# US 2025/0085331 A1 · Carrier-resolved hall measurement with multi-harmonic magnetoresistance analysis

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- Number: US 2025/0085331 A1
- Document ID: US20250085331A1
- Status: Published application
- Kind: A1
- Application number: 18/464,481
- Filed: 2023-09-11
- Published: 2025-03-13
- Earliest claimed: 2023-09-11
- Assignees: International Business Machines Corporation
- Applicants: International Business Machines Corporation
- Inventors: Oki GUNAWAN; Ching-Tzu Chen; Josephine Monica
- Web page: https://patentgrep.com/patent/US20250085331A1

## CPC

- G01R 31/2648: {Characterising semiconductor materials (testing of materials or semi-finished products G01R31/2831; testing during manufacture H10P74/00)}
- G01R 31/26: Testing of individual semiconductor devices (testing of photovoltaic devices H02S50/10; testing or measuring during manufacture or treatment {H10P74/00})
- G01R 31/2601: {Apparatus or methods therefor (G01R31/2607, G01R31/2642 take precedence)}
- G01R 31/2831: {Testing of materials or semi-finished products, e.g. semiconductor wafers or substrates (G01R31/318511 takes precedence; testing during manufacture H10P74/00)}
- H10P 74/20: characterised by the properties tested or measured, e.g. structural or electrical properties
- H10P 74/207: {Electrical properties, e.g. testing or measuring of resistance, deep levels or capacitance-voltage characteristics}
- H10P 74/277: {Circuits for electrically characterising or monitoring manufacturing processes, e.g. circuits in tested chips or circuits in testing wafers}
- G01R: MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES (indicating correct tuning of resonant circuits H03J3/12)
- H10P: GENERIC PROCESSES OR APPARATUS FOR THE MANUFACTURE OR TREATMENT OF DEVICES COVERED BY CLASS H10

## Abstract

A method, system, and a computer program product characterize a semiconductor. The method includes receiving longitudinal and transverse magnetoresistance measurements of a sample with first and second channels in a parallel magnetic field and receiving longitudinal and transverse magnetoresistance measurements of the sample in a perpendicular magnetic field. The method also includes subtracting the magnetoresistances measured in the parallel magnetic field from the magnetoresistances measured in the perpendicular field to obtain longitudinal and transverse magnetoresistance values Rxx and Rxy. The method further includes determining, based on the values Rxx and Rxy, harmonic magnetoresistance coefficients a0, and a2, b1, and b3. A method and system for semiconductor testing and characterization are disclosed as well.

## Claims

### Claim 1

A computer-implemented method of semiconductor characterization, comprising:
receiving, at a processor communicatively coupled to a memory, first longitudinal and transverse magnetoresistance measurements from a sample in a parallel magnetic field, the sample comprising a first channel and a second channel;
receiving, at the processor, second longitudinal and transverse magnetoresistance measurements from the sample in a perpendicular magnetic field;
subtracting, by the processor, the first longitudinal and transverse magnetoresistance measurements from the second longitudinal and transverse magnetoresistance measurements obtain a longitudinal magnetoresistance value Rxx and a transverse magnetoresistance value Rxy; and
determining, based on the longitudinal magnetoresistance value Rxx and the transverse magnetoresistance value Rxy, harmonic magnetoresistance coefficients a0, a2, b1, and b3.

### Claim 2 (depends on 1)

The method of claim 1, wherein the determining the harmonic magnetoresistance coefficients a0 and a2 comprises using Equations (Eqs.) 17-22:
ρ
xx
(
B
)
=
σ
1
+
σ
2
+
σ
1
⁢
σ
2
(
σ
⁢
H
1
2
+
σ
2
⁢
H
2
2
)
⁢
B
2
(
σ
1
+
σ
2
)
2
+
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
⁢
B
2
(
17
)
ρ
xx
(
B
)
=
σ
1
+
σ
2
+
σ
1
⁢
σ
2
(
σ
⁢
H
1
2
+
σ
2
⁢
H
2
2
)
⁢
B
2
(
σ
1
+
σ
2
)
2
[
1
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
⁢
B
2
(
σ
1
+
σ
2
)
2
]
(
18
)
ρ
xx
(
B
)
=
a
0
+
a
2
⁢
B
2
(
19
)
a
0
=
1
σ
1
+
σ
2
(
20
)
a
2
=
σ
1
⁢
σ
2
(
σ
⁢
H
1
2
+
σ
2
⁢
H
2
2
)
(
σ
1
+
σ
2
)
2
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
(
σ
1
+
σ
2
)
2
(
21
)
a
2
=
σ
1
⁢
σ
2
(
σ
1
+
σ
2
)
3
⁢
(
μ
1
-
μ
2
)
2
.
(
22
)

### Claim 3 (depends on 1)

The method of claim 1, wherein the determining the harmonic magnetoresistance coefficients b1 and b3 comprises using Eqs. 23-28:
ρ
xy
(
B
)
=
B
⁡
(
H
1
⁢
σ
1
2
+
H
2
⁢
σ
2
2
)
+
B
3
⁢
σ
1
2
⁢
σ
2
2
⁢
H
1
⁢
H
2
(
H
1
+
H
2
)
(
σ
1
+
σ
2
)
2
+
B
2
⁢
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
(
23
)
ρ
xy
(
B
)
=
(
H
1
⁢
σ
1
2
+
H
2
⁢
σ
2
2
)
⁢
B
+
σ
1
2
⁢
σ
2
2
⁢
H
1
⁢
H
2
(
H
1
+
H
2
)
⁢
B
3
(
σ
1
+
σ
2
)
2
[
1
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
⁢
B
2
(
σ
1
+
σ
2
)
2
]
(
24
)
ρ
xy
=
(
B
)
=
b
1
⁢
B
+
b
3
⁢
B
3
(
25
)
b
1
=
H
1
⁢
σ
1
2
+
H
2
⁢
σ
2
2
(
σ
1
+
σ
2
)
2
(
26
)
b
3
=
σ
1
2
⁢
σ
2
2
(
σ
1
⁢
σ
2
)
2
⁢
H
1
⁢
H
2
(
H
1
+
H
2
)
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
⁢
(
σ
1
2
⁢
H
1
+
σ
2
2
⁢
H
2
)
(
σ
1
+
σ
2
)
4
(
27
)
b
3
=
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
(
σ
1
+
σ
2
)
4
⁢
(
μ
1
-
μ
2
)
2
(
28
)

### Claim 4 (depends on 1)

The method of claim 1, further comprising using the harmonic magnetoresistance coefficients a0, a2, b1, and b3 to determine conductivities σc and Hall coefficients Hc of the first and second channels.

### Claim 5 (depends on 4)

The method of claim 4, wherein the determining the conductivities σc and the Hall coefficients Hc of the first and second channels comprises using Eqs. 29-32:
σ
1
=
d
1
+
a
2
⁢
b
1
+
a
0
⁢
b
3
2
⁢
a
0
⁢
d
1
(
29
)
σ
2
=
d
1
-
a
2
⁢
b
1
-
a
0
⁢
b
3
2
⁢
a
0
⁢
d
1
(
30
)
H
1
=
(
-
a
0
⁢
b
3
(
a
0
⁢
b
3
2
+
4
⁢
a
2
3
+
2
⁢
a
2
⁢
b
1
⁢
b
3
+
b
3
⁢
d
1
)
+
a
2
(
(
2
⁢
a
2
2
+
b
1
⁢
b
3
)
⁢
d
1
-
a
2
⁢
b
1
2
⁢
b
3
)
)
(
2
⁢
a
2
4
)
(
31
)
H
2
=
-
a
0
⁢
b
3
(
a
0
⁢
b
3
2
+
4
⁢
a
2
3
+
2
⁢
a
2
⁢
b
1
⁢
b
3
+
b
3
⁢
d
1
)
+
a
2
(
(
2
⁢
a
2
2
+
b
1
⁢
b
3
)
⁢
d
1
+
a
2
⁢
b
1
2
⁢
b
3
)
2
⁢
a
2
4
(
32
)
where
⁢
d
1
⁢
is
:
d
1
=
4
⁢
a
0
⁢
a
2
3
+
a
2
2
⁢
b
1
2
+
2
⁢
a
0
⁢
a
2
⁢
b
1
⁢
b
3
+
a
0
2
⁢
b
3
2
(
33
)

### Claim 6 (depends on 4)

The method of claim 4, further comprising determining, based on the conductivities σc and the Hall coefficients Hc of the first and second channels, carrier densities nc and mobilities μc of charge carriers in the first and second channels.

### Claim 7 (depends on 6)

The method of claim 6, wherein the determining the carrier densities nc and the mobilities μc of the charge carriers in the first and second channels comprises using Eqs. 34 and 35:
μ
c
=
σ
c
⁢
H
c
(
34
)
n
c
=
1
eH
c
.
(
35
)

### Claim 8 (depends on 1)

The method of claim 1, further comprising extracting harmonic coefficients α0, α2, β1 and β3 using Fourier analysis and lock-in detection.

### Claim 9 (depends on 8)

The method of claim 8, wherein the determining the harmonic magnetoresistance coefficients a0, a2, b1, and b3 comprises using Eqs. 38-41:
a
0
=
wd
l
⁢
(
α
0
-
α
2
)
(
38
)
a
2
=
2
⁢
wd
l
⁢
α
2
B
m
2
(
39
)
b
1
=
wd
l
⁢
β
1
-
3
⁢
β
3
B
m
(
40
)
b
3
=
4
⁢
wd
l
⁢
β
3
B
m
2
,
(
41
)
where w, l, and d are width, length, and thickness, respectively, of an active area of the sample.

### Claim 10

A system for semiconductor characterization, comprising:
a memory; and
a processor communicatively coupled to the memory, wherein the processor is configured perform a method comprising:
receiving first longitudinal and transverse magnetoresistance measurements from a sample in a parallel magnetic field, the sample comprising a first channel and a second channel;
receiving second longitudinal and transverse magnetoresistance measurements from the sample in a perpendicular magnetic field;
subtracting the first longitudinal and transverse magnetoresistance measurements from the second longitudinal and transverse magnetoresistance measurements obtain a longitudinal magnetoresistance value Rxx and a transverse magnetoresistance value Rxy; and
determining, based on the longitudinal magnetoresistance value Rxx and the transverse magnetoresistance value Rxy, harmonic magnetoresistance coefficients a0, a2, b1, and b3.

### Claim 11 (depends on 10)

The system of claim 10, wherein the determining the harmonic magnetoresistance coefficients a0 and a2 comprises using Equations (Eqs.) 17-22:
ρ
xx
(
B
)
=
σ
1
+
σ
2
+
σ
1
⁢
σ
2
(
σ
⁢
H
1
2
+
σ
2
⁢
H
2
2
)
⁢
B
2
(
σ
1
+
σ
2
)
2
+
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
⁢
B
2
(
17
)
ρ
xx
(
B
)
=
σ
1
+
σ
2
+
σ
1
⁢
σ
2
(
σ
⁢
H
1
2
+
σ
2
⁢
H
2
2
)
⁢
B
2
(
σ
1
+
σ
2
)
2
[
1
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
⁢
B
2
(
σ
1
+
σ
2
)
2
]
(
18
)
ρ
xx
(
B
)
=
a
0
+
a
2
⁢
B
2
(
19
)
a
0
=
1
σ
1
+
σ
2
(
20
)
a
2
=
σ
1
⁢
σ
2
(
σ
⁢
H
1
2
+
σ
2
⁢
H
2
2
)
(
σ
1
+
σ
2
)
2
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
(
σ
1
+
σ
2
)
2
(
21
)
a
2
=
σ
1
⁢
σ
2
(
σ
1
+
σ
2
)
3
⁢
(
μ
1
-
μ
2
)
2
.
(
22
)

### Claim 12 (depends on 10)

The system of claim 10, wherein the determining the harmonic magnetoresistance coefficients b1 and b3 comprises using Eqs. 23-28:
ρ
xy
(
B
)
=
B
⁡
(
H
1
⁢
σ
1
2
+
H
2
⁢
σ
2
2
)
+
B
3
⁢
σ
1
2
⁢
σ
2
2
⁢
H
1
⁢
H
2
(
H
1
+
H
2
)
(
σ
1
+
σ
2
)
2
+
B
2
⁢
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
(
23
)
ρ
xy
(
B
)
=
(
H
1
⁢
σ
1
2
+
H
2
⁢
σ
2
2
)
⁢
B
+
σ
1
2
⁢
σ
2
2
⁢
H
1
⁢
H
2
(
H
1
+
H
2
)
⁢
B
3
(
σ
1
+
σ
2
)
2
[
1
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
⁢
B
2
(
σ
1
+
σ
2
)
2
]
(
24
)
ρ
xy
=
(
B
)
=
b
1
⁢
B
+
b
3
⁢
B
3
(
25
)
b
1
=
H
1
⁢
σ
1
2
+
H
2
⁢
σ
2
2
(
σ
1
+
σ
2
)
2
(
26
)
b
3
=
σ
1
2
⁢
σ
2
2
(
σ
1
⁢
σ
2
)
2
⁢
H
1
⁢
H
2
(
H
1
+
H
2
)
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
2
⁢
(
σ
1
2
⁢
H
1
+
σ
2
2
⁢
H
2
)
(
σ
1
+
σ
2
)
4
(
27
)
b
3
=
-
σ
1
2
⁢
σ
2
2
(
H
1
+
H
2
)
(
σ
1
+
σ
2
)
4
⁢
(
μ
1
-
μ
2
)
2
.
(
28
)

### Claim 13 (depends on 10)

The system of claim 10, wherein the method further comprises using the harmonic magnetoresistance coefficients a0, a2, b1, and b3 to determine conductivities ac and Hall coefficients Hc of the first and second channels.

### Claim 14 (depends on 13)

The system of claim 13, wherein the determining the conductivities σc and the Hall coefficients Hc of the first and second channels comprises using Eqs. 29-32:
σ
1
=
d
1
+
a
2
⁢
b
1
+
a
0
⁢
b
3
2
⁢
a
0
⁢
d
1
(
29
)
σ
2
=
d
1
-
a
2
⁢
b
1
-
a
0
⁢
b
3
2
⁢
a
0
⁢
d
1
(
30
)
H
1
=
(
-
a
0
⁢
b
3
(
a
0
⁢
b
3
2
+
4
⁢
a
2
3
+
2
⁢
a
2
⁢
b
1
⁢
b
3
+
b
3
⁢
d
1
)
+
a
2
(
(
2
⁢
a
2
2
+
b
1
⁢
b
3
)
⁢
d
1
-
a
2
⁢
b
1
2
⁢
b
3
)
)
(
2
⁢
a
2
4
)
(
31
)
H
2
=
-
a
0
⁢
b
3
(
a
0
⁢
b
3
2
+
4
⁢
a
2
3
+
2
⁢
a
2
⁢
b
1
⁢
b
3
+
b
3
⁢
d
1
)
+
a
2
(
(
2
⁢
a
2
2
+
b
1
⁢
b
3
)
⁢
d
1
+
a
2
⁢
b
1
2
⁢
b
3
)
2
⁢
a
2
4
(
32
)
where
⁢
d
1
⁢
is
:
d
1
=
4
⁢
a
0
⁢
a
2
3
+
a
2
2
⁢
b
1
2
+
2
⁢
a
0
⁢
a
2
⁢
b
1
⁢
b
3
+
a
0
2
⁢
b
3
2
(
33
)

### Claim 15 (depends on 13)

The system of claim 13, wherein the method further comprises determining, based on the conductivities σc and the Hall coefficients Hc of the first and second channels, carrier densities nc and mobilities μc of charge carriers in the first and second channels.

### Claim 16 (depends on 15)

The system of claim 15, wherein the determining the carrier densities nc and the mobilities μc of the charge carriers in the first and second channels comprises using Eqs. 34 and 35:
μ
c
=
σ
c
⁢
H
c
(
34
)
n
c
=
1
eH
c
.
(
35
)

### Claim 17 (depends on 10)

The system of claim 10, further comprising extracting harmonic coefficients α0, α2, β1 and β3 using Fourier analysis and lock-in detection.

### Claim 18 (depends on 17)

The system of claim 17, wherein the determining the harmonic magnetoresistance coefficients a0, a2, b1, and b3 comprises using Eqs. 38-41:
a
0
=
wd
l
⁢
(
α
0
-
α
2
)
(
38
)
a
2
=
2
⁢
wd
l
⁢
α
2
B
m
2
(
39
)
b
1
=
wd
l
⁢
β
1
-
3
⁢
β
3
B
m
(
40
)
b
3
=
4
⁢
wd
l
⁢
β
3
B
m
2
,
(
41
)
where w, l, and d are width, length, and thickness, respectively, of an active area of the sample.

### Claim 19

A computer program product for semiconductor characterization, the computer program product comprising a computer readable storage medium having program instructions embodied therewith, the program instructions executable by a processor to cause a device to perform a method comprising:
receiving first longitudinal and transverse magnetoresistance measurements from a sample in a parallel magnetic field, the sample comprising a first channel and a second channel;
receiving second longitudinal and transverse magnetoresistance measurements from the sample in a perpendicular magnetic field;
subtracting the first longitudinal and transverse magnetoresistance measurements from the second longitudinal and transverse magnetoresistance measurements to obtain a longitudinal magnetoresistance value Rxx and a transverse magnetoresistance value Rxy; and
determining, based on the longitudinal magnetoresistance value Rxx and the transverse magnetoresistance value Rxy, harmonic magnetoresistance coefficients a0, a2, b1, and b3.

### Claim 20 (depends on 19)

The computer program product of claim 19, wherein the method further comprises determining, based on the harmonic magnetoresistance coefficients a0, a2, b1, and b3, conductivities σc and Hall coefficients Hc of the first and second channels.

### Claim 21

A method of semiconductor testing and characterization, comprising:
obtaining a sample comprising a first channel and a second channel;
measuring, in a parallel magnetic field, a first longitudinal magnetoresistance and a first transverse magnetoresistance of the sample;
measuring, in a perpendicular magnetic field, a second longitudinal magnetoresistance and a second transverse magnetoresistance of the sample;
calculating, by a processor communicatively coupled to a memory, longitudinal and transverse magnetoresistance values Rxx and Rxy, wherein the calculating comprises subtracting the first longitudinal magnetoresistance and the first transverse magnetoresistance from the second longitudinal magnetoresistance and the second transverse magnetoresistance; and
determining, based on the longitudinal magnetoresistance value Rxx and the transverse magnetoresistance value Rxy, harmonic magnetoresistance coefficients a0, a2, b1, and b3.

### Claim 22 (depends on 21)

The method of claim 21, further comprising determining, based on the harmonic magnetoresistance coefficients a0, a2, b1, and b3, conductivities σc and Hall coefficients Hc of the first and second channels.

### Claim 23 (depends on 21)

The method of claim 21, further comprising determining, based on the conductivities σc and the Hall coefficients Hc of the first and second channels, carrier densities nc and mobilities μc of charge carriers in the first and second channels.

### Claim 24

A system for semiconductor testing and characterization, comprising:
a testing module, the testing module comprising dipole-line magnets; and
a processor communicatively coupled to a memory, wherein the processor is configured to carry out a method comprising:
applying, using the dipole-line magnets, a parallel magnetic field at a sample in the testing module;
measuring, in the parallel magnetic field, a first longitudinal magnetoresistance and a first transverse magnetoresistance of the sample;
applying, using the dipole-line magnets, a perpendicular magnetic field at the sample in the testing module;
measuring, in the perpendicular magnetic field, a second longitudinal magnetoresistance and a second transverse magnetoresistance of the sample;
subtracting the first longitudinal magnetoresistance and the first transverse magnetoresistance from the second longitudinal magnetoresistance and the second transverse magnetoresistance to obtain longitudinal and transverse magnetoresistance values Rxx and Rxy; and
determining, based on the longitudinal magnetoresistance value Rxx and the transverse magnetoresistance value Rxy, harmonic magnetoresistance coefficients a0, a2, b1, and b3.

### Claim 25 (depends on 24)

The system of claim 24, the method further comprising:
determining, based on the harmonic magnetoresistance coefficients a0, a2, b1, and b3, conductivities σc and Hall coefficients Hc of the first and second channels; and
determining, based on the conductivities σc and the Hall coefficients Hc of the first and second channels, carrier densities nc and mobilities μc of charge carriers in the first and second channels.