜂 − 2𝛼(𝐾2 + 𝐾4 )𝛽) = 𝑀1 𝜂 + 𝛼𝑀2 𝛽
=
𝑑𝑡 1 + 𝛼 2
(S21)
𝑑𝜂
𝛾0
(−2𝛼𝐾1 𝜂 + 2(𝐾2 + 𝐾4 )𝛽) = 𝑀1 𝛼𝜂 − 𝑀2 𝛽
=
𝑑𝑡 1 + 𝛼 2
(S22)
𝛾
𝛾
0
0
where, we have used: 𝑀1 = −2 1+𝛼
2 𝐾1 and 𝑀2 = −2 1+𝛼2 (𝐾2 + 𝐾4 ). Now in order to solve
the problem, we use matrixes of derivatives:
𝛼𝑀
𝛽̇
[ ]=[ 2
−𝑀2
𝜂̇
which, simply means 𝐴 = [
𝛼𝑀2
−𝑀2
𝑀1 𝛽
][ ]
𝛼𝑀1 𝜂
(S23)
𝑀1
]. Now by setting det(𝑢𝐼 − 𝐴) = 0 we can find the
𝛼𝑀1
eigenvalues for the differential equations:
𝑢 − 𝛼𝑀2
det(𝑢𝐼 − 𝐴) = |
𝑀2
−𝑀1
|
𝑢 − 𝛼𝑀1
(S24)
= 𝑢2 − 𝛼(𝑀1 + 𝑀2 )𝑢 + ((1 + 𝛼 2 )𝑀1 𝑀2 ) = 0
This leads to 𝑢 =
𝛼(𝑀1 +𝑀2 )±√𝛼2 (𝑀1 +𝑀2 )2 −4(1+𝛼2 )𝑀1 𝑀2
2
. This leads to 𝜆1 and 𝜆1 that are
1
conjugative values. Therefore, comparing to (S17), we can simply say 𝜁𝜔0 = − 2 𝛼(𝑀1 + 𝑀2 ),
and therefore we will have:
𝜔0 =
𝜁=
𝜃(𝑡) =
√4(1 + 𝛼 2 )𝑀1 𝑀2 − 𝛼 2 (𝑀1 + 𝑀2 )2
2
(S25)
𝛼(𝑀1 + 𝑀2 )
√4(1 + 𝛼 2 )𝑀1 𝑀2 − 𝛼 2 (𝑀1 + 𝑀2 )2
𝜋 𝜋 − 𝜁𝜔 𝑡
0 cos(𝜔 𝑡),
− 𝑒
𝑑
2 2
𝜔𝑑 = 𝜔0 √1 − 𝜁 2
(S26)
(S27)
As a result, the general damping factor, 𝜁, is a function of the Gilbert damping factor, the
applied voltage, and the material properties. By having the value of 𝜃(𝑡), the tunnel
magnetoresistance can be obtained at any time since:
𝑅𝑀𝑇𝐽 =
1
{𝑅𝑚 + 2 (𝑅𝑀 − 𝑅𝑚 ) × (1 − cos𝜃(𝑡))}
𝑉𝑀𝑇𝐽 2
1+( 𝑉 )
ℎ
(S28)
where, 𝑅𝑀 is the high resistance state, in which free and pinned layers have anti-parallel (AP)
magnetization orientation, 𝑅𝑚 is the low resistance state, in which they have parallel (P)
orientation, and 𝑉ℎ is the voltage at which the resistance is half of its value at zero bias.